Research ArticleHelmholtz Theorem for Nondifferentiable Hamiltonian Systemsin the Framework of Cressonrsquos Quantum Calculus

Freacutedeacuteric Pierret1 and Delfim F M Torres2

1 Institut de Mecanique Celeste et de Calcul des Ephemerides Observatoire de Paris 75014 Paris France2Center for Research and Development in Mathematics and Applications (CIDMA) Department of MathematicsUniversity of Aveiro 3810-193 Aveiro Portugal

Correspondence should be addressed to Delfim F M Torres delfimuapt

Received 23 January 2016 Accepted 28 April 2016

Academic Editor Taher S Hassan

Copyright copy 2016 F Pierret and D F M TorresThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

We derive the Helmholtz theorem for nondifferentiable Hamiltonian systems in the framework of Cressonrsquos quantum calculusPrecisely we give a theorem characterizing nondifferentiable equations admitting a Hamiltonian formulation Moreover in theaffirmative case we give the associated Hamiltonian

1 Introduction

Several types of quantum calculus are available in the lit-erature including Jacksonrsquos quantum calculus [1 2] Hahnrsquosquantum calculus [3ndash5] the time-scale 119902-calculus [6 7] thepower quantum calculus [8] and the symmetric quantumcalculus [9ndash11] Cresson introduced in 2005 his quantum cal-culus on a set of Holder functions [12]This calculus attractedattention due to its applications in physics and the calculusof variations and has been further developed by several dif-ferent authors (see [13ndash16] and references therein) Cressonrsquoscalculus of 2005 [12] presents however some difficulties andin 2011 Cresson and Greff improved it [17 18] Indeed thequantum calculus of [12] let a free parameter which is presentin all the computations Such parameter is certainly difficultto interpretThe new calculus of [17 18] bypasses the problemby considering a quantity that is free of extra parameters andreduces to the classical derivative for differentiable functionsIt is this new version of 2011 that we consider here with abrief review of it being given in Section 2 Along the textby Cressonrsquos calculus we mean this quantum version of 2011[17 18] For the state of the art on the quantum calculusof variations we refer the reader to the recent book [19]With respect to Cressonrsquos approach the quantum calculusof variations is still in its infancy see [13 17 18 20ndash22] In[17] nondifferentiable Euler-Lagrange equations are used in

the study of PDEs Euler-Lagrange equations for variationalfunctionals with Lagrangians containing multiple quantumderivatives depending on a parameter or containing higher-order quantum derivatives are studied in [20] Variationalproblems with constraints with one and more than oneindependent variable of first and higher-order type are inves-tigated in [21] Recently problems of the calculus of variationsand optimal control with time delay were considered [22]In [18] a Noether type theorem is proved but only withthe momentum term This result is further extended in [23]by considering invariance transformations that also changethe time variable thus obtaining not only the generalizedmomentum term of [18] but also a new energy term In [13]nondifferentiable variational problems with a free terminalpoint with or without constraints of first and higher-orderare investigated Here we continue to develop Cressonrsquosquantum calculus in obtaining a result for Hamiltoniansystems and by considering the so-called inverse problem ofthe calculus of variations

A classical problem in analysis is the well-known Helm-holtzrsquos inverse problem of the calculus of variations find anecessary and sufficient condition under which a (system of)differential equation(s) can be written as an Euler-Lagrangeor a Hamiltonian equation and in the affirmative case findall possible Lagrangian or Hamiltonian formulations Thiscondition is usually called the Helmholtz condition The

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 8073023 8 pageshttpdxdoiorg10115520168073023

2 Discrete Dynamics in Nature and Society

Lagrangian Helmholtz problem has been studied and solvedby Douglas [24] Mayer [25] and Hirsch [26 27]TheHamil-tonian Helmholtz problem has been studied and solved upto our knowledge by Santilli in his book [28] Generalizationof this problem in the discrete calculus of variations frame-work has been done in [29 30] in the discrete Lagrangiancase In the case of time-scale calculus that is a mixingbetween continuous and discrete subintervals of time see [31]for a necessary condition for a dynamic integrodifferentialequation to be an Euler-Lagrange equation on time scalesFor the Hamiltonian case it has been done for the discretecalculus of variations in [32] using the framework of [33]and in [34] using a discrete embedding procedure derivedin [35] In the case of time-scale calculus it has been donein [36] for the Stratonovich stochastic calculus see [37] Herewe give the Helmholtz theorem for Hamiltonian systemsin the case of nondifferentiable Hamiltonian systems in theframework of Cressonrsquos quantum calculus By definition thenondifferentiable calculus extends the differentiable calculusSuch as in the discrete time-scale and stochastic cases werecover the same conditions of existence of a Hamiltonianstructure

The paper is organized as follows In Section 2 we givesome generalities and notions about the nondifferentiablecalculus introduced in [17] the so-called Cressonrsquos quantumcalculus In Section 3 we remind definitions and resultsabout classical and nondifferentiableHamiltonian systems InSection 4 we give a brief survey of the classical HelmholtzHamiltonian problem and then we prove the main resultof this papermdashthe nondifferentiable Hamiltonian Helmholtztheorem Finally we give two applications of our results inSection 5 and we end in Section 6 with conclusions andfuture work

2 Cressonrsquos Quantum Calculus

We briefly review the necessary concepts and results of thequantum calculus [17]

21 Definitions Let X119889 denote the set R119889 or C119889 119889 isin N andlet 119868 be an open set in R with [119886 119887] sub 119868 119886 lt 119887 We denote byF(119868X119889) the set of functions 119891 119868 rarr X119889 and by C0(119868X119889)the subset of functions ofF(119868X119889) which are continuous

Definition 1 (Holderian functions [17]) Let 119891 isin C0(119868R119889)Let 119905 isin 119868 Function 119891 is said to be 120572-Holderian 0 lt 120572 lt 1 atpoint 119905 if there exist positive constants 120598 gt 0 and 119888 gt 0 suchthat |119905 minus 119905

1015840| ⩽ 120598 implies 119891(119905) minus 119891(119905

1015840) ⩽ 119888|119905 minus 119905

1015840|120572 for all 1199051015840 isin 119868

where sdot is a norm on R119889

The set of Holderian functions of Holder exponent 120572 forsome 120572 is denoted by 119867

120572(119868R119889) The quantum derivative is

defined as follows

Definition 2 (the 120598-left and 120598-right quantum derivatives [17])Let 119891 isin C0(119868R119889) For all 120598 gt 0 the 120598-left and 120598-right

quantum derivatives of 119891 denoted respectively by 119889minus

120598119891 and

119889+

120598119891 are defined by

119889minus

120598119891 (119905) =

119891 (119905) minus 119891 (119905 minus 120598)

120598

119889+

120598119891 (119905) =

119891 (119905 + 120598) minus 119891 (119905)

120598

(1)

Remark 3 The 120598-left and 120598-right quantum derivatives of acontinuous function 119891 correspond to the classical derivativeof the 120598-mean function 119891

120590

120598defined by

119891120590

120598(119905) =

120590

120598int

119905+120590120598

119905

119891 (119904) 119889119904 120590 = plusmn (2)

The next operator generalizes the classical derivative

Definition 4 (the 120598-scale derivative [17]) Let 119891 isin C0(119868R119889)For all 120598 gt 0 the 120598-scale derivative of 119891 denoted by ◻120598119891◻119905is defined by

◻120598119891

◻119905=

1

2[(119889+

120598119891 + 119889minus

120598119891) + 119894120583 (119889

+

120598119891 minus 119889minus

120598119891)] (3)

where 119894 is the imaginary unit and 120583 isin minus1 1 0 minus119894 119894

Remark 5 If 119891 is differentiable then one can take the limit ofthe scale derivative when 120598 goes to zero We then obtain theclassical derivative 119889119891119889119905 of 119891

We also need to extend the scale derivative to complexvalued functions

Definition 6 (see [17]) Let 119891 isin C0(119868C119889) be a continuouscomplex valued function For all 120598 gt 0 the 120598-scale derivativeof 119891 denoted by ◻120598119891◻119905 is defined by

◻120598119891

◻119905=

◻120598Re (119891)◻119905

+ 119894◻120598 Im (119891)

◻119905 (4)

where Re(119891) and Im(119891) denote the real and imaginary partof 119891 respectively

In Definition 4 the 120598-scale derivative depends on 120598which is a free parameter related to the smoothing orderof the function This brings many difficulties in applicationsto physics when one is interested in particular equationsthat do not depend on an extra parameter To solve theseproblems the authors of [17] introduced a procedure toextract information independent of 120598 but related with themean behavior of the function

Definition 7 (see [17]) Let C0conv(119868 times ]0 1]R119889) sube C0(119868 times ]0

1]R119889) be such that for any function119891 isin C0conv(119868 times ]0 1]R119889)

the lim120598rarr0119891(119905 120598) exists for any 119905 isin 119868 We denote by 119864 a com-plementary space ofC0conv(119868 times ]0 1]R119889) inC0(119868 times ]0 1]R119889)We define the projection map 120587 by

120587 C0

conv (119868 times ]0 1] R119889) oplus 119864 997888rarr C

0

conv (119868 times ]0 1] R119889)

119891conv + 119891119864 997891997888rarr 119891conv

(5)

Discrete Dynamics in Nature and Society 3

and the operator ⟨sdot⟩ by

⟨sdot⟩ C0(119868 times ]0 1] R

119889) 997888rarr C

0(119868R119889)

119891 997891997888rarr ⟨119891⟩ 119905 997891997888rarr lim120598rarr0

120587 (119891) (119905 120598)

(6)

The quantum derivative of 119891 without the dependence of120598 is introduced in [17]

Definition 8 (see [17]) The quantum derivative of 119891 in thespaceC0(119868R119889) is given by

◻119891

◻119905= ⟨

◻120598119891

◻119905⟩ (7)

The quantum derivative (7) has some nice propertiesNamely it satisfies a Leibniz rule and a version of thefundamental theorem of calculus

Theorem 9 (the quantum Leibniz rule [17]) Let 120572 + 120573 gt 1For 119891 isin 119867

120572(119868R119889) and 119892 isin 119867

120573(119868R119889) one has

◻

◻119905(119891 sdot 119892) (119905) =

◻119891 (119905)

◻119905sdot 119892 (119905) + 119891 (119905) sdot

◻119892 (119905)

◻119905 (8)

Remark 10 For 119891 isin C1(119868R119889) and 119892 isin C1(119868R119889) oneobtains from (8) the classical Leibniz rule (119891 sdot 119892)

1015840= 1198911015840sdot 119892 +

119891 sdot 1198921015840

Definition 11 We denote by C1◻the set of continuous func-

tions 119902 isin C0([119886 119887]R119889) such that ◻119902◻119905 isin C0(119868R119889)

Theorem 12 (the quantum version of the fundamental theo-rem of calculus [17]) Let 119891 isin C1

◻([119886 119887]R119889) be such that

lim120598rarr0

int

119887

119886

(◻120598119891

◻119905)

119864

(119905) 119889119905 = 0 (9)

Then

int

119887

119886

◻119891

◻119905(119905) 119889119905 = 119891 (119887) minus 119891 (119886) (10)

22 Nondifferentiable Calculus of Variations In [17] thecalculus of variations with quantum derivatives is introducedand respective Euler-Lagrange equations derived without thedependence of 120598

Definition 13 An admissible Lagrangian 119871 is a continuousfunction 119871 R times R119889 times C119889 rarr C such that 119871(119905 119909 V) is holo-morphic with respect to V and differentiable with respect to119909 Moreover 119871(119905 119909 V) isin R when V isin R119889 119871(119905 119909 V) isin C whenV isin C119889

An admissible Lagrangian function 119871 RtimesRdtimesC119889 rarr C

defines a functional onC1(119868R119889) denoted by

L C1(119868R119889) 997888rarr R

119902 997891997888rarr int

119887

119886

119871 (119905 119902 (119905) (119905)) 119889119905

(11)

Extremals of the functional L can be characterized by thewell-known Euler-Lagrange equation (see eg [38])

Theorem 14 The extremals 119902 isin C1(119868R119889) ofL coincide withthe solutions of the Euler-Lagrange equation

119889

119889119905[120597119871

120597V(119905 119902 (119905) (119905))] =

120597119871

120597119909(119905 119902 (119905) (119905)) (12)

The nondifferentiable embedding procedure allows us todefine a natural extension of the classical Euler-Lagrangeequation in the nondifferentiable context

Definition 15 (see [17]) The nondifferentiable LagrangianfunctionalL◻ associated withL is given by

L◻ C1

◻(119868R119889) 997888rarr R

119902 997891997888rarr int

119887

119886

119871(119904 119902 (119904) ◻119902 (119904)

◻119905) 119889119904

(13)

Let 1198671205730

fl ℎ isin 119867120573(119868R119889) ℎ(119886) = ℎ(119887) = 0 and 119902 isin

119867120572(119868R119889) with 120572 + 120573 gt 1 A 119867

120573

0-variation of 119902 is a function

of the form 119902 + ℎ where ℎ isin 119867120573

119874 We denote by 119863L◻(119902)(ℎ)

the quantity

lim120598rarr0

L◻ (119902 + 120598ℎ) minusL◻ (119902)

120598(14)

if there exists the so-called Frechet derivative ofL◻ at point119902 in direction ℎ

Definition 16 (nondifferentiable extremals) A 119867120573

0-extremal

curve of the functionalL◻ is a curve 119902 isin 119867120572(119868R119889) satisfying

119863L◻(119902)(ℎ) = 0 for any ℎ isin 119867120573

0

Theorem 17 (nondifferentiable Euler-Lagrange equations[17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be an admissibleLagrangian of class C2 We assume that 120574 isin 119867

120572(119868R119889) such

that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that 119871(119905 120574(119905)

◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin 119867120573

0(119868R119889)

A curve 120574 satisfying the nondifferentiable Euler-Lagrangeequation

◻

◻119905[120597119871

120597V(119905 120574 (119905)

◻120574 (119905)

◻119905)] =

120597119871

120597119909(119905 120574 (119905)

◻120574 (119905)

◻119905) (15)

is an extremal curve of functional (13)

3 Reminder about Hamiltonian Systems

We now recall the main concepts and results of both classicaland Cressonrsquos nondifferentiable Hamiltonian systems

31 Classical Hamiltonian Systems Let 119871 be an admissibleLagrangian function If 119871 satisfies the so-called Legendreproperty then we can associate to 119871 a Hamiltonian functiondenoted by119867

4 Discrete Dynamics in Nature and Society

Definition 18 Let 119871 be an admissible Lagrangian functionTheLagrangian119871 is said to satisfy the Legendre property if themapping V 997891rarr (120597119871120597V)(119905 119909 V) is invertible for any (119905 119902 V) isin

119868 timesR119889 times C119889

If we introduce a new variable

119901 =120597119871

120597V(119905 119902 V) (16)

and 119871 satisfies the Legendre property then we can find afunction 119891 such that

V = 119891 (119905 119902 119901) (17)

Using this notation we have the following definition

Definition 19 Let 119871 be an admissible Lagrangian functionsatisfying the Legendre property The Hamiltonian function119867 associated with 119871 is given by

119867 R timesR119889times C119889997888rarr C

(119905 119902 119901) 997891997888rarr 119867(119905 119902 119901) = 119901119891 (119905 119902 119901) minus 119871 (119905 119902 119891 (119905 119902 119901))

(18)

We have the following theorem (see eg [38])

Theorem 20 (Hamiltonrsquos least-action principle) The curve(119902 119901) isin C(119868R119889)timesC(119868C119889) is an extremal of the Hamiltonianfunctional

H (119902 119901) = int

119887

119886

119901 (119905) (119905) minus 119867 (119905 119902 (119905) 119901 (119905)) 119889119905 (19)

if and only if it satisfies the Hamiltonian system associated with119867 given by

(119905) =120597119867 (119905 119902 (119905) 119901 (119905))

120597119901

(119905) = minus120597119867 (119905 119902 (119905) 119901 (119905))

120597119902

(20)

called the Hamiltonian equations

A vectorial notation is obtained for the Hamiltonianequations in posing 119911 = (119902 119901)

⊤ and nabla119867 = (120597119867120597119902 120597119867120597119901)⊤

where ⊤ denotes the transposition The Hamiltonian equa-tions are then written as

119889119911 (119905)

119889119905= 119869 sdot nabla119867 (119905 119911 (119905)) (21)

where

119869 = (0 119868119889

minus119868119889 0) (22)

denotes the symplectic matrix with 119868119889 being the identitymatrix on R119889

32 NondifferentiableHamiltonian Systems Thenondifferen-tiable embedding induces a change in the phase space withrespect to the classical case As a consequence we have towork with variables (119909 119901) that belong to R119889 times C119889 and notonly to R119889 timesR119889 as usual

Definition 21 (nondifferentiable embedding of Hamiltoniansystems [17]) The nondifferentiable embedded Hamiltoniansystem (20) is given by

◻119902 (119905)

◻119905=

120597119867 (119905 119902 (119905) 119901 (119905))

120597119901

◻119901 (119905)

◻119905= minus

120597119867 (119905 119902 (119905) 119901 (119905))

120597119902

(23)

and the embedded Hamiltonian functionalH◻ is defined on119867120572(119868R119889) times 119867

120572(119868C119889) by

H◻ (119902 119901) = int

119887

119886

(119901 (119905)◻119902 (119905)

◻119905minus 119867 (119905 119902 (119905) 119901 (119905))) 119889119905 (24)

The nondifferentiable calculus of variations allows us toderive the extremals forH◻

Theorem 22 (nondifferentiable Hamiltonrsquos least-action prin-ciple [17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be anadmissible C2-Lagrangian We assume that 120574 isin 119867

120572(119868R119889)

such that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that

119871(119905 120574(119905) ◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin

119867120573

0(119868R119889) Let 119867 be the corresponding Hamiltonian defined

by (18) A curve 120574 997891rarr (119905 119902(119905) 119901(119905)) isin 119868 times R119889 times C119889 solution ofthe nondifferentiable Hamiltonian system (23) is an extremalof functional (24) over the space of variations119881 = 119867

120573

0(119868R119889)times

119867120573

0(119868C119889)

4 Nondifferentiable Helmholtz Problem

In this section we solve the inverse problem of the nondif-ferentiable calculus of variations in the Hamiltonian case Wefirst recall the usual way to derive the Helmholtz conditionsfollowing the presentation made by Santilli [28] Two mainderivations are available

(i) The first is related to the characterization of Hamilto-nian systems via the symplectic two-differential formand the fact that by duality the associated one-differential form to a Hamiltonian vector field isclosedmdashthe so-called integrability conditions

(ii) The second uses the characterization of Hamiltoniansystems via the self-adjointness of the Frechet deriva-tive of the differential operator associated with theequationmdashthe so-called Helmholtz conditions

Of course we have coincidence of the two procedures in theclassical case As there is no analogous of differential formin the framework of Cressonrsquos quantum calculus we followthe second way to obtain the nondifferentiable analogue of

Discrete Dynamics in Nature and Society 5

the Helmholtz conditions For simplicity we consider a time-independent Hamiltonian The time-dependent case can bedone in the same way

41 Helmholtz Conditions for Classical Hamiltonian SystemsIn this section we work on R2119889 119889 ge 1 119889 isin N

411 Symplectic Scalar Product The symplectic scalar product⟨sdot sdot⟩119869 is defined by

⟨119883 119884⟩119869 = ⟨119883 119869 sdot 119884⟩ (25)

for all 119883119884 isin R2119889 where ⟨sdot sdot⟩ denotes the usual scalarproduct and 119869 is the symplectic matrix (22) We also considerthe 1198712 symplectic scalar product induced by ⟨sdot sdot⟩119869 defined for119891 119892 isin C0([119886 119887]R2119889) by

⟨119891 119892⟩1198712 119869

= int

119887

119886

⟨119891 (119905) 119892 (119905)⟩119869119889119905 (26)

412 Adjoint of a Differential Operator In the following weconsider first-order differential equations of the form

119889

119889119905(119902

119901) = (

119883119902 (119902 119901)

119883119901 (119902 119901)) (27)

where the vector fields 119883119902 and 119883119901 are C1 with respect to 119902

and 119901 The associated differential operator is written as

119874119883 (119902 119901) = (

minus 119883119902 (119902 119901)

minus 119883119901 (119902 119901)) (28)

A natural notion of adjoint for a differential operator is thendefined as follows

Definition 23 Let 119860 C1([119886 119887]R2119889) rarr C1([119886 119887]R2119889) Wedefine the adjoint 119860lowast

119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by

⟨119860 sdot 119891 119892⟩1198712119869

= ⟨119860lowast

119869sdot 119892 119891⟩

1198712 119869 (29)

An operator 119860 will be called self-adjoint if 119860 = 119860lowast

119869with

respect to the 1198712 symplectic scalar product

413 Hamiltonian Helmholtz Conditions The Helmholtzconditions in the Hamiltonian case are given by the followingresult (see Theorem 3121 p 176-177 in [28])

Theorem 24 (Hamiltonian Helmholtz theorem) Let119883(119902 119901)

be a vector field defined by 119883(119902 119901)⊤

= (119883119902(119902 119901) 119883119901(119902 119901))The differential equation (27) is Hamiltonian if and only if theassociated differential operator 119874119883 given by (28) has a self-adjoint Frechet derivative with respect to the 119871

2 symplecticscalar product In this case the Hamiltonian is given by

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (30)

The conditions for the self-adjointness of the differentialoperator can bemade explicitThey coincide with the integra-bility conditions characterizing the exactness of the one-formassociated with the vector field by duality (see [28] Theorem273 p 88)

Theorem 25 (integrability conditions) Let 119883(119902 119901)⊤

=

(119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differential operator119874119883 given by (28) has a self-adjoint Frechet derivative withrespect to the 1198712 symplectic scalar product if and only if

120597119883119902

120597119902+ (

120597119883119901

120597119901)

⊤

= 0120597119883119902

120597119901120597119883119901

120597119902119886119903119890 119904119910119898119898119890119905119903119894119888 (31)

42 Helmholtz Conditions for Nondifferentiable HamiltonianSystems The previous scalar products extend naturally tocomplex valued functions Let 0 lt 120572 lt 1 and let (119902 119901) isin

119867120572(119868R119889) times 119867

120572(119868C119889) such that ◻119902◻119905 isin 119867

120572(119868C119889) and

◻119901◻119905 isin 119867120572(119868C119889) We consider first-order nondifferential

equations of the form

◻

◻119905(119902

119901) = (

119883119902 (119902 119901)

119883119901 (119902 119901)) (32)

The associated quantum differential operator is written as

119874◻119883 (119902 119901) = (

◻119902

◻119905minus 119883119902 (119902 119901)

◻119901

◻119905minus 119883119901 (119902 119901)

) (33)

A natural notion of adjoint for a quantum differential opera-tor is then defined

Definition 26 Let 119860 C1◻([119886 119887]C2119889) rarr C1

◻([119886 119887]C2119889) We

define the adjoint 119860lowast119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by

⟨119860 sdot 119891 119892⟩1198712 119869

= ⟨119860lowast

119869sdot 119892 119891⟩

1198712 119869 (34)

An operator 119860 will be called self-adjoint if 119860 = 119860lowast

119869with

respect to the 1198712 symplectic scalar product We can now

obtain the adjoint operator associated with 119874◻119883

Proposition 27 Let 120573 be such that 120572 + 120573 gt 1 Let (119906 V) isin

119867120573

0(119868R119889) times 119867

120573

0(119868C119889) such that ◻119906◻119905 isin 119867

120572(119868C119889) and

◻V◻119905 isin 119867120572(119868C119889) The Frechet derivative 119863119874◻119883 of (33) at

(119902 119901) along (119906 V) is then given by

119863119874◻119883 (119902 119901) (119906 V) = (

◻119906

◻119905minus

120597119883119902

120597119902sdot 119906 minus

120597119883119902

120597119901sdot V

◻V◻119905

minus120597119883119901

120597119902sdot 119906 minus

120597119883119901

120597119901sdot V

) (35)

6 Discrete Dynamics in Nature and Society

Assume that 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin

119867120573

0(119868C119889) In consequence the adjoint 119863119874

lowast

◻119883of 119863119874◻119883(119902 119901)

with respect to the 1198712 symplectic scalar product is given by

119863119874lowast

◻119883(119902 119901) (119906 V)

= (

◻119906

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119906 minus (120597119883119902

120597119901)

⊤

sdot V

◻V◻119905

minus (120597119883119901

120597119902)

⊤

sdot 119906 + (120597119883119902

120597119902)

⊤

sdot V

)

(36)

Proof The expression for the Frechet derivative of (33) at(119902 119901) along (119906 V) is a simple computation Let (119908 119909) isin

119867120573

0(119868R119889) times 119867

120573

0(119868C119889) be such that ◻119908◻119905 isin 119867

120572(119868C119889) and

◻119909◻119905 isin 119867120572(119868C119889) By definition we have

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[◻119906

◻119905sdot 119909

minus (120597119883119902

120597119902sdot 119906) sdot 119909 minus (

120597119883119902

120597119901sdot V) sdot 119909 minus

◻V◻119905

sdot 119908

+ (120597119883119901

120597119902sdot 119906) sdot 119908 + (

120597119883119901

120597119901sdot V) sdot 119908]119889119905

(37)

As 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin 119867120573

0(119868C119889)

using the quantum Leibniz rule and the quantum version ofthe fundamental theorem of calculus we obtain

int

119887

119886

◻119906

◻119905sdot 119887 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

int

119887

119886

◻V◻119905

sdot 119886 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

(38)

Then

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[minus119906

sdot (◻119909

◻119905minus (

120597119883119901

120597119902)

⊤

sdot 119908 + (120597119883119902

120597119902)

⊤

sdot 119909) + V

sdot (◻119908

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119908 minus (120597119883119902

120597119901)

⊤

sdot 119909)]119889119905

(39)

By definition we obtain the expression of the adjoint 119863119874lowast

◻119883

of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar

product

In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions

Theorem 28 (nondifferentiable integrability conditions) Let119883(119902 119901)

⊤= (119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differ-

ential operator 119874◻119883 given by (33) has a self-adjoint Frechet

derivative with respect to the symplectic scalar product if andonly if

120597119883119902

120597119902+ (

120597119883119901

120597119901)

⊤

= 0 (HC1)

120597119883119902

120597119901120597119883119901

120597119902119886119903119890 119904119910119898119898119890119905119903119894119888 (HC2)

Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions

We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems

Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)

⊤=

(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (40)

Proof If 119883 is Hamiltonian then there exists a function 119867

R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (41)

that is we must show that

119883119902 (119902 119901) =120597119867 (119902 119901)

120597119901

119883119901 (119902 119901) = minus120597119867 (119902 119901)

120597119902

(42)

We have

120597119867 (119902 119901)

120597119902= int

1

0

[119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119902minus 119883119901 (120582119902 120582119901)

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119902] 119889120582

Discrete Dynamics in Nature and Society 7

120597119867 (119902 119901)

120597119901= int

1

0

[119883119901 (120582119902 120582119901) + 119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119901

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119901] 119889120582

(43)

Using the nondifferentiable integrability conditions weobtain

120597119867 (119902 119901)

120597119902= int

1

0

minus120597 (120582119883119901 (120582119902 120582119901))

120597120582119889120582 = minus119883119901 (119902 119901)

120597119867 (119902 119901)

120597119902= int

1

0

120597 (120582119883119902 (120582119902 120582119901))

120597120582119889120582 = 119883119902 (119902 119901)

(44)

which concludes the proof

5 Applications

We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]

Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867

120572(119868C119889) be

such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867

120572(119868C119889)

51 The Linear Case Let us consider the discrete nondiffer-entiable system

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 + 120575119901

(45)

where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 minus 120572119901

(46)

Using formula (40) we compute explicitly the Hamiltonianwhich is given by

119867(119902 119901) =1

2(1205731199012minus 1205741199022) + 120572119902 sdot 119901 (47)

52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby

=119901

119898

= minus1198801015840(119902)

(48)

with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by

119867(119902 119901) =1

21198981199012+ 119880 (119902) (49)

Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by

◻119902

◻119905=

119901

119898

◻119901

◻119905= minus1198801015840(119902)

(50)

The Hamiltonian Helmholtz conditions are clearly satisfied

Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)

6 Conclusion

We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions

References

[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002

[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012

[3] AMC Brito daCruzNMartins andD FMTorres ldquoHigher-order Hahnrsquos quantum variational calculusrdquoNonlinear AnalysisTheory Methods amp Applications vol 75 no 3 pp 1147ndash11572012

[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013

8 Discrete Dynamics in Nature and Society

[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010

[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001

[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011

[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012

[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013

[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013

[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013

[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005

[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015

[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009

[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010

[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009

[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011

[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011

[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014

[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010

[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011

[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013

[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014

[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941

[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896

[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897

[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898

[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978

[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013

[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004

[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014

[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999

[33] J EMarsden andMWest ldquoDiscretemechanics and variationalintegratorsrdquo Acta Numerica vol 10 pp 357ndash514 2001

[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203

[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117

[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015

[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015

[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

Lagrangian Helmholtz problem has been studied and solvedby Douglas [24] Mayer [25] and Hirsch [26 27]TheHamil-tonian Helmholtz problem has been studied and solved upto our knowledge by Santilli in his book [28] Generalizationof this problem in the discrete calculus of variations frame-work has been done in [29 30] in the discrete Lagrangiancase In the case of time-scale calculus that is a mixingbetween continuous and discrete subintervals of time see [31]for a necessary condition for a dynamic integrodifferentialequation to be an Euler-Lagrange equation on time scalesFor the Hamiltonian case it has been done for the discretecalculus of variations in [32] using the framework of [33]and in [34] using a discrete embedding procedure derivedin [35] In the case of time-scale calculus it has been donein [36] for the Stratonovich stochastic calculus see [37] Herewe give the Helmholtz theorem for Hamiltonian systemsin the case of nondifferentiable Hamiltonian systems in theframework of Cressonrsquos quantum calculus By definition thenondifferentiable calculus extends the differentiable calculusSuch as in the discrete time-scale and stochastic cases werecover the same conditions of existence of a Hamiltonianstructure

The paper is organized as follows In Section 2 we givesome generalities and notions about the nondifferentiablecalculus introduced in [17] the so-called Cressonrsquos quantumcalculus In Section 3 we remind definitions and resultsabout classical and nondifferentiableHamiltonian systems InSection 4 we give a brief survey of the classical HelmholtzHamiltonian problem and then we prove the main resultof this papermdashthe nondifferentiable Hamiltonian Helmholtztheorem Finally we give two applications of our results inSection 5 and we end in Section 6 with conclusions andfuture work

2 Cressonrsquos Quantum Calculus

We briefly review the necessary concepts and results of thequantum calculus [17]

21 Definitions Let X119889 denote the set R119889 or C119889 119889 isin N andlet 119868 be an open set in R with [119886 119887] sub 119868 119886 lt 119887 We denote byF(119868X119889) the set of functions 119891 119868 rarr X119889 and by C0(119868X119889)the subset of functions ofF(119868X119889) which are continuous

Definition 1 (Holderian functions [17]) Let 119891 isin C0(119868R119889)Let 119905 isin 119868 Function 119891 is said to be 120572-Holderian 0 lt 120572 lt 1 atpoint 119905 if there exist positive constants 120598 gt 0 and 119888 gt 0 suchthat |119905 minus 119905

1015840| ⩽ 120598 implies 119891(119905) minus 119891(119905

1015840) ⩽ 119888|119905 minus 119905

1015840|120572 for all 1199051015840 isin 119868

where sdot is a norm on R119889

The set of Holderian functions of Holder exponent 120572 forsome 120572 is denoted by 119867

120572(119868R119889) The quantum derivative is

defined as follows

Definition 2 (the 120598-left and 120598-right quantum derivatives [17])Let 119891 isin C0(119868R119889) For all 120598 gt 0 the 120598-left and 120598-right

quantum derivatives of 119891 denoted respectively by 119889minus

120598119891 and

119889+

120598119891 are defined by

119889minus

120598119891 (119905) =

119891 (119905) minus 119891 (119905 minus 120598)

120598

119889+

120598119891 (119905) =

119891 (119905 + 120598) minus 119891 (119905)

120598

(1)

Remark 3 The 120598-left and 120598-right quantum derivatives of acontinuous function 119891 correspond to the classical derivativeof the 120598-mean function 119891

120590

120598defined by

119891120590

120598(119905) =

120590

120598int

119905+120590120598

119905

119891 (119904) 119889119904 120590 = plusmn (2)

The next operator generalizes the classical derivative

Definition 4 (the 120598-scale derivative [17]) Let 119891 isin C0(119868R119889)For all 120598 gt 0 the 120598-scale derivative of 119891 denoted by ◻120598119891◻119905is defined by

◻120598119891

◻119905=

1

2[(119889+

120598119891 + 119889minus

120598119891) + 119894120583 (119889

+

120598119891 minus 119889minus

120598119891)] (3)

where 119894 is the imaginary unit and 120583 isin minus1 1 0 minus119894 119894

Remark 5 If 119891 is differentiable then one can take the limit ofthe scale derivative when 120598 goes to zero We then obtain theclassical derivative 119889119891119889119905 of 119891

We also need to extend the scale derivative to complexvalued functions

Definition 6 (see [17]) Let 119891 isin C0(119868C119889) be a continuouscomplex valued function For all 120598 gt 0 the 120598-scale derivativeof 119891 denoted by ◻120598119891◻119905 is defined by

◻120598119891

◻119905=

◻120598Re (119891)◻119905

+ 119894◻120598 Im (119891)

◻119905 (4)

where Re(119891) and Im(119891) denote the real and imaginary partof 119891 respectively

In Definition 4 the 120598-scale derivative depends on 120598which is a free parameter related to the smoothing orderof the function This brings many difficulties in applicationsto physics when one is interested in particular equationsthat do not depend on an extra parameter To solve theseproblems the authors of [17] introduced a procedure toextract information independent of 120598 but related with themean behavior of the function

Definition 7 (see [17]) Let C0conv(119868 times ]0 1]R119889) sube C0(119868 times ]0

1]R119889) be such that for any function119891 isin C0conv(119868 times ]0 1]R119889)

the lim120598rarr0119891(119905 120598) exists for any 119905 isin 119868 We denote by 119864 a com-plementary space ofC0conv(119868 times ]0 1]R119889) inC0(119868 times ]0 1]R119889)We define the projection map 120587 by

120587 C0

conv (119868 times ]0 1] R119889) oplus 119864 997888rarr C

0

conv (119868 times ]0 1] R119889)

119891conv + 119891119864 997891997888rarr 119891conv

(5)

Discrete Dynamics in Nature and Society 3

and the operator ⟨sdot⟩ by

⟨sdot⟩ C0(119868 times ]0 1] R

119889) 997888rarr C

0(119868R119889)

119891 997891997888rarr ⟨119891⟩ 119905 997891997888rarr lim120598rarr0

120587 (119891) (119905 120598)

(6)

The quantum derivative of 119891 without the dependence of120598 is introduced in [17]

Definition 8 (see [17]) The quantum derivative of 119891 in thespaceC0(119868R119889) is given by

◻119891

◻119905= ⟨

◻120598119891

◻119905⟩ (7)

The quantum derivative (7) has some nice propertiesNamely it satisfies a Leibniz rule and a version of thefundamental theorem of calculus

Theorem 9 (the quantum Leibniz rule [17]) Let 120572 + 120573 gt 1For 119891 isin 119867

120572(119868R119889) and 119892 isin 119867

120573(119868R119889) one has

◻

◻119905(119891 sdot 119892) (119905) =

◻119891 (119905)

◻119905sdot 119892 (119905) + 119891 (119905) sdot

◻119892 (119905)

◻119905 (8)

Remark 10 For 119891 isin C1(119868R119889) and 119892 isin C1(119868R119889) oneobtains from (8) the classical Leibniz rule (119891 sdot 119892)

1015840= 1198911015840sdot 119892 +

119891 sdot 1198921015840

Definition 11 We denote by C1◻the set of continuous func-

tions 119902 isin C0([119886 119887]R119889) such that ◻119902◻119905 isin C0(119868R119889)

Theorem 12 (the quantum version of the fundamental theo-rem of calculus [17]) Let 119891 isin C1

◻([119886 119887]R119889) be such that

lim120598rarr0

int

119887

119886

(◻120598119891

◻119905)

119864

(119905) 119889119905 = 0 (9)

Then

int

119887

119886

◻119891

◻119905(119905) 119889119905 = 119891 (119887) minus 119891 (119886) (10)

22 Nondifferentiable Calculus of Variations In [17] thecalculus of variations with quantum derivatives is introducedand respective Euler-Lagrange equations derived without thedependence of 120598

Definition 13 An admissible Lagrangian 119871 is a continuousfunction 119871 R times R119889 times C119889 rarr C such that 119871(119905 119909 V) is holo-morphic with respect to V and differentiable with respect to119909 Moreover 119871(119905 119909 V) isin R when V isin R119889 119871(119905 119909 V) isin C whenV isin C119889

An admissible Lagrangian function 119871 RtimesRdtimesC119889 rarr C

defines a functional onC1(119868R119889) denoted by

L C1(119868R119889) 997888rarr R

119902 997891997888rarr int

119887

119886

119871 (119905 119902 (119905) (119905)) 119889119905

(11)

Extremals of the functional L can be characterized by thewell-known Euler-Lagrange equation (see eg [38])

Theorem 14 The extremals 119902 isin C1(119868R119889) ofL coincide withthe solutions of the Euler-Lagrange equation

119889

119889119905[120597119871

120597V(119905 119902 (119905) (119905))] =

120597119871

120597119909(119905 119902 (119905) (119905)) (12)

The nondifferentiable embedding procedure allows us todefine a natural extension of the classical Euler-Lagrangeequation in the nondifferentiable context

Definition 15 (see [17]) The nondifferentiable LagrangianfunctionalL◻ associated withL is given by

L◻ C1

◻(119868R119889) 997888rarr R

119902 997891997888rarr int

119887

119886

119871(119904 119902 (119904) ◻119902 (119904)

◻119905) 119889119904

(13)

Let 1198671205730

fl ℎ isin 119867120573(119868R119889) ℎ(119886) = ℎ(119887) = 0 and 119902 isin

119867120572(119868R119889) with 120572 + 120573 gt 1 A 119867

120573

0-variation of 119902 is a function

of the form 119902 + ℎ where ℎ isin 119867120573

119874 We denote by 119863L◻(119902)(ℎ)

the quantity

lim120598rarr0

L◻ (119902 + 120598ℎ) minusL◻ (119902)

120598(14)

if there exists the so-called Frechet derivative ofL◻ at point119902 in direction ℎ

Definition 16 (nondifferentiable extremals) A 119867120573

0-extremal

curve of the functionalL◻ is a curve 119902 isin 119867120572(119868R119889) satisfying

119863L◻(119902)(ℎ) = 0 for any ℎ isin 119867120573

0

Theorem 17 (nondifferentiable Euler-Lagrange equations[17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be an admissibleLagrangian of class C2 We assume that 120574 isin 119867

120572(119868R119889) such

that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that 119871(119905 120574(119905)

◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin 119867120573

0(119868R119889)

A curve 120574 satisfying the nondifferentiable Euler-Lagrangeequation

◻

◻119905[120597119871

120597V(119905 120574 (119905)

◻120574 (119905)

◻119905)] =

120597119871

120597119909(119905 120574 (119905)

◻120574 (119905)

◻119905) (15)

is an extremal curve of functional (13)

3 Reminder about Hamiltonian Systems

We now recall the main concepts and results of both classicaland Cressonrsquos nondifferentiable Hamiltonian systems

31 Classical Hamiltonian Systems Let 119871 be an admissibleLagrangian function If 119871 satisfies the so-called Legendreproperty then we can associate to 119871 a Hamiltonian functiondenoted by119867

4 Discrete Dynamics in Nature and Society

Definition 18 Let 119871 be an admissible Lagrangian functionTheLagrangian119871 is said to satisfy the Legendre property if themapping V 997891rarr (120597119871120597V)(119905 119909 V) is invertible for any (119905 119902 V) isin

119868 timesR119889 times C119889

If we introduce a new variable

119901 =120597119871

120597V(119905 119902 V) (16)

and 119871 satisfies the Legendre property then we can find afunction 119891 such that

V = 119891 (119905 119902 119901) (17)

Using this notation we have the following definition

Definition 19 Let 119871 be an admissible Lagrangian functionsatisfying the Legendre property The Hamiltonian function119867 associated with 119871 is given by

119867 R timesR119889times C119889997888rarr C

(119905 119902 119901) 997891997888rarr 119867(119905 119902 119901) = 119901119891 (119905 119902 119901) minus 119871 (119905 119902 119891 (119905 119902 119901))

(18)

We have the following theorem (see eg [38])

Theorem 20 (Hamiltonrsquos least-action principle) The curve(119902 119901) isin C(119868R119889)timesC(119868C119889) is an extremal of the Hamiltonianfunctional

H (119902 119901) = int

119887

119886

119901 (119905) (119905) minus 119867 (119905 119902 (119905) 119901 (119905)) 119889119905 (19)

if and only if it satisfies the Hamiltonian system associated with119867 given by

(119905) =120597119867 (119905 119902 (119905) 119901 (119905))

120597119901

(119905) = minus120597119867 (119905 119902 (119905) 119901 (119905))

120597119902

(20)

called the Hamiltonian equations

A vectorial notation is obtained for the Hamiltonianequations in posing 119911 = (119902 119901)

⊤ and nabla119867 = (120597119867120597119902 120597119867120597119901)⊤

where ⊤ denotes the transposition The Hamiltonian equa-tions are then written as

119889119911 (119905)

119889119905= 119869 sdot nabla119867 (119905 119911 (119905)) (21)

where

119869 = (0 119868119889

minus119868119889 0) (22)

denotes the symplectic matrix with 119868119889 being the identitymatrix on R119889

32 NondifferentiableHamiltonian Systems Thenondifferen-tiable embedding induces a change in the phase space withrespect to the classical case As a consequence we have towork with variables (119909 119901) that belong to R119889 times C119889 and notonly to R119889 timesR119889 as usual

Definition 21 (nondifferentiable embedding of Hamiltoniansystems [17]) The nondifferentiable embedded Hamiltoniansystem (20) is given by

◻119902 (119905)

◻119905=

120597119867 (119905 119902 (119905) 119901 (119905))

120597119901

◻119901 (119905)

◻119905= minus

120597119867 (119905 119902 (119905) 119901 (119905))

120597119902

(23)

and the embedded Hamiltonian functionalH◻ is defined on119867120572(119868R119889) times 119867

120572(119868C119889) by

H◻ (119902 119901) = int

119887

119886

(119901 (119905)◻119902 (119905)

◻119905minus 119867 (119905 119902 (119905) 119901 (119905))) 119889119905 (24)

The nondifferentiable calculus of variations allows us toderive the extremals forH◻

Theorem 22 (nondifferentiable Hamiltonrsquos least-action prin-ciple [17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be anadmissible C2-Lagrangian We assume that 120574 isin 119867

120572(119868R119889)

such that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that

119871(119905 120574(119905) ◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin

119867120573

0(119868R119889) Let 119867 be the corresponding Hamiltonian defined

by (18) A curve 120574 997891rarr (119905 119902(119905) 119901(119905)) isin 119868 times R119889 times C119889 solution ofthe nondifferentiable Hamiltonian system (23) is an extremalof functional (24) over the space of variations119881 = 119867

120573

0(119868R119889)times

119867120573

0(119868C119889)

4 Nondifferentiable Helmholtz Problem

In this section we solve the inverse problem of the nondif-ferentiable calculus of variations in the Hamiltonian case Wefirst recall the usual way to derive the Helmholtz conditionsfollowing the presentation made by Santilli [28] Two mainderivations are available

(i) The first is related to the characterization of Hamilto-nian systems via the symplectic two-differential formand the fact that by duality the associated one-differential form to a Hamiltonian vector field isclosedmdashthe so-called integrability conditions

(ii) The second uses the characterization of Hamiltoniansystems via the self-adjointness of the Frechet deriva-tive of the differential operator associated with theequationmdashthe so-called Helmholtz conditions

Of course we have coincidence of the two procedures in theclassical case As there is no analogous of differential formin the framework of Cressonrsquos quantum calculus we followthe second way to obtain the nondifferentiable analogue of

Discrete Dynamics in Nature and Society 5

the Helmholtz conditions For simplicity we consider a time-independent Hamiltonian The time-dependent case can bedone in the same way

41 Helmholtz Conditions for Classical Hamiltonian SystemsIn this section we work on R2119889 119889 ge 1 119889 isin N

411 Symplectic Scalar Product The symplectic scalar product⟨sdot sdot⟩119869 is defined by

⟨119883 119884⟩119869 = ⟨119883 119869 sdot 119884⟩ (25)

for all 119883119884 isin R2119889 where ⟨sdot sdot⟩ denotes the usual scalarproduct and 119869 is the symplectic matrix (22) We also considerthe 1198712 symplectic scalar product induced by ⟨sdot sdot⟩119869 defined for119891 119892 isin C0([119886 119887]R2119889) by

⟨119891 119892⟩1198712 119869

= int

119887

119886

⟨119891 (119905) 119892 (119905)⟩119869119889119905 (26)

412 Adjoint of a Differential Operator In the following weconsider first-order differential equations of the form

119889

119889119905(119902

119901) = (

119883119902 (119902 119901)

119883119901 (119902 119901)) (27)

where the vector fields 119883119902 and 119883119901 are C1 with respect to 119902

and 119901 The associated differential operator is written as

119874119883 (119902 119901) = (

minus 119883119902 (119902 119901)

minus 119883119901 (119902 119901)) (28)

A natural notion of adjoint for a differential operator is thendefined as follows

Definition 23 Let 119860 C1([119886 119887]R2119889) rarr C1([119886 119887]R2119889) Wedefine the adjoint 119860lowast

119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by

⟨119860 sdot 119891 119892⟩1198712119869

= ⟨119860lowast

119869sdot 119892 119891⟩

1198712 119869 (29)

An operator 119860 will be called self-adjoint if 119860 = 119860lowast

119869with

respect to the 1198712 symplectic scalar product

413 Hamiltonian Helmholtz Conditions The Helmholtzconditions in the Hamiltonian case are given by the followingresult (see Theorem 3121 p 176-177 in [28])

Theorem 24 (Hamiltonian Helmholtz theorem) Let119883(119902 119901)

be a vector field defined by 119883(119902 119901)⊤

= (119883119902(119902 119901) 119883119901(119902 119901))The differential equation (27) is Hamiltonian if and only if theassociated differential operator 119874119883 given by (28) has a self-adjoint Frechet derivative with respect to the 119871

2 symplecticscalar product In this case the Hamiltonian is given by

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (30)

The conditions for the self-adjointness of the differentialoperator can bemade explicitThey coincide with the integra-bility conditions characterizing the exactness of the one-formassociated with the vector field by duality (see [28] Theorem273 p 88)

Theorem 25 (integrability conditions) Let 119883(119902 119901)⊤

=

(119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differential operator119874119883 given by (28) has a self-adjoint Frechet derivative withrespect to the 1198712 symplectic scalar product if and only if

120597119883119902

120597119902+ (

120597119883119901

120597119901)

⊤

= 0120597119883119902

120597119901120597119883119901

120597119902119886119903119890 119904119910119898119898119890119905119903119894119888 (31)

42 Helmholtz Conditions for Nondifferentiable HamiltonianSystems The previous scalar products extend naturally tocomplex valued functions Let 0 lt 120572 lt 1 and let (119902 119901) isin

119867120572(119868R119889) times 119867

120572(119868C119889) such that ◻119902◻119905 isin 119867

120572(119868C119889) and

◻119901◻119905 isin 119867120572(119868C119889) We consider first-order nondifferential

equations of the form

◻

◻119905(119902

119901) = (

119883119902 (119902 119901)

119883119901 (119902 119901)) (32)

The associated quantum differential operator is written as

119874◻119883 (119902 119901) = (

◻119902

◻119905minus 119883119902 (119902 119901)

◻119901

◻119905minus 119883119901 (119902 119901)

) (33)

A natural notion of adjoint for a quantum differential opera-tor is then defined

Definition 26 Let 119860 C1◻([119886 119887]C2119889) rarr C1

◻([119886 119887]C2119889) We

define the adjoint 119860lowast119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by

⟨119860 sdot 119891 119892⟩1198712 119869

= ⟨119860lowast

119869sdot 119892 119891⟩

1198712 119869 (34)

An operator 119860 will be called self-adjoint if 119860 = 119860lowast

119869with

respect to the 1198712 symplectic scalar product We can now

obtain the adjoint operator associated with 119874◻119883

Proposition 27 Let 120573 be such that 120572 + 120573 gt 1 Let (119906 V) isin

119867120573

0(119868R119889) times 119867

120573

0(119868C119889) such that ◻119906◻119905 isin 119867

120572(119868C119889) and

◻V◻119905 isin 119867120572(119868C119889) The Frechet derivative 119863119874◻119883 of (33) at

(119902 119901) along (119906 V) is then given by

119863119874◻119883 (119902 119901) (119906 V) = (

◻119906

◻119905minus

120597119883119902

120597119902sdot 119906 minus

120597119883119902

120597119901sdot V

◻V◻119905

minus120597119883119901

120597119902sdot 119906 minus

120597119883119901

120597119901sdot V

) (35)

6 Discrete Dynamics in Nature and Society

Assume that 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin

119867120573

0(119868C119889) In consequence the adjoint 119863119874

lowast

◻119883of 119863119874◻119883(119902 119901)

with respect to the 1198712 symplectic scalar product is given by

119863119874lowast

◻119883(119902 119901) (119906 V)

= (

◻119906

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119906 minus (120597119883119902

120597119901)

⊤

sdot V

◻V◻119905

minus (120597119883119901

120597119902)

⊤

sdot 119906 + (120597119883119902

120597119902)

⊤

sdot V

)

(36)

Proof The expression for the Frechet derivative of (33) at(119902 119901) along (119906 V) is a simple computation Let (119908 119909) isin

119867120573

0(119868R119889) times 119867

120573

0(119868C119889) be such that ◻119908◻119905 isin 119867

120572(119868C119889) and

◻119909◻119905 isin 119867120572(119868C119889) By definition we have

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[◻119906

◻119905sdot 119909

minus (120597119883119902

120597119902sdot 119906) sdot 119909 minus (

120597119883119902

120597119901sdot V) sdot 119909 minus

◻V◻119905

sdot 119908

+ (120597119883119901

120597119902sdot 119906) sdot 119908 + (

120597119883119901

120597119901sdot V) sdot 119908]119889119905

(37)

As 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin 119867120573

0(119868C119889)

using the quantum Leibniz rule and the quantum version ofthe fundamental theorem of calculus we obtain

int

119887

119886

◻119906

◻119905sdot 119887 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

int

119887

119886

◻V◻119905

sdot 119886 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

(38)

Then

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[minus119906

sdot (◻119909

◻119905minus (

120597119883119901

120597119902)

⊤

sdot 119908 + (120597119883119902

120597119902)

⊤

sdot 119909) + V

sdot (◻119908

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119908 minus (120597119883119902

120597119901)

⊤

sdot 119909)]119889119905

(39)

By definition we obtain the expression of the adjoint 119863119874lowast

◻119883

of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar

product

In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions

Theorem 28 (nondifferentiable integrability conditions) Let119883(119902 119901)

⊤= (119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differ-

ential operator 119874◻119883 given by (33) has a self-adjoint Frechet

derivative with respect to the symplectic scalar product if andonly if

120597119883119902

120597119902+ (

120597119883119901

120597119901)

⊤

= 0 (HC1)

120597119883119902

120597119901120597119883119901

120597119902119886119903119890 119904119910119898119898119890119905119903119894119888 (HC2)

Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions

We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems

Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)

⊤=

(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (40)

Proof If 119883 is Hamiltonian then there exists a function 119867

R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (41)

that is we must show that

119883119902 (119902 119901) =120597119867 (119902 119901)

120597119901

119883119901 (119902 119901) = minus120597119867 (119902 119901)

120597119902

(42)

We have

120597119867 (119902 119901)

120597119902= int

1

0

[119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119902minus 119883119901 (120582119902 120582119901)

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119902] 119889120582

Discrete Dynamics in Nature and Society 7

120597119867 (119902 119901)

120597119901= int

1

0

[119883119901 (120582119902 120582119901) + 119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119901

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119901] 119889120582

(43)

Using the nondifferentiable integrability conditions weobtain

120597119867 (119902 119901)

120597119902= int

1

0

minus120597 (120582119883119901 (120582119902 120582119901))

120597120582119889120582 = minus119883119901 (119902 119901)

120597119867 (119902 119901)

120597119902= int

1

0

120597 (120582119883119902 (120582119902 120582119901))

120597120582119889120582 = 119883119902 (119902 119901)

(44)

which concludes the proof

5 Applications

We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]

Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867

120572(119868C119889) be

such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867

120572(119868C119889)

51 The Linear Case Let us consider the discrete nondiffer-entiable system

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 + 120575119901

(45)

where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 minus 120572119901

(46)

Using formula (40) we compute explicitly the Hamiltonianwhich is given by

119867(119902 119901) =1

2(1205731199012minus 1205741199022) + 120572119902 sdot 119901 (47)

52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby

=119901

119898

= minus1198801015840(119902)

(48)

with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by

119867(119902 119901) =1

21198981199012+ 119880 (119902) (49)

Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by

◻119902

◻119905=

119901

119898

◻119901

◻119905= minus1198801015840(119902)

(50)

The Hamiltonian Helmholtz conditions are clearly satisfied

Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)

6 Conclusion

We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions

References

[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002

[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012

[3] AMC Brito daCruzNMartins andD FMTorres ldquoHigher-order Hahnrsquos quantum variational calculusrdquoNonlinear AnalysisTheory Methods amp Applications vol 75 no 3 pp 1147ndash11572012

[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013

8 Discrete Dynamics in Nature and Society

[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010

[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001

[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011

[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012

[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013

[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013

[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013

[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005

[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015

[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009

[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010

[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009

[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011

[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011

[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014

[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010

[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011

[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013

[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014

[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941

[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896

[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897

[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898

[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978

[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013

[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004

[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014

[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999

[33] J EMarsden andMWest ldquoDiscretemechanics and variationalintegratorsrdquo Acta Numerica vol 10 pp 357ndash514 2001

[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203

[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117

[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015

[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015

[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

and the operator ⟨sdot⟩ by

⟨sdot⟩ C0(119868 times ]0 1] R

119889) 997888rarr C

0(119868R119889)

119891 997891997888rarr ⟨119891⟩ 119905 997891997888rarr lim120598rarr0

120587 (119891) (119905 120598)

(6)

The quantum derivative of 119891 without the dependence of120598 is introduced in [17]

Definition 8 (see [17]) The quantum derivative of 119891 in thespaceC0(119868R119889) is given by

◻119891

◻119905= ⟨

◻120598119891

◻119905⟩ (7)

The quantum derivative (7) has some nice propertiesNamely it satisfies a Leibniz rule and a version of thefundamental theorem of calculus

Theorem 9 (the quantum Leibniz rule [17]) Let 120572 + 120573 gt 1For 119891 isin 119867

120572(119868R119889) and 119892 isin 119867

120573(119868R119889) one has

◻

◻119905(119891 sdot 119892) (119905) =

◻119891 (119905)

◻119905sdot 119892 (119905) + 119891 (119905) sdot

◻119892 (119905)

◻119905 (8)

Remark 10 For 119891 isin C1(119868R119889) and 119892 isin C1(119868R119889) oneobtains from (8) the classical Leibniz rule (119891 sdot 119892)

1015840= 1198911015840sdot 119892 +

119891 sdot 1198921015840

Definition 11 We denote by C1◻the set of continuous func-

tions 119902 isin C0([119886 119887]R119889) such that ◻119902◻119905 isin C0(119868R119889)

Theorem 12 (the quantum version of the fundamental theo-rem of calculus [17]) Let 119891 isin C1

◻([119886 119887]R119889) be such that

lim120598rarr0

int

119887

119886

(◻120598119891

◻119905)

119864

(119905) 119889119905 = 0 (9)

Then

int

119887

119886

◻119891

◻119905(119905) 119889119905 = 119891 (119887) minus 119891 (119886) (10)

22 Nondifferentiable Calculus of Variations In [17] thecalculus of variations with quantum derivatives is introducedand respective Euler-Lagrange equations derived without thedependence of 120598

Definition 13 An admissible Lagrangian 119871 is a continuousfunction 119871 R times R119889 times C119889 rarr C such that 119871(119905 119909 V) is holo-morphic with respect to V and differentiable with respect to119909 Moreover 119871(119905 119909 V) isin R when V isin R119889 119871(119905 119909 V) isin C whenV isin C119889

An admissible Lagrangian function 119871 RtimesRdtimesC119889 rarr C

defines a functional onC1(119868R119889) denoted by

L C1(119868R119889) 997888rarr R

119902 997891997888rarr int

119887

119886

119871 (119905 119902 (119905) (119905)) 119889119905

(11)

Extremals of the functional L can be characterized by thewell-known Euler-Lagrange equation (see eg [38])

Theorem 14 The extremals 119902 isin C1(119868R119889) ofL coincide withthe solutions of the Euler-Lagrange equation

119889

119889119905[120597119871

120597V(119905 119902 (119905) (119905))] =

120597119871

120597119909(119905 119902 (119905) (119905)) (12)

The nondifferentiable embedding procedure allows us todefine a natural extension of the classical Euler-Lagrangeequation in the nondifferentiable context

Definition 15 (see [17]) The nondifferentiable LagrangianfunctionalL◻ associated withL is given by

L◻ C1

◻(119868R119889) 997888rarr R

119902 997891997888rarr int

119887

119886

119871(119904 119902 (119904) ◻119902 (119904)

◻119905) 119889119904

(13)

Let 1198671205730

fl ℎ isin 119867120573(119868R119889) ℎ(119886) = ℎ(119887) = 0 and 119902 isin

119867120572(119868R119889) with 120572 + 120573 gt 1 A 119867

120573

0-variation of 119902 is a function

of the form 119902 + ℎ where ℎ isin 119867120573

119874 We denote by 119863L◻(119902)(ℎ)

the quantity

lim120598rarr0

L◻ (119902 + 120598ℎ) minusL◻ (119902)

120598(14)

if there exists the so-called Frechet derivative ofL◻ at point119902 in direction ℎ

Definition 16 (nondifferentiable extremals) A 119867120573

0-extremal

curve of the functionalL◻ is a curve 119902 isin 119867120572(119868R119889) satisfying

119863L◻(119902)(ℎ) = 0 for any ℎ isin 119867120573

0

Theorem 17 (nondifferentiable Euler-Lagrange equations[17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be an admissibleLagrangian of class C2 We assume that 120574 isin 119867

120572(119868R119889) such

that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that 119871(119905 120574(119905)

◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin 119867120573

0(119868R119889)

A curve 120574 satisfying the nondifferentiable Euler-Lagrangeequation

◻

◻119905[120597119871

120597V(119905 120574 (119905)

◻120574 (119905)

◻119905)] =

120597119871

120597119909(119905 120574 (119905)

◻120574 (119905)

◻119905) (15)

is an extremal curve of functional (13)

3 Reminder about Hamiltonian Systems

We now recall the main concepts and results of both classicaland Cressonrsquos nondifferentiable Hamiltonian systems

31 Classical Hamiltonian Systems Let 119871 be an admissibleLagrangian function If 119871 satisfies the so-called Legendreproperty then we can associate to 119871 a Hamiltonian functiondenoted by119867

4 Discrete Dynamics in Nature and Society

Definition 18 Let 119871 be an admissible Lagrangian functionTheLagrangian119871 is said to satisfy the Legendre property if themapping V 997891rarr (120597119871120597V)(119905 119909 V) is invertible for any (119905 119902 V) isin

119868 timesR119889 times C119889

If we introduce a new variable

119901 =120597119871

120597V(119905 119902 V) (16)

and 119871 satisfies the Legendre property then we can find afunction 119891 such that

V = 119891 (119905 119902 119901) (17)

Using this notation we have the following definition

Definition 19 Let 119871 be an admissible Lagrangian functionsatisfying the Legendre property The Hamiltonian function119867 associated with 119871 is given by

119867 R timesR119889times C119889997888rarr C

(119905 119902 119901) 997891997888rarr 119867(119905 119902 119901) = 119901119891 (119905 119902 119901) minus 119871 (119905 119902 119891 (119905 119902 119901))

(18)

We have the following theorem (see eg [38])

Theorem 20 (Hamiltonrsquos least-action principle) The curve(119902 119901) isin C(119868R119889)timesC(119868C119889) is an extremal of the Hamiltonianfunctional

H (119902 119901) = int

119887

119886

119901 (119905) (119905) minus 119867 (119905 119902 (119905) 119901 (119905)) 119889119905 (19)

if and only if it satisfies the Hamiltonian system associated with119867 given by

(119905) =120597119867 (119905 119902 (119905) 119901 (119905))

120597119901

(119905) = minus120597119867 (119905 119902 (119905) 119901 (119905))

120597119902

(20)

called the Hamiltonian equations

A vectorial notation is obtained for the Hamiltonianequations in posing 119911 = (119902 119901)

⊤ and nabla119867 = (120597119867120597119902 120597119867120597119901)⊤

where ⊤ denotes the transposition The Hamiltonian equa-tions are then written as

119889119911 (119905)

119889119905= 119869 sdot nabla119867 (119905 119911 (119905)) (21)

where

119869 = (0 119868119889

minus119868119889 0) (22)

denotes the symplectic matrix with 119868119889 being the identitymatrix on R119889

32 NondifferentiableHamiltonian Systems Thenondifferen-tiable embedding induces a change in the phase space withrespect to the classical case As a consequence we have towork with variables (119909 119901) that belong to R119889 times C119889 and notonly to R119889 timesR119889 as usual

Definition 21 (nondifferentiable embedding of Hamiltoniansystems [17]) The nondifferentiable embedded Hamiltoniansystem (20) is given by

◻119902 (119905)

◻119905=

120597119867 (119905 119902 (119905) 119901 (119905))

120597119901

◻119901 (119905)

◻119905= minus

120597119867 (119905 119902 (119905) 119901 (119905))

120597119902

(23)

and the embedded Hamiltonian functionalH◻ is defined on119867120572(119868R119889) times 119867

120572(119868C119889) by

H◻ (119902 119901) = int

119887

119886

(119901 (119905)◻119902 (119905)

◻119905minus 119867 (119905 119902 (119905) 119901 (119905))) 119889119905 (24)

The nondifferentiable calculus of variations allows us toderive the extremals forH◻

Theorem 22 (nondifferentiable Hamiltonrsquos least-action prin-ciple [17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be anadmissible C2-Lagrangian We assume that 120574 isin 119867

120572(119868R119889)

such that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that

119871(119905 120574(119905) ◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin

119867120573

0(119868R119889) Let 119867 be the corresponding Hamiltonian defined

by (18) A curve 120574 997891rarr (119905 119902(119905) 119901(119905)) isin 119868 times R119889 times C119889 solution ofthe nondifferentiable Hamiltonian system (23) is an extremalof functional (24) over the space of variations119881 = 119867

120573

0(119868R119889)times

119867120573

0(119868C119889)

4 Nondifferentiable Helmholtz Problem

In this section we solve the inverse problem of the nondif-ferentiable calculus of variations in the Hamiltonian case Wefirst recall the usual way to derive the Helmholtz conditionsfollowing the presentation made by Santilli [28] Two mainderivations are available

(i) The first is related to the characterization of Hamilto-nian systems via the symplectic two-differential formand the fact that by duality the associated one-differential form to a Hamiltonian vector field isclosedmdashthe so-called integrability conditions

(ii) The second uses the characterization of Hamiltoniansystems via the self-adjointness of the Frechet deriva-tive of the differential operator associated with theequationmdashthe so-called Helmholtz conditions

Of course we have coincidence of the two procedures in theclassical case As there is no analogous of differential formin the framework of Cressonrsquos quantum calculus we followthe second way to obtain the nondifferentiable analogue of

Discrete Dynamics in Nature and Society 5

the Helmholtz conditions For simplicity we consider a time-independent Hamiltonian The time-dependent case can bedone in the same way

41 Helmholtz Conditions for Classical Hamiltonian SystemsIn this section we work on R2119889 119889 ge 1 119889 isin N

411 Symplectic Scalar Product The symplectic scalar product⟨sdot sdot⟩119869 is defined by

⟨119883 119884⟩119869 = ⟨119883 119869 sdot 119884⟩ (25)

for all 119883119884 isin R2119889 where ⟨sdot sdot⟩ denotes the usual scalarproduct and 119869 is the symplectic matrix (22) We also considerthe 1198712 symplectic scalar product induced by ⟨sdot sdot⟩119869 defined for119891 119892 isin C0([119886 119887]R2119889) by

⟨119891 119892⟩1198712 119869

= int

119887

119886

⟨119891 (119905) 119892 (119905)⟩119869119889119905 (26)

412 Adjoint of a Differential Operator In the following weconsider first-order differential equations of the form

119889

119889119905(119902

119901) = (

119883119902 (119902 119901)

119883119901 (119902 119901)) (27)

where the vector fields 119883119902 and 119883119901 are C1 with respect to 119902

and 119901 The associated differential operator is written as

119874119883 (119902 119901) = (

minus 119883119902 (119902 119901)

minus 119883119901 (119902 119901)) (28)

A natural notion of adjoint for a differential operator is thendefined as follows

Definition 23 Let 119860 C1([119886 119887]R2119889) rarr C1([119886 119887]R2119889) Wedefine the adjoint 119860lowast

119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by

⟨119860 sdot 119891 119892⟩1198712119869

= ⟨119860lowast

119869sdot 119892 119891⟩

1198712 119869 (29)

An operator 119860 will be called self-adjoint if 119860 = 119860lowast

119869with

respect to the 1198712 symplectic scalar product

413 Hamiltonian Helmholtz Conditions The Helmholtzconditions in the Hamiltonian case are given by the followingresult (see Theorem 3121 p 176-177 in [28])

Theorem 24 (Hamiltonian Helmholtz theorem) Let119883(119902 119901)

be a vector field defined by 119883(119902 119901)⊤

= (119883119902(119902 119901) 119883119901(119902 119901))The differential equation (27) is Hamiltonian if and only if theassociated differential operator 119874119883 given by (28) has a self-adjoint Frechet derivative with respect to the 119871

2 symplecticscalar product In this case the Hamiltonian is given by

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (30)

The conditions for the self-adjointness of the differentialoperator can bemade explicitThey coincide with the integra-bility conditions characterizing the exactness of the one-formassociated with the vector field by duality (see [28] Theorem273 p 88)

Theorem 25 (integrability conditions) Let 119883(119902 119901)⊤

=

(119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differential operator119874119883 given by (28) has a self-adjoint Frechet derivative withrespect to the 1198712 symplectic scalar product if and only if

120597119883119902

120597119902+ (

120597119883119901

120597119901)

⊤

= 0120597119883119902

120597119901120597119883119901

120597119902119886119903119890 119904119910119898119898119890119905119903119894119888 (31)

42 Helmholtz Conditions for Nondifferentiable HamiltonianSystems The previous scalar products extend naturally tocomplex valued functions Let 0 lt 120572 lt 1 and let (119902 119901) isin

119867120572(119868R119889) times 119867

120572(119868C119889) such that ◻119902◻119905 isin 119867

120572(119868C119889) and

◻119901◻119905 isin 119867120572(119868C119889) We consider first-order nondifferential

equations of the form

◻

◻119905(119902

119901) = (

119883119902 (119902 119901)

119883119901 (119902 119901)) (32)

The associated quantum differential operator is written as

119874◻119883 (119902 119901) = (

◻119902

◻119905minus 119883119902 (119902 119901)

◻119901

◻119905minus 119883119901 (119902 119901)

) (33)

A natural notion of adjoint for a quantum differential opera-tor is then defined

Definition 26 Let 119860 C1◻([119886 119887]C2119889) rarr C1

◻([119886 119887]C2119889) We

define the adjoint 119860lowast119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by

⟨119860 sdot 119891 119892⟩1198712 119869

= ⟨119860lowast

119869sdot 119892 119891⟩

1198712 119869 (34)

An operator 119860 will be called self-adjoint if 119860 = 119860lowast

119869with

respect to the 1198712 symplectic scalar product We can now

obtain the adjoint operator associated with 119874◻119883

Proposition 27 Let 120573 be such that 120572 + 120573 gt 1 Let (119906 V) isin

119867120573

0(119868R119889) times 119867

120573

0(119868C119889) such that ◻119906◻119905 isin 119867

120572(119868C119889) and

◻V◻119905 isin 119867120572(119868C119889) The Frechet derivative 119863119874◻119883 of (33) at

(119902 119901) along (119906 V) is then given by

119863119874◻119883 (119902 119901) (119906 V) = (

◻119906

◻119905minus

120597119883119902

120597119902sdot 119906 minus

120597119883119902

120597119901sdot V

◻V◻119905

minus120597119883119901

120597119902sdot 119906 minus

120597119883119901

120597119901sdot V

) (35)

6 Discrete Dynamics in Nature and Society

Assume that 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin

119867120573

0(119868C119889) In consequence the adjoint 119863119874

lowast

◻119883of 119863119874◻119883(119902 119901)

with respect to the 1198712 symplectic scalar product is given by

119863119874lowast

◻119883(119902 119901) (119906 V)

= (

◻119906

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119906 minus (120597119883119902

120597119901)

⊤

sdot V

◻V◻119905

minus (120597119883119901

120597119902)

⊤

sdot 119906 + (120597119883119902

120597119902)

⊤

sdot V

)

(36)

Proof The expression for the Frechet derivative of (33) at(119902 119901) along (119906 V) is a simple computation Let (119908 119909) isin

119867120573

0(119868R119889) times 119867

120573

0(119868C119889) be such that ◻119908◻119905 isin 119867

120572(119868C119889) and

◻119909◻119905 isin 119867120572(119868C119889) By definition we have

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[◻119906

◻119905sdot 119909

minus (120597119883119902

120597119902sdot 119906) sdot 119909 minus (

120597119883119902

120597119901sdot V) sdot 119909 minus

◻V◻119905

sdot 119908

+ (120597119883119901

120597119902sdot 119906) sdot 119908 + (

120597119883119901

120597119901sdot V) sdot 119908]119889119905

(37)

As 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin 119867120573

0(119868C119889)

using the quantum Leibniz rule and the quantum version ofthe fundamental theorem of calculus we obtain

int

119887

119886

◻119906

◻119905sdot 119887 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

int

119887

119886

◻V◻119905

sdot 119886 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

(38)

Then

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[minus119906

sdot (◻119909

◻119905minus (

120597119883119901

120597119902)

⊤

sdot 119908 + (120597119883119902

120597119902)

⊤

sdot 119909) + V

sdot (◻119908

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119908 minus (120597119883119902

120597119901)

⊤

sdot 119909)]119889119905

(39)

By definition we obtain the expression of the adjoint 119863119874lowast

◻119883

of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar

product

In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions

Theorem 28 (nondifferentiable integrability conditions) Let119883(119902 119901)

⊤= (119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differ-

ential operator 119874◻119883 given by (33) has a self-adjoint Frechet

derivative with respect to the symplectic scalar product if andonly if

120597119883119902

120597119902+ (

120597119883119901

120597119901)

⊤

= 0 (HC1)

120597119883119902

120597119901120597119883119901

120597119902119886119903119890 119904119910119898119898119890119905119903119894119888 (HC2)

Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions

We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems

Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)

⊤=

(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (40)

Proof If 119883 is Hamiltonian then there exists a function 119867

R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (41)

that is we must show that

119883119902 (119902 119901) =120597119867 (119902 119901)

120597119901

119883119901 (119902 119901) = minus120597119867 (119902 119901)

120597119902

(42)

We have

120597119867 (119902 119901)

120597119902= int

1

0

[119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119902minus 119883119901 (120582119902 120582119901)

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119902] 119889120582

Discrete Dynamics in Nature and Society 7

120597119867 (119902 119901)

120597119901= int

1

0

[119883119901 (120582119902 120582119901) + 119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119901

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119901] 119889120582

(43)

Using the nondifferentiable integrability conditions weobtain

120597119867 (119902 119901)

120597119902= int

1

0

minus120597 (120582119883119901 (120582119902 120582119901))

120597120582119889120582 = minus119883119901 (119902 119901)

120597119867 (119902 119901)

120597119902= int

1

0

120597 (120582119883119902 (120582119902 120582119901))

120597120582119889120582 = 119883119902 (119902 119901)

(44)

which concludes the proof

5 Applications

We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]

Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867

120572(119868C119889) be

such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867

120572(119868C119889)

51 The Linear Case Let us consider the discrete nondiffer-entiable system

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 + 120575119901

(45)

where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 minus 120572119901

(46)

Using formula (40) we compute explicitly the Hamiltonianwhich is given by

119867(119902 119901) =1

2(1205731199012minus 1205741199022) + 120572119902 sdot 119901 (47)

52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby

=119901

119898

= minus1198801015840(119902)

(48)

with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by

119867(119902 119901) =1

21198981199012+ 119880 (119902) (49)

Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by

◻119902

◻119905=

119901

119898

◻119901

◻119905= minus1198801015840(119902)

(50)

The Hamiltonian Helmholtz conditions are clearly satisfied

Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)

6 Conclusion

We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions

References

[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002

[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012

[3] AMC Brito daCruzNMartins andD FMTorres ldquoHigher-order Hahnrsquos quantum variational calculusrdquoNonlinear AnalysisTheory Methods amp Applications vol 75 no 3 pp 1147ndash11572012

[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013

8 Discrete Dynamics in Nature and Society

[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010

[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001

[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011

[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012

[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013

[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013

[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013

[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005

[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015

[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009

[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010

[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009

[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011

[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011

[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014

[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010

[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011

[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013

[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014

[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941

[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896

[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897

[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898

[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978

[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013

[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004

[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014

[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999

[33] J EMarsden andMWest ldquoDiscretemechanics and variationalintegratorsrdquo Acta Numerica vol 10 pp 357ndash514 2001

[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203

[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117

[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015

[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015

[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

Definition 18 Let 119871 be an admissible Lagrangian functionTheLagrangian119871 is said to satisfy the Legendre property if themapping V 997891rarr (120597119871120597V)(119905 119909 V) is invertible for any (119905 119902 V) isin

119868 timesR119889 times C119889

If we introduce a new variable

119901 =120597119871

120597V(119905 119902 V) (16)

and 119871 satisfies the Legendre property then we can find afunction 119891 such that

V = 119891 (119905 119902 119901) (17)

Using this notation we have the following definition

Definition 19 Let 119871 be an admissible Lagrangian functionsatisfying the Legendre property The Hamiltonian function119867 associated with 119871 is given by

119867 R timesR119889times C119889997888rarr C

(119905 119902 119901) 997891997888rarr 119867(119905 119902 119901) = 119901119891 (119905 119902 119901) minus 119871 (119905 119902 119891 (119905 119902 119901))

(18)

We have the following theorem (see eg [38])

Theorem 20 (Hamiltonrsquos least-action principle) The curve(119902 119901) isin C(119868R119889)timesC(119868C119889) is an extremal of the Hamiltonianfunctional

H (119902 119901) = int

119887

119886

119901 (119905) (119905) minus 119867 (119905 119902 (119905) 119901 (119905)) 119889119905 (19)

if and only if it satisfies the Hamiltonian system associated with119867 given by

(119905) =120597119867 (119905 119902 (119905) 119901 (119905))

120597119901

(119905) = minus120597119867 (119905 119902 (119905) 119901 (119905))

120597119902

(20)

called the Hamiltonian equations

A vectorial notation is obtained for the Hamiltonianequations in posing 119911 = (119902 119901)

⊤ and nabla119867 = (120597119867120597119902 120597119867120597119901)⊤

where ⊤ denotes the transposition The Hamiltonian equa-tions are then written as

119889119911 (119905)

119889119905= 119869 sdot nabla119867 (119905 119911 (119905)) (21)

where

119869 = (0 119868119889

minus119868119889 0) (22)

denotes the symplectic matrix with 119868119889 being the identitymatrix on R119889

32 NondifferentiableHamiltonian Systems Thenondifferen-tiable embedding induces a change in the phase space withrespect to the classical case As a consequence we have towork with variables (119909 119901) that belong to R119889 times C119889 and notonly to R119889 timesR119889 as usual

Definition 21 (nondifferentiable embedding of Hamiltoniansystems [17]) The nondifferentiable embedded Hamiltoniansystem (20) is given by

◻119902 (119905)

◻119905=

120597119867 (119905 119902 (119905) 119901 (119905))

120597119901

◻119901 (119905)

◻119905= minus

120597119867 (119905 119902 (119905) 119901 (119905))

120597119902

(23)

and the embedded Hamiltonian functionalH◻ is defined on119867120572(119868R119889) times 119867

120572(119868C119889) by

H◻ (119902 119901) = int

119887

119886

(119901 (119905)◻119902 (119905)

◻119905minus 119867 (119905 119902 (119905) 119901 (119905))) 119889119905 (24)

The nondifferentiable calculus of variations allows us toderive the extremals forH◻

Theorem 22 (nondifferentiable Hamiltonrsquos least-action prin-ciple [17]) Let 0 lt 120572 120573 lt 1 with 120572 + 120573 gt 1 Let 119871 be anadmissible C2-Lagrangian We assume that 120574 isin 119867

120572(119868R119889)

such that ◻120574◻119905 isin 119867120572(119868R119889) Moreover we assume that

119871(119905 120574(119905) ◻120574(119905)◻119905)ℎ(119905) satisfies condition (9) for all ℎ isin

119867120573

0(119868R119889) Let 119867 be the corresponding Hamiltonian defined

by (18) A curve 120574 997891rarr (119905 119902(119905) 119901(119905)) isin 119868 times R119889 times C119889 solution ofthe nondifferentiable Hamiltonian system (23) is an extremalof functional (24) over the space of variations119881 = 119867

120573

0(119868R119889)times

119867120573

0(119868C119889)

4 Nondifferentiable Helmholtz Problem

In this section we solve the inverse problem of the nondif-ferentiable calculus of variations in the Hamiltonian case Wefirst recall the usual way to derive the Helmholtz conditionsfollowing the presentation made by Santilli [28] Two mainderivations are available

(i) The first is related to the characterization of Hamilto-nian systems via the symplectic two-differential formand the fact that by duality the associated one-differential form to a Hamiltonian vector field isclosedmdashthe so-called integrability conditions

(ii) The second uses the characterization of Hamiltoniansystems via the self-adjointness of the Frechet deriva-tive of the differential operator associated with theequationmdashthe so-called Helmholtz conditions

Of course we have coincidence of the two procedures in theclassical case As there is no analogous of differential formin the framework of Cressonrsquos quantum calculus we followthe second way to obtain the nondifferentiable analogue of

Discrete Dynamics in Nature and Society 5

the Helmholtz conditions For simplicity we consider a time-independent Hamiltonian The time-dependent case can bedone in the same way

41 Helmholtz Conditions for Classical Hamiltonian SystemsIn this section we work on R2119889 119889 ge 1 119889 isin N

411 Symplectic Scalar Product The symplectic scalar product⟨sdot sdot⟩119869 is defined by

⟨119883 119884⟩119869 = ⟨119883 119869 sdot 119884⟩ (25)

for all 119883119884 isin R2119889 where ⟨sdot sdot⟩ denotes the usual scalarproduct and 119869 is the symplectic matrix (22) We also considerthe 1198712 symplectic scalar product induced by ⟨sdot sdot⟩119869 defined for119891 119892 isin C0([119886 119887]R2119889) by

⟨119891 119892⟩1198712 119869

= int

119887

119886

⟨119891 (119905) 119892 (119905)⟩119869119889119905 (26)

412 Adjoint of a Differential Operator In the following weconsider first-order differential equations of the form

119889

119889119905(119902

119901) = (

119883119902 (119902 119901)

119883119901 (119902 119901)) (27)

where the vector fields 119883119902 and 119883119901 are C1 with respect to 119902

and 119901 The associated differential operator is written as

119874119883 (119902 119901) = (

minus 119883119902 (119902 119901)

minus 119883119901 (119902 119901)) (28)

A natural notion of adjoint for a differential operator is thendefined as follows

Definition 23 Let 119860 C1([119886 119887]R2119889) rarr C1([119886 119887]R2119889) Wedefine the adjoint 119860lowast

119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by

⟨119860 sdot 119891 119892⟩1198712119869

= ⟨119860lowast

119869sdot 119892 119891⟩

1198712 119869 (29)

An operator 119860 will be called self-adjoint if 119860 = 119860lowast

119869with

respect to the 1198712 symplectic scalar product

413 Hamiltonian Helmholtz Conditions The Helmholtzconditions in the Hamiltonian case are given by the followingresult (see Theorem 3121 p 176-177 in [28])

Theorem 24 (Hamiltonian Helmholtz theorem) Let119883(119902 119901)

be a vector field defined by 119883(119902 119901)⊤

= (119883119902(119902 119901) 119883119901(119902 119901))The differential equation (27) is Hamiltonian if and only if theassociated differential operator 119874119883 given by (28) has a self-adjoint Frechet derivative with respect to the 119871

2 symplecticscalar product In this case the Hamiltonian is given by

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (30)

The conditions for the self-adjointness of the differentialoperator can bemade explicitThey coincide with the integra-bility conditions characterizing the exactness of the one-formassociated with the vector field by duality (see [28] Theorem273 p 88)

Theorem 25 (integrability conditions) Let 119883(119902 119901)⊤

=

(119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differential operator119874119883 given by (28) has a self-adjoint Frechet derivative withrespect to the 1198712 symplectic scalar product if and only if

120597119883119902

120597119902+ (

120597119883119901

120597119901)

⊤

= 0120597119883119902

120597119901120597119883119901

120597119902119886119903119890 119904119910119898119898119890119905119903119894119888 (31)

42 Helmholtz Conditions for Nondifferentiable HamiltonianSystems The previous scalar products extend naturally tocomplex valued functions Let 0 lt 120572 lt 1 and let (119902 119901) isin

119867120572(119868R119889) times 119867

120572(119868C119889) such that ◻119902◻119905 isin 119867

120572(119868C119889) and

◻119901◻119905 isin 119867120572(119868C119889) We consider first-order nondifferential

equations of the form

◻

◻119905(119902

119901) = (

119883119902 (119902 119901)

119883119901 (119902 119901)) (32)

The associated quantum differential operator is written as

119874◻119883 (119902 119901) = (

◻119902

◻119905minus 119883119902 (119902 119901)

◻119901

◻119905minus 119883119901 (119902 119901)

) (33)

A natural notion of adjoint for a quantum differential opera-tor is then defined

Definition 26 Let 119860 C1◻([119886 119887]C2119889) rarr C1

◻([119886 119887]C2119889) We

define the adjoint 119860lowast119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by

⟨119860 sdot 119891 119892⟩1198712 119869

= ⟨119860lowast

119869sdot 119892 119891⟩

1198712 119869 (34)

An operator 119860 will be called self-adjoint if 119860 = 119860lowast

119869with

respect to the 1198712 symplectic scalar product We can now

obtain the adjoint operator associated with 119874◻119883

Proposition 27 Let 120573 be such that 120572 + 120573 gt 1 Let (119906 V) isin

119867120573

0(119868R119889) times 119867

120573

0(119868C119889) such that ◻119906◻119905 isin 119867

120572(119868C119889) and

◻V◻119905 isin 119867120572(119868C119889) The Frechet derivative 119863119874◻119883 of (33) at

(119902 119901) along (119906 V) is then given by

119863119874◻119883 (119902 119901) (119906 V) = (

◻119906

◻119905minus

120597119883119902

120597119902sdot 119906 minus

120597119883119902

120597119901sdot V

◻V◻119905

minus120597119883119901

120597119902sdot 119906 minus

120597119883119901

120597119901sdot V

) (35)

6 Discrete Dynamics in Nature and Society

Assume that 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin

119867120573

0(119868C119889) In consequence the adjoint 119863119874

lowast

◻119883of 119863119874◻119883(119902 119901)

with respect to the 1198712 symplectic scalar product is given by

119863119874lowast

◻119883(119902 119901) (119906 V)

= (

◻119906

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119906 minus (120597119883119902

120597119901)

⊤

sdot V

◻V◻119905

minus (120597119883119901

120597119902)

⊤

sdot 119906 + (120597119883119902

120597119902)

⊤

sdot V

)

(36)

Proof The expression for the Frechet derivative of (33) at(119902 119901) along (119906 V) is a simple computation Let (119908 119909) isin

119867120573

0(119868R119889) times 119867

120573

0(119868C119889) be such that ◻119908◻119905 isin 119867

120572(119868C119889) and

◻119909◻119905 isin 119867120572(119868C119889) By definition we have

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[◻119906

◻119905sdot 119909

minus (120597119883119902

120597119902sdot 119906) sdot 119909 minus (

120597119883119902

120597119901sdot V) sdot 119909 minus

◻V◻119905

sdot 119908

+ (120597119883119901

120597119902sdot 119906) sdot 119908 + (

120597119883119901

120597119901sdot V) sdot 119908]119889119905

(37)

As 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin 119867120573

0(119868C119889)

using the quantum Leibniz rule and the quantum version ofthe fundamental theorem of calculus we obtain

int

119887

119886

◻119906

◻119905sdot 119887 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

int

119887

119886

◻V◻119905

sdot 119886 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

(38)

Then

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[minus119906

sdot (◻119909

◻119905minus (

120597119883119901

120597119902)

⊤

sdot 119908 + (120597119883119902

120597119902)

⊤

sdot 119909) + V

sdot (◻119908

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119908 minus (120597119883119902

120597119901)

⊤

sdot 119909)]119889119905

(39)

By definition we obtain the expression of the adjoint 119863119874lowast

◻119883

of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar

product

In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions

Theorem 28 (nondifferentiable integrability conditions) Let119883(119902 119901)

⊤= (119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differ-

ential operator 119874◻119883 given by (33) has a self-adjoint Frechet

derivative with respect to the symplectic scalar product if andonly if

120597119883119902

120597119902+ (

120597119883119901

120597119901)

⊤

= 0 (HC1)

120597119883119902

120597119901120597119883119901

120597119902119886119903119890 119904119910119898119898119890119905119903119894119888 (HC2)

Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions

We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems

Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)

⊤=

(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (40)

Proof If 119883 is Hamiltonian then there exists a function 119867

R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (41)

that is we must show that

119883119902 (119902 119901) =120597119867 (119902 119901)

120597119901

119883119901 (119902 119901) = minus120597119867 (119902 119901)

120597119902

(42)

We have

120597119867 (119902 119901)

120597119902= int

1

0

[119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119902minus 119883119901 (120582119902 120582119901)

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119902] 119889120582

Discrete Dynamics in Nature and Society 7

120597119867 (119902 119901)

120597119901= int

1

0

[119883119901 (120582119902 120582119901) + 119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119901

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119901] 119889120582

(43)

Using the nondifferentiable integrability conditions weobtain

120597119867 (119902 119901)

120597119902= int

1

0

minus120597 (120582119883119901 (120582119902 120582119901))

120597120582119889120582 = minus119883119901 (119902 119901)

120597119867 (119902 119901)

120597119902= int

1

0

120597 (120582119883119902 (120582119902 120582119901))

120597120582119889120582 = 119883119902 (119902 119901)

(44)

which concludes the proof

5 Applications

We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]

Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867

120572(119868C119889) be

such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867

120572(119868C119889)

51 The Linear Case Let us consider the discrete nondiffer-entiable system

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 + 120575119901

(45)

where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 minus 120572119901

(46)

Using formula (40) we compute explicitly the Hamiltonianwhich is given by

119867(119902 119901) =1

2(1205731199012minus 1205741199022) + 120572119902 sdot 119901 (47)

52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby

=119901

119898

= minus1198801015840(119902)

(48)

with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by

119867(119902 119901) =1

21198981199012+ 119880 (119902) (49)

Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by

◻119902

◻119905=

119901

119898

◻119901

◻119905= minus1198801015840(119902)

(50)

The Hamiltonian Helmholtz conditions are clearly satisfied

Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)

6 Conclusion

We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions

References

[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002

[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012

[3] AMC Brito daCruzNMartins andD FMTorres ldquoHigher-order Hahnrsquos quantum variational calculusrdquoNonlinear AnalysisTheory Methods amp Applications vol 75 no 3 pp 1147ndash11572012

[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013

8 Discrete Dynamics in Nature and Society

[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010

[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001

[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011

[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012

[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013

[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013

[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013

[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005

[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015

[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009

[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010

[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009

[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011

[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011

[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014

[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010

[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011

[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013

[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014

[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941

[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896

[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897

[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898

[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978

[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013

[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004

[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014

[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999

[33] J EMarsden andMWest ldquoDiscretemechanics and variationalintegratorsrdquo Acta Numerica vol 10 pp 357ndash514 2001

[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203

[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117

[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015

[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015

[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

the Helmholtz conditions For simplicity we consider a time-independent Hamiltonian The time-dependent case can bedone in the same way

41 Helmholtz Conditions for Classical Hamiltonian SystemsIn this section we work on R2119889 119889 ge 1 119889 isin N

411 Symplectic Scalar Product The symplectic scalar product⟨sdot sdot⟩119869 is defined by

⟨119883 119884⟩119869 = ⟨119883 119869 sdot 119884⟩ (25)

for all 119883119884 isin R2119889 where ⟨sdot sdot⟩ denotes the usual scalarproduct and 119869 is the symplectic matrix (22) We also considerthe 1198712 symplectic scalar product induced by ⟨sdot sdot⟩119869 defined for119891 119892 isin C0([119886 119887]R2119889) by

⟨119891 119892⟩1198712 119869

= int

119887

119886

⟨119891 (119905) 119892 (119905)⟩119869119889119905 (26)

412 Adjoint of a Differential Operator In the following weconsider first-order differential equations of the form

119889

119889119905(119902

119901) = (

119883119902 (119902 119901)

119883119901 (119902 119901)) (27)

where the vector fields 119883119902 and 119883119901 are C1 with respect to 119902

and 119901 The associated differential operator is written as

119874119883 (119902 119901) = (

minus 119883119902 (119902 119901)

minus 119883119901 (119902 119901)) (28)

A natural notion of adjoint for a differential operator is thendefined as follows

Definition 23 Let 119860 C1([119886 119887]R2119889) rarr C1([119886 119887]R2119889) Wedefine the adjoint 119860lowast

119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by

⟨119860 sdot 119891 119892⟩1198712119869

= ⟨119860lowast

119869sdot 119892 119891⟩

1198712 119869 (29)

An operator 119860 will be called self-adjoint if 119860 = 119860lowast

119869with

respect to the 1198712 symplectic scalar product

413 Hamiltonian Helmholtz Conditions The Helmholtzconditions in the Hamiltonian case are given by the followingresult (see Theorem 3121 p 176-177 in [28])

Theorem 24 (Hamiltonian Helmholtz theorem) Let119883(119902 119901)

be a vector field defined by 119883(119902 119901)⊤

= (119883119902(119902 119901) 119883119901(119902 119901))The differential equation (27) is Hamiltonian if and only if theassociated differential operator 119874119883 given by (28) has a self-adjoint Frechet derivative with respect to the 119871

2 symplecticscalar product In this case the Hamiltonian is given by

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (30)

The conditions for the self-adjointness of the differentialoperator can bemade explicitThey coincide with the integra-bility conditions characterizing the exactness of the one-formassociated with the vector field by duality (see [28] Theorem273 p 88)

Theorem 25 (integrability conditions) Let 119883(119902 119901)⊤

=

(119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differential operator119874119883 given by (28) has a self-adjoint Frechet derivative withrespect to the 1198712 symplectic scalar product if and only if

120597119883119902

120597119902+ (

120597119883119901

120597119901)

⊤

= 0120597119883119902

120597119901120597119883119901

120597119902119886119903119890 119904119910119898119898119890119905119903119894119888 (31)

42 Helmholtz Conditions for Nondifferentiable HamiltonianSystems The previous scalar products extend naturally tocomplex valued functions Let 0 lt 120572 lt 1 and let (119902 119901) isin

119867120572(119868R119889) times 119867

120572(119868C119889) such that ◻119902◻119905 isin 119867

120572(119868C119889) and

◻119901◻119905 isin 119867120572(119868C119889) We consider first-order nondifferential

equations of the form

◻

◻119905(119902

119901) = (

119883119902 (119902 119901)

119883119901 (119902 119901)) (32)

The associated quantum differential operator is written as

119874◻119883 (119902 119901) = (

◻119902

◻119905minus 119883119902 (119902 119901)

◻119901

◻119905minus 119883119901 (119902 119901)

) (33)

A natural notion of adjoint for a quantum differential opera-tor is then defined

Definition 26 Let 119860 C1◻([119886 119887]C2119889) rarr C1

◻([119886 119887]C2119889) We

define the adjoint 119860lowast119869of 119860 with respect to ⟨sdot sdot⟩1198712 119869 by

⟨119860 sdot 119891 119892⟩1198712 119869

= ⟨119860lowast

119869sdot 119892 119891⟩

1198712 119869 (34)

An operator 119860 will be called self-adjoint if 119860 = 119860lowast

119869with

respect to the 1198712 symplectic scalar product We can now

obtain the adjoint operator associated with 119874◻119883

Proposition 27 Let 120573 be such that 120572 + 120573 gt 1 Let (119906 V) isin

119867120573

0(119868R119889) times 119867

120573

0(119868C119889) such that ◻119906◻119905 isin 119867

120572(119868C119889) and

◻V◻119905 isin 119867120572(119868C119889) The Frechet derivative 119863119874◻119883 of (33) at

(119902 119901) along (119906 V) is then given by

119863119874◻119883 (119902 119901) (119906 V) = (

◻119906

◻119905minus

120597119883119902

120597119902sdot 119906 minus

120597119883119902

120597119901sdot V

◻V◻119905

minus120597119883119901

120597119902sdot 119906 minus

120597119883119901

120597119901sdot V

) (35)

6 Discrete Dynamics in Nature and Society

Assume that 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin

119867120573

0(119868C119889) In consequence the adjoint 119863119874

lowast

◻119883of 119863119874◻119883(119902 119901)

with respect to the 1198712 symplectic scalar product is given by

119863119874lowast

◻119883(119902 119901) (119906 V)

= (

◻119906

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119906 minus (120597119883119902

120597119901)

⊤

sdot V

◻V◻119905

minus (120597119883119901

120597119902)

⊤

sdot 119906 + (120597119883119902

120597119902)

⊤

sdot V

)

(36)

Proof The expression for the Frechet derivative of (33) at(119902 119901) along (119906 V) is a simple computation Let (119908 119909) isin

119867120573

0(119868R119889) times 119867

120573

0(119868C119889) be such that ◻119908◻119905 isin 119867

120572(119868C119889) and

◻119909◻119905 isin 119867120572(119868C119889) By definition we have

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[◻119906

◻119905sdot 119909

minus (120597119883119902

120597119902sdot 119906) sdot 119909 minus (

120597119883119902

120597119901sdot V) sdot 119909 minus

◻V◻119905

sdot 119908

+ (120597119883119901

120597119902sdot 119906) sdot 119908 + (

120597119883119901

120597119901sdot V) sdot 119908]119889119905

(37)

As 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin 119867120573

0(119868C119889)

using the quantum Leibniz rule and the quantum version ofthe fundamental theorem of calculus we obtain

int

119887

119886

◻119906

◻119905sdot 119887 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

int

119887

119886

◻V◻119905

sdot 119886 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

(38)

Then

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[minus119906

sdot (◻119909

◻119905minus (

120597119883119901

120597119902)

⊤

sdot 119908 + (120597119883119902

120597119902)

⊤

sdot 119909) + V

sdot (◻119908

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119908 minus (120597119883119902

120597119901)

⊤

sdot 119909)]119889119905

(39)

By definition we obtain the expression of the adjoint 119863119874lowast

◻119883

of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar

product

In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions

Theorem 28 (nondifferentiable integrability conditions) Let119883(119902 119901)

⊤= (119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differ-

ential operator 119874◻119883 given by (33) has a self-adjoint Frechet

derivative with respect to the symplectic scalar product if andonly if

120597119883119902

120597119902+ (

120597119883119901

120597119901)

⊤

= 0 (HC1)

120597119883119902

120597119901120597119883119901

120597119902119886119903119890 119904119910119898119898119890119905119903119894119888 (HC2)

Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions

We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems

Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)

⊤=

(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (40)

Proof If 119883 is Hamiltonian then there exists a function 119867

R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (41)

that is we must show that

119883119902 (119902 119901) =120597119867 (119902 119901)

120597119901

119883119901 (119902 119901) = minus120597119867 (119902 119901)

120597119902

(42)

We have

120597119867 (119902 119901)

120597119902= int

1

0

[119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119902minus 119883119901 (120582119902 120582119901)

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119902] 119889120582

Discrete Dynamics in Nature and Society 7

120597119867 (119902 119901)

120597119901= int

1

0

[119883119901 (120582119902 120582119901) + 119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119901

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119901] 119889120582

(43)

Using the nondifferentiable integrability conditions weobtain

120597119867 (119902 119901)

120597119902= int

1

0

minus120597 (120582119883119901 (120582119902 120582119901))

120597120582119889120582 = minus119883119901 (119902 119901)

120597119867 (119902 119901)

120597119902= int

1

0

120597 (120582119883119902 (120582119902 120582119901))

120597120582119889120582 = 119883119902 (119902 119901)

(44)

which concludes the proof

5 Applications

We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]

Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867

120572(119868C119889) be

such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867

120572(119868C119889)

51 The Linear Case Let us consider the discrete nondiffer-entiable system

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 + 120575119901

(45)

where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 minus 120572119901

(46)

Using formula (40) we compute explicitly the Hamiltonianwhich is given by

119867(119902 119901) =1

2(1205731199012minus 1205741199022) + 120572119902 sdot 119901 (47)

52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby

=119901

119898

= minus1198801015840(119902)

(48)

with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by

119867(119902 119901) =1

21198981199012+ 119880 (119902) (49)

Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by

◻119902

◻119905=

119901

119898

◻119901

◻119905= minus1198801015840(119902)

(50)

The Hamiltonian Helmholtz conditions are clearly satisfied

Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)

6 Conclusion

We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions

References

[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002

[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012

[3] AMC Brito daCruzNMartins andD FMTorres ldquoHigher-order Hahnrsquos quantum variational calculusrdquoNonlinear AnalysisTheory Methods amp Applications vol 75 no 3 pp 1147ndash11572012

[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013

8 Discrete Dynamics in Nature and Society

[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010

[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001

[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011

[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012

[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013

[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013

[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013

[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005

[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015

[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009

[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010

[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009

[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011

[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011

[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014

[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010

[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011

[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013

[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014

[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941

[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896

[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897

[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898

[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978

[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013

[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004

[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014

[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999

[33] J EMarsden andMWest ldquoDiscretemechanics and variationalintegratorsrdquo Acta Numerica vol 10 pp 357ndash514 2001

[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203

[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117

[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015

[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015

[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Assume that 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin

119867120573

0(119868C119889) In consequence the adjoint 119863119874

lowast

◻119883of 119863119874◻119883(119902 119901)

with respect to the 1198712 symplectic scalar product is given by

119863119874lowast

◻119883(119902 119901) (119906 V)

= (

◻119906

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119906 minus (120597119883119902

120597119901)

⊤

sdot V

◻V◻119905

minus (120597119883119901

120597119902)

⊤

sdot 119906 + (120597119883119902

120597119902)

⊤

sdot V

)

(36)

Proof The expression for the Frechet derivative of (33) at(119902 119901) along (119906 V) is a simple computation Let (119908 119909) isin

119867120573

0(119868R119889) times 119867

120573

0(119868C119889) be such that ◻119908◻119905 isin 119867

120572(119868C119889) and

◻119909◻119905 isin 119867120572(119868C119889) By definition we have

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[◻119906

◻119905sdot 119909

minus (120597119883119902

120597119902sdot 119906) sdot 119909 minus (

120597119883119902

120597119901sdot V) sdot 119909 minus

◻V◻119905

sdot 119908

+ (120597119883119901

120597119902sdot 119906) sdot 119908 + (

120597119883119901

120597119901sdot V) sdot 119908]119889119905

(37)

As 119906 sdot ℎ and V sdot ℎ satisfy condition (9) for any ℎ isin 119867120573

0(119868C119889)

using the quantum Leibniz rule and the quantum version ofthe fundamental theorem of calculus we obtain

int

119887

119886

◻119906

◻119905sdot 119887 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

int

119887

119886

◻V◻119905

sdot 119886 119889119905 = int

119887

119886

minus119906 sdot◻119887

◻119905119889119905

(38)

Then

⟨119863119874◻119883 (119902 119901) (119906 V) (119908 119909)⟩1198712 119869

= int

119887

119886

[minus119906

sdot (◻119909

◻119905minus (

120597119883119901

120597119902)

⊤

sdot 119908 + (120597119883119902

120597119902)

⊤

sdot 119909) + V

sdot (◻119908

◻119905+ (

120597119883119901

120597119901)

⊤

sdot 119908 minus (120597119883119902

120597119901)

⊤

sdot 119909)]119889119905

(39)

By definition we obtain the expression of the adjoint 119863119874lowast

◻119883

of 119863119874◻119883(119902 119901) with respect to the 1198712 symplectic scalar

product

In consequence from a direct identification we obtainthe nondifferentiable self-adjointess conditions called Helm-holtzrsquos conditions As in the classical case we call these con-ditions nondifferentiable integrability conditions

Theorem 28 (nondifferentiable integrability conditions) Let119883(119902 119901)

⊤= (119883119902(119902 119901) 119883119901(119902 119901)) be a vector field The differ-

ential operator 119874◻119883 given by (33) has a self-adjoint Frechet

derivative with respect to the symplectic scalar product if andonly if

120597119883119902

120597119902+ (

120597119883119901

120597119901)

⊤

= 0 (HC1)

120597119883119902

120597119901120597119883119901

120597119902119886119903119890 119904119910119898119898119890119905119903119894119888 (HC2)

Remark 29 One can see that the Helmholtz conditions arethe same as in the classical discrete time-scale and stochasticcases We expected such a result because Cressonrsquos quantumcalculus provides a quantum Leibniz rule and a quantumversion of the fundamental theorem of calculus If suchproperties of an underlying calculus exist then theHelmholtzconditions will always be the same up to some conditions onthe working space of functions

We now obtain the main result of this paper which is theHelmholtz theorem for nondifferentiable Hamiltonian systems

Theorem 30 (nondifferentiable Hamiltonian Helmholtz the-orem) Let 119883(119902 119901) be a vector field defined by 119883(119902 119901)

⊤=

(119883119902(119902 119901) 119883119901(119902 119901)) The nondifferentiable system of (32) isHamiltonian if and only if the associated quantum differentialoperator119874◻119883 given by (33) has a self-adjoint Frechet derivativewith respect to the 1198712 symplectic scalar product In this case theHamiltonian is given by

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (40)

Proof If 119883 is Hamiltonian then there exists a function 119867

R119889 times C119889 rarr C such that119867(119902 119901) is holomorphic with respectto V and differentiable with respect to 119902 and119883119902 = 120597119867120597119901 and119883119901 = minus120597119867120597119902The nondifferentiable integrability conditionsare clearly verified using Schwarzrsquos lemma Reciprocally weassume that 119883 satisfies the nondifferentiable integrabilityconditions We will show that 119883 is Hamiltonian with respectto the Hamiltonian

119867(119902 119901) = int

1

0

[119901 sdot 119883119902 (120582119902 120582119901) minus 119902 sdot 119883119901 (120582119902 120582119901)] 119889120582 (41)

that is we must show that

119883119902 (119902 119901) =120597119867 (119902 119901)

120597119901

119883119901 (119902 119901) = minus120597119867 (119902 119901)

120597119902

(42)

We have

120597119867 (119902 119901)

120597119902= int

1

0

[119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119902minus 119883119901 (120582119902 120582119901)

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119902] 119889120582

Discrete Dynamics in Nature and Society 7

120597119867 (119902 119901)

120597119901= int

1

0

[119883119901 (120582119902 120582119901) + 119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119901

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119901] 119889120582

(43)

Using the nondifferentiable integrability conditions weobtain

120597119867 (119902 119901)

120597119902= int

1

0

minus120597 (120582119883119901 (120582119902 120582119901))

120597120582119889120582 = minus119883119901 (119902 119901)

120597119867 (119902 119901)

120597119902= int

1

0

120597 (120582119883119902 (120582119902 120582119901))

120597120582119889120582 = 119883119902 (119902 119901)

(44)

which concludes the proof

5 Applications

We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]

Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867

120572(119868C119889) be

such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867

120572(119868C119889)

51 The Linear Case Let us consider the discrete nondiffer-entiable system

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 + 120575119901

(45)

where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 minus 120572119901

(46)

Using formula (40) we compute explicitly the Hamiltonianwhich is given by

119867(119902 119901) =1

2(1205731199012minus 1205741199022) + 120572119902 sdot 119901 (47)

52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby

=119901

119898

= minus1198801015840(119902)

(48)

with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by

119867(119902 119901) =1

21198981199012+ 119880 (119902) (49)

Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by

◻119902

◻119905=

119901

119898

◻119901

◻119905= minus1198801015840(119902)

(50)

The Hamiltonian Helmholtz conditions are clearly satisfied

Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)

6 Conclusion

We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions

References

[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002

[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012

[3] AMC Brito daCruzNMartins andD FMTorres ldquoHigher-order Hahnrsquos quantum variational calculusrdquoNonlinear AnalysisTheory Methods amp Applications vol 75 no 3 pp 1147ndash11572012

[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013

8 Discrete Dynamics in Nature and Society

[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010

[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001

[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011

[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012

[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013

[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013

[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013

[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005

[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015

[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009

[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010

[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009

[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011

[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011

[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014

[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010

[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011

[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013

[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014

[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941

[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896

[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897

[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898

[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978

[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013

[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004

[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014

[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999

[33] J EMarsden andMWest ldquoDiscretemechanics and variationalintegratorsrdquo Acta Numerica vol 10 pp 357ndash514 2001

[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203

[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117

[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015

[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015

[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

120597119867 (119902 119901)

120597119901= int

1

0

[119883119901 (120582119902 120582119901) + 119901 sdot 120582120597119883119902 (120582119902 120582119901)

120597119901

minus 119902 sdot 120582120597119883119901 (120582119902 120582119901)

120597119901] 119889120582

(43)

Using the nondifferentiable integrability conditions weobtain

120597119867 (119902 119901)

120597119902= int

1

0

minus120597 (120582119883119901 (120582119902 120582119901))

120597120582119889120582 = minus119883119901 (119902 119901)

120597119867 (119902 119901)

120597119902= int

1

0

120597 (120582119883119902 (120582119902 120582119901))

120597120582119889120582 = 119883119902 (119902 119901)

(44)

which concludes the proof

5 Applications

We now provide two illustrative examples of our results onewith the formulation of dynamical systems with linear partsand another with Newtonrsquos equation which is particularlyuseful to study partial differentiable equations such as theNavier-Stokes equation Indeed the Navier-Stokes equationcan be recovered from a Lagrangian structure with Cressonrsquosquantum calculus [17] For more applications see [34]

Let 0 lt 120572 lt 1 and let (119902 119901) isin 119867120572(119868R119889) times 119867

120572(119868C119889) be

such that ◻119902◻119905 isin 119867120572(119868C119889) and ◻119902◻119905 isin 119867

120572(119868C119889)

51 The Linear Case Let us consider the discrete nondiffer-entiable system

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 + 120575119901

(45)

where 120572 120573 120574 and 120575 are constants The Helmholtz condition(HC2) is clearly satisfied However system (45) satisfies thecondition (HC1) if and only if 120572 + 120575 = 0 As a consequencelinear Hamiltonian nondifferentiable equations are of theform

◻119902

◻119905= 120572119902 + 120573119901

◻119901

◻119905= 120574119902 minus 120572119901

(46)

Using formula (40) we compute explicitly the Hamiltonianwhich is given by

119867(119902 119901) =1

2(1205731199012minus 1205741199022) + 120572119902 sdot 119901 (47)

52 Newtonrsquos Equation Newtonrsquos equation (see [38]) is givenby

=119901

119898

= minus1198801015840(119902)

(48)

with119898 isin R+ and 119902 119901 isin R119889This equation possesses a naturalHamiltonian structure with the Hamiltonian given by

119867(119902 119901) =1

21198981199012+ 119880 (119902) (49)

Using Cressonrsquos quantum calculus we obtain a natural non-differentiable system given by

◻119902

◻119905=

119901

119898

◻119901

◻119905= minus1198801015840(119902)

(50)

The Hamiltonian Helmholtz conditions are clearly satisfied

Remark 31 Itmust be noted thatHamiltonian (49) associatedwith (50) is recovered by formula (40)

6 Conclusion

We proved a Helmholtz theorem for nondifferentiable equa-tions which gives necessary and sufficient conditions for theexistence of a Hamiltonian structure In the affirmative casethe Hamiltonian is given Our result extends the results ofthe classical case when restricting attention to differentiablefunctions An important complementary result for the non-differentiable case is to obtain the Helmholtz theorem in theLagrangian case This is nontrivial and will be subject offuture research

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by FCT and CIDMA throughproject UIDMAT041062013 The first author is grateful toCIDMA and DMat-UA for the hospitality and good workingconditions during his visit at University of Aveiro Theauthors would like to thank an anonymous referee for carefulreading of the submitted paper and for useful suggestions

References

[1] V Kac and P Cheung Quantum Calculus Universitext Sprin-ger New York NY USA 2002

[2] N Martins and D F M Torres ldquoHigher-order infinite horizonvariational problems in discrete quantum calculusrdquo Computersamp Mathematics with Applications vol 64 no 7 pp 2166ndash21752012

[3] AMC Brito daCruzNMartins andD FMTorres ldquoHigher-order Hahnrsquos quantum variational calculusrdquoNonlinear AnalysisTheory Methods amp Applications vol 75 no 3 pp 1147ndash11572012

[4] A B Malinowska and N Martins ldquoGeneralized transversalityconditions for the Hahn quantum variational calculusrdquo Opti-mization vol 62 no 3 pp 323ndash344 2013

8 Discrete Dynamics in Nature and Society

[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010

[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001

[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011

[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012

[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013

[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013

[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013

[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005

[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015

[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009

[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010

[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009

[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011

[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011

[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014

[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010

[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011

[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013

[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014

[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941

[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896

[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897

[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898

[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978

[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013

[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004

[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014

[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999

[33] J EMarsden andMWest ldquoDiscretemechanics and variationalintegratorsrdquo Acta Numerica vol 10 pp 357ndash514 2001

[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203

[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117

[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015

[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015

[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

[5] A B Malinowska and D F M Torres ldquoThe Hahn quantumvariational calculusrdquo Journal of OptimizationTheory and Appli-cations vol 147 no 3 pp 419ndash442 2010

[6] M Bohner and A PetersonDynamic Equations on Time ScalesAn Introduction with Applications Birkhaauser Boston MassUSA 2001

[7] NMartins and D F M Torres ldquoLrsquoHopital-type rules for mono-tonicity with application to quantum calculusrdquo InternationalJournal of Mathematics and Computation vol 10 no M11 pp99ndash106 2011

[8] K A Aldwoah A B Malinowska and D F M Torres ldquoThepower quantum calculus and variational problemsrdquo Dynamicsof Continuous Discrete amp Impulsive Systems Series B Applica-tions amp Algorithms vol 19 no 1-2 pp 93ndash116 2012

[9] AM C Brito da Cruz NMartins and D F M Torres ldquoHahnrsquossymmetric quantum variational calculusrdquo Numerical AlgebraControl and Optimization vol 3 no 1 pp 77ndash94 2013

[10] A M C Brito da Cruz N Martins and D F M TorresldquoA symmetric Norlund sum with application to inequalitiesrdquoin Differential and Difference Equations with Applications SPinelas M Chipot and Z Dosla Eds vol 47 of SpringerProceedings in Mathematics amp Statistics pp 495ndash503 SpringerNew York NY USA 2013

[11] A M C Brito da Cruz N Martins and D F M Torres ldquoAsymmetric quantum calculusrdquo in Differential and DifferenceEquations with Applications vol 47 of Springer Proceedings inMathematics amp Statistics pp 359ndash366 Springer New York NYUSA 2013

[12] J Cresson ldquoNon-differentiable variational principlesrdquo Journalof Mathematical Analysis and Applications vol 307 no 1 pp48ndash64 2005

[13] R Almeida and N Martins ldquoVariational problems for Hold-erian functions with free terminal pointrdquo Mathematical Meth-ods in the Applied Sciences vol 38 no 6 pp 1059ndash1069 2015

[14] R Almeida and D F M Torres ldquoHolderian variational prob-lems subject to integral constraintsrdquo Journal of MathematicalAnalysis and Applications vol 359 no 2 pp 674ndash681 2009

[15] C Castro ldquoOn nonlinear quantum mechanics noncommuta-tive phase spaces fractal-scale calculus and vacuum energyrdquoFoundations of Physics vol 40 no 11 pp 1712ndash1730 2010

[16] J Cresson G S F Frederico and D F M Torres ldquoConstants ofmotion for non-differentiable quantum variational problemsrdquoTopologicalMethods inNonlinearAnalysis vol 33 no 2 pp 217ndash231 2009

[17] J Cresson and I Greff ldquoNon-differentiable embedding ofLagrangian systems and partial differential equationsrdquo Journalof Mathematical Analysis and Applications vol 384 no 2 pp626ndash646 2011

[18] J Cresson and I Greff ldquoA non-differentiable Noetherrsquos theo-remrdquo Journal of Mathematical Physics vol 52 no 2 Article ID023513 10 pages 2011

[19] A B Malinowska and D F M Torres Quantum VariationalCalculus Springer Briefs in Control Automation and RoboticsSpringer New York NY USA 2014

[20] R Almeida and D F M Torres ldquoGeneralized Euler-Lagrangeequations for variational problems with scale derivativesrdquoLetters in Mathematical Physics vol 92 no 3 pp 221ndash229 2010

[21] R Almeida and D F M Torres ldquoNondifferentiable variationalprinciples in terms of a quantum operatorrdquo MathematicalMethods in the Applied Sciences vol 34 no 18 pp 2231ndash22412011

[22] G S F Frederico and D F M Torres ldquoA nondifferentiablequantum variational embedding in presence of time delaysrdquoInternational Journal of Difference Equations vol 8 no 1 pp49ndash62 2013

[23] G S F Frederico and D F M Torres ldquoNoetherrsquos theorem withmomentumand energy terms forCressonrsquos quantumvariationalproblemsrdquo Advances in Dynamical Systems and Applicationsvol 9 no 2 pp 179ndash189 2014

[24] J Douglas ldquoSolution of the inverse problem of the calculus ofvariationsrdquo Transactions of the American Mathematical Societyvol 50 pp 71ndash128 1941

[25] A Mayer ldquoDie existenzbedingungen eines kinetischen poten-tialesrdquo Bericht Verhand Konig Sachs Gesell WissLeipzigMath-Phys Klasse vol 84 pp 519ndash529 1896

[26] A Hirsch ldquoUeber eine charakteristische Eigenschaft der Dif-ferentialgleichungen der Variationsrechnungrdquo MathematischeAnnalen vol 49 no 1 pp 49ndash72 1897

[27] A Hirsch ldquoDie existenzbedingungen des verallgemeinertenkinetischen potentialsrdquo Mathematische Annalen vol 50 no 2pp 429ndash441 1898

[28] R Santilli Foundations of Theoretical Mechanics The InverseProblem in Newtonian Mechanics Texts and Monographs inPhysics Springer New York NY USA 1978

[29] L Bourdin and J Cresson ldquoHelmholtzrsquos inverse problem of thediscrete calculus of variationsrdquo Journal of Difference Equationsand Applications vol 19 no 9 pp 1417ndash1436 2013

[30] P E Hydon and E L Mansfield ldquoA variational complex for dif-ference equationsrdquo Foundations of Computational Mathematicsvol 4 no 2 pp 187ndash217 2004

[31] M Dryl and D F M Torres ldquoNecessary condition for anEuler-Lagrange equation on time scalesrdquo Abstract and AppliedAnalysis vol 2014 Article ID 631281 7 pages 2014

[32] I D Albu andDOpris ldquoHelmholtz type condition formechan-ical integratorsrdquoNovi Sad Journal of Mathematics vol 29 no 3pp 11ndash21 1999

[33] J EMarsden andMWest ldquoDiscretemechanics and variationalintegratorsrdquo Acta Numerica vol 10 pp 357ndash514 2001

[34] J Cresson and F Pierret ldquoContinuous versusdiscrete structuresIImdashdiscrete Hamiltonian systems and Helmholtz conditionsrdquohttparxivorgabs150103203

[35] J Cresson and F Pierret ldquoContinuous versus discretestructuresImdashdiscrete embeddings and ordinary differential equationsrdquohttparxivorgabs14117117

[36] F Pierret ldquoHelmholtz theorem forHamiltonian systems on timescalesrdquo International Journal of Difference Equations vol 10 no1 pp 121ndash135 2015

[37] F Pierret ldquoHelmholtz theorem for stochastic Hamiltoniansystemsrdquo Advances in Dynamical Systems and Applications vol10 no 2 pp 201ndash214 2015

[38] V I Arnold Mathematical Methods of Classical MechanicsGraduate Texts in Mathematics Springer New York NY USA1978

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of