The Skinner-Rusk Approach for Vakonomic

8/13/2019 The Skinner-Rusk Approach for Vakonomic

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The Skinner-Rusk approach for vakonomicand nonholonomic eld theories

F. Cantrijn, J. Vankerschaver 1

Department of Mathematical Physics and Astronomy, Ghent University,Krijgslaan 281, B-9000 Ghent, Belgium

Abstract

We extend the Skinner-Rusk formalism to eld theories with nonholonomic andvakonomic constraints. This framework is then used to study the relation betweenboth types of constraints.

Key words: classical eld theories, constraints, Skinner-Rusk theory1991 MSC: 70S05, 58A20, 53C05

Dedicated to Willy Sarlet on the occasion of his sixtieth birthday.

1 Introduction

During the last decade, eld theories with nonholonomic constraints have beenstudied from different points of view (see [3,18,10,11]). At the same time anextensive study has been made of vakonomic methods in eld theory (see[8,1,2]). In this paper, we study the relation between both approaches, inthe case where the constraints are affine. Even though affine constraints areadmittedly rather exceptional in classical eld theory, this case is neverthelessquite interesting, as it allows a thorough comparison between vakonomic andnonholonomic dynamics. Indeed, in this paper, we will follow the work of Cortes et al. [5], who used the so-called formulation of Skinner and Rusk torecast both models in a form which allows comparison more easily.

In [15,16], Skinner and Rusk reformulated the equations of motion of a me-chanical system as a presymplectic system on T QT Q . Their idea in studyingthis rst-order system was to obtain a common framework for both regularand singular dynamics. Over the years, the framework of Skinner and Rusk

Email address: [email protected], [email protected] .1 Research Assistant of the Research Foundation Flanders (FWO-Vlaanderen)


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The Skinner-Rusk approach for vakonomic and nonholonomic eld theories 2

was extended in many directions: for our purposes, the most important con-tributions are [6,7], where the authors developed a Skinner-Rusk formalismfor classical eld theories.

This formulation will be briey recalled in section 2. In section 3, we willthen show how the vakonomic model can be described in the Skinner-Ruskframework, and we will do the same thing for the nonholonomic dynamics insection 4. Finally, in section 5 we propose a simple extension of a procedureby Cortes et al. [5] to compare both formulations, and we prove that they areequivalent in the case of integrable constraints.

2 Classical eld theories

2.1 The bundle framework

Classical elds are modeled as sections of a bre bundle : Y X of rankm , with (n + 1)-dimensional orientable base space X . On X , we consider axed volume form . Throughout this paper, we will use a coordinate system(x ), = 1 , . . . , n + 1, on X adapted to , i.e. such that dn +1 x := d x1 dxn +1 . Furthermore, on Y a coordinate system ( x , yA ), A = 1 , . . . , m ,adapted to is used.

Let

n +12 Y be the bundle of (n + 1)-forms on Y satisfying the following prop-

erty: ( n +12 Y )y if iv iw = 0 for all v, w (V )y .

In coordinates, an element of n +12 Y can be represented as = p

A dyA

dn x + pdn +1 x . Hence, on n +12 Y , we have a coordinate system ( x , yA ; p

A , p).

The bundle n +12 Y is of fundamental interest in classical eld theory, because

it can be equipped with a natural multisymplectic form, which is the general-isation to higher degree of the symplectic form on a cotangent bundle. If weintroduce rst the ( n + 1)-form as

( )(v1, . . . , v n +1 ) = (T (v1), . . . , T (vn +1 )) ,

where v1, . . . , v n +1 T ( n +12 Y ) and where : n +12 Y Y is the bundle

projection, then this multisymplectic form is dened by setting := d(see [4]).

The central stage for Skinner-Rusk theories is the product bundle J 1 n +12 Y Y . On this bundle, there exists a duality pairing , : J 1 n +12 Y R , which is reminiscent of the obvious pairing by duality on

T Q T Q , the bundle originally considered by Skinner and Rusk. This pair-ing is dened as follows: let y ( n +12 Y )y and j 1x J 1 , such that1,0( j 1x ) = y. Now, consider an (n + 1)-form on Y extending y , i.e. such

that (y) = y . The pullback (

)(x) is then a form at x of maximal degree,and hence a multiple a(x) of the volume form: ( )(x) = a(x)x . We now


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dene the duality pairing as

j 1x,

:= a

(x

).

(1)One can easily check that this denition is independent of the extension of .In coordinates, we have that a(x) = pA yA + p.

2.2 Affine constraints

Throughout this paper, we consider mainly eld theories with affine con-straints. These constraints are modeled by considering a distribution D onY of corank k. The distribution D is said to be weakly horizontal (see [9,p. 40]) if D is complementary to a subbundle of the vertical bundle V . Notethat this implies that k m .

A weakly horizontal distribution determines an affine subspace C of J 1 bysetting

C = { j 1x J 1 : ImT x D (x )}.If the annihilator of D is spanned by the linear independent 1-forms =AA dyA + A dx ( = 1 , . . . , k ), weak horizontality implies that the matrix A Ahas maximal rank k. In terms of these coordinate forms for , C is determinedby the vanishing of the k(n + 1) functions = A A yA + A . Conversely, it canbe seen that any affine subbundle C J 1 determines a weakly horizontaldistribution D on Y .

Let us go one step further, and assume moreover that there exists a bration : Y Q of Y over a new manifold Q, which is bered in turn over X (see(2)). The constraint distribution D will then be taken to be the horizontaldistribution of a connection on . See the commutative diagram below:

Y

Q

X

(2)

Consider a system of bundle coordinates ( x , y a ) on Q , where = 1 , . . . , n + 1and a = 1 , . . . , m k, and assume as before that there exists bundle coordi-nates on Y adapted to both and , i.e. coordinates (x ; ya , y ), collectivelydenoted by ( x , yA ), such that is locally given by (x , yA ) = ( x , y a ). Innonholonomic mechanics, a similar setup was studied in [14].

The constraint distribution D will be taken to be the horizontal distributionof a connection in the bre bundle : Y Q and hence D V = T Y .Since V is a subbundle of V , D is also weakly horizontal and determines asubmanifold C of J 1 as before. Since the coefficient matrix AA has maximalrank k , there exists (locally at least) a basis of the annihilator D spanned by

:= d y B a dya B dx .


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This basis is generally more suited for our purposes.

Remark 1 If the distribution D is integrable, then Y is foliated by integral submanifolds of D , in which case we say that the linear constraints are holo-nomic. The theory of holonomic and linear nonholonomic constraints was alsotreated in great detail in [10].

3 Skinner-Rusk formulation of vakonomic eld theories

Let : C J 1 be a constraint submanifold of codimension k(n + 1) in J 1 ,locally annihilated by k(n + 1) functionally independent constraint functions , where = 1 , . . . , k and = 1 , . . . , n + 1. Further on, C will be inducedby a weakly horizontal distribution as in section 2.2, but for now this is notrequired. We assume that ( 1,0) |C is a bration, such that it is possible tochoose locally an adapted coordinate system ( x ; yA ; ya , y ) on J 1 , and func-tions (x , yA , ya ) such that C is locally determined by the following set of k(n + 1) equations:

y (x

, yA , ya ) = 0 . (3)Hence, (x ; yA ; ya ) dene coordinates on C . We now redene as y (x , yA , ya ); note that the zero level set of these functions is still C .

3.1 Direct derivation

The vakonomic approach to the constrained problem specied by a LagrangianL and a constraint manifold C consists of looking for extremals of the followingaugmented Lagrangian: Lvak = L + (see [12]), where the functions are Lagrange multipliers. In other words, we impose the constraints on thespace of sections where the action is dened, rather than on the variations, aswill be the case in nonholonomic eld theory.

Let L := L : C R be the induced Lagrangian on C . By looking for ex-tremals of the action associated to Lvak , and rewriting the resulting extremalityconditions in terms of L , we obtain the following vakonomic eld equations :

ddx

Ly a

y a

= Ly a

y a

(4)

together withd dx

= Ly

y

and y = . (5)

3.2 Skinner-Rusk formulation

Consider now the Cartesian product bundle W 0 : W 0 := C n +12 Y Y .Dene also the projection 0 : W 0 X by putting 0 = W 0 . The given


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Lagrangian L induces a function H vak , called generalized Hamiltonian , on W 0,dened as follows:

H vak ( j 1x , ) = j 1, L( j 1x ), for all ( j 1x , ) (W 0)y , (6)

where , is the pairing between J 1 and n +12 Y dened in (1), and L = L

is again the restriction of L to C . In coordinates, we have H vak = pa ya + p + p L(x , yA , ya , ).

The multisymplectic form on n +12 Y can be used, together with the gener-

alized Hamiltonian Hvak , to dene a pre-multisymplectic form H vak on W 0:

H vak = + d H vak .

In terms of this form, the Skinner-Rusk eld equations are given by

i h H vak = nH vak , (7)

where h is the horizontal projector of a connection on 0 (see [6,7]). We willshow that these equations are equivalent to the vakonomic eld equations (4)and (5). In brief, we will construct a sequence of submanifolds

. . . W 3 W 2 W 1 W 0 = J 1 n +12 Y .

where W 1, W 2 and W 3 admit the following interpretation:

(1) W 1 consists of points where a solution h of (7) exists;(2) W 2 contains the points of W 1 where the image of the solution h is tangent

to W 1;(3) W 3 is dened by an additional technical assumption, to be specied later

on.

Under a certain regularity condition, W 1 and W 2 coincide and only the mani-folds W 0, W 1 and W 3 come into play. In the general case, one needs to applysome form of Gotays constraint algorithm to formulate the dynamics on anal constraint submanifold W , but this will not be considered here.

Let us now turn to the construction of W 1, W 2, and W 3. Notice that the eldequation (7) does not necessarily have a solution on the whole of W 0. Hence,we introduce a subset W 1 W 0, dened as the set of points of W 0 for whichthere does exist a horizontal projector of a connection on 0 : C n +12 Y X solving equation (7). If h has the following coordinate expression:

h = d x x

+ AA

y A + B

p

+ C A

p A+ D a

y a

, (8)

for unknown functions AA , B , C A , and D a , then a brief coordinate calcu-


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lation shows that W 1 is determined by the following equations:

pa = p y a +

Ly a

= p y a

+ Ly a

+ Ly

y a

. (9)

In addition, the connection coefficients have to satisfy the following con-straints:

A = , A

a = y

a

C A + p

y A

Ly A

= 0 . (10)

Let us now assume that W 1 is a manifold. This is a very restrictive assumption,but for the sake of clarity, we adopt it nevertheless. When dealing with real-world applications, it should be veried by calculations, and it can be expectedthat interesting behaviour may occur in the points where W 1 fails to be amanifold.

Secondly, we dene W 2 as the submanifold of W 1 where the image of thehorizontal projector h solving (7) is tangent to W 1. This is expressed by thefollowing equation:

h x

p a Ly a + p

y a = 0 .

In coordinates, this implies the following for the connection coefficients of h :

C a D Ly a

+ C y a

+ p D y a

= 0 , (11)

where D is the operator dened as

D =

x + y

a

ya +

y + D

a

ya

.

Equation (11) uniquely determines the coefficients D a if the following matrixis nonsingular:

C ab = 2Ly a y b

p 2

y a y b .

This we now assume. Hence, W 2 is the whole of W 1. If C ab is singular, addi-tional steps in the constraint algorithm are necessary. For this procedure,we refer to [6].

We end this section by giving a meaning to the coordinate p, and, at the sametime, xing the remaining connection coefficient B . This we do by considering


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the submanifold W 3 of W 2 dened as

W 3 := W 2 {Hvak (x

, yA

, ya ; p

A ) = 0}.

Demanding that a horizontal projector h on W 2 solving (7) is tangent to W 3leads to the following condition for B :

B + C a ya + C

+ D

a p

a + p

D (

) D (L) = 0

which allows for the determination of B in terms of the other connectioncoefficients as well as the momenta pA .

Let us now proceed to derive the vakonomic eld equations. On W 3, theSkinner-Rusk equation (7) can be locally written as

dya

dx =

H vakp a

and d pa

dx =

H vaky a

,

where H vak is dened on W 3 as H vak := p = pa ya + p L . By substitutingthis expression, we nally obtain the following eld equations:

Ly a

p y a

= ddx

Ly a

p y a

as well as

d pdx

= Ly

p y a

and y = (x

, yA , ya ).

If we identify the momenta p with the Lagrange multipliers , then theseequations are precisely the vakonomic eld equations (4) and (5).

Note in passing that, if L is regular, then W 3 is a multisymplectic manifold,with multisymplectic form W 3 := j3,0H vak , where j 3,0 : W 3 W 0 is thecanonical injection. This can be veried by a routine coordinate calculation.


Let D be the horizontal distribution of a connection on as in section 2.2.Recall that we may assume that the annihilator D is locally spanned by thefollowing k forms:

:= d y B a dya B dx

.

In case of affine constraints, the coefficients D a are determined by the follow-


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ing expression:

D b 2 Ly a y b

= 2 Lx y a

yb 2 Ly by a

b 2 Ly by a

+ Ly a + B

a Ly

+ pB ax

+ ybB ay b

+ B ay

B a y

y a

, (12)

where = B a ya + B . The expression between brackets in equation (12) isclosely related to the curvature of D . Indeed, we recall that the curvature Rof D is a section of

2 Y T Y , locally dened as R = Rab dya dyb y +R a dya dx y , where

R ab = Ba

y b Bb

y a + B b B

a

y B a B

b

y

R a = B ax

B y a

+ B B ay

B aB y

.

Bearing this in mind, one then obtains for the coefficients D a the followingexpression:

D b 2L

y a y b =

2Lx y a

yb 2L

y by a b

2Ly by a

+ Ly a

+ B a Ly

+ p (R

ab y

b + R

a ).

(13)

These expressions will play an important role in the comparison between vako-nomic and nonholonomic dynamics below in section 5.

4 Skinner-Rusk formulation of nonholonomic eld theories

A similar, but slightly more involved method can be used to cast the non-holonomic eld equations into Skinner-Rusk form. We consider a constraint

submanifold C of codimension k(n + 1), determined by similar expressions asin (3). The nonholonomic eld equations will be recast as a Skinner-Rusk typesystem on the bundle W 0 : W 0 := J 1

n +12 Y Y .

4.1 The nonholonomic eld equations

Let us rst briey recall the nonholonomic eld equations. For a more detailedtreatment, see [3,18].

Assume as before that C is a constraint submanifold of J 1 . In addition, let

F be a bundle of reaction forces . For the sake of deniteness, we asssume thatF is the subbundle of n +1 (T J 1 ) spanned by = S (d ), where S is


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the vertical endomorphism on J 1 (see [13]). In coordinates, we have

=

y a (dya

ya dx

) dn

x . (14)

Other choices F are also possible (see [17]) but will not be considered here.

In the presence of nonholonomic constraints, the eld equations become

x

Ly a

( j 1) Ly a

=

y a, (15)

together with the constraint that j 1 C , which serves to determine the un-known multipliers . These equations can be cast into the following intrinsicform:

i h L nL I (F ) and Im h T C , (16)where I (F ) is the ideal generated by F . The terms on the right-hand sideof (15) and (16) represent the constraint forces that keep the section j 1constrained to C .

4.2 Skinner-Rusk formulation

Consider rst the bundle of reaction forces F spanned by the ( n + 1)-forms dened in (14). We again denote by I (F ) the ideal in (J 1 ) generated

by F and we use the same notation to denote the pullback of this ideal to W 0.In the nonholonomic case, the generalized Hamiltonian is dened as

H nh := pr1, pr2 pr2L.

Note that Hnh involves the values of L on the whole of J 1 and not just onC as in the vakonomic approach. The pre-multisymplectic form H nh is thendened as in section 3 by putting H nh := + d H nh .

The nonholonomic eld equations are now

(i k H nh nH nh ) |Cn +12 Y I (F ) and (Im k ) |C

n +12 Y T (C

n +12 Y )

(17)for a horizontal projector k on 0 := W 0 ; notice the similarity between theseequations and the nonholonomic eld equations (16). A similar computationas in section 3 shows us that a horizontal projector, with coordinate expression

k = d x x

+ AA

y A + B

p

+ C A

p A+ D A

y A

, (18)

is a solution of the nonholonomic eld equations if and only if

AA = yA , pA = Ly A and C

A = Ly A +

y A , (19)


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where = y and the are a set of Lagrange multipliers, to be deter-mined by imposing the second part of (17). Let us now dene a submanifold

W 1 of W 0, specied by the relations (compare with (9)):

pA = Ly A

. (20)

Again as with vakonomic dynamics, we dene the submanifold W 2 W 1as the set of points where the image of the solution k determined by (19) istangent to W 1. This leads to the following conditions:

C A 2Lx y A

yB 2L

y B y A

D B 2L

y B yA

= 0 , (21)

as well asD

x

yA y A

D a y a

= 0 . (22)

It is easily seen that, in the case of a regular Lagrangian, these conditions donot restrict the submanifold W 1 any further, i.e. W 2 = W 1.

Finally, we dene the submanifold W 3 as (compare with the denition of W 3in the vakonomic case):

W 3 := W 2 {Hnh (x , yA , y a ; pA ) = 0}.

Demanding that a connection k whose image is tangent to W 2 has an imagetangent to W 3 imposes an additional condition on the the connection coeffi-cient B :

B + C A yA + D

A p

A

Lx

+ AALy A

+ D A Ly A

= 0 .

If we now dene H nh along W 3 as H nh := p = pA yA L , then the nonholo-nomic Skinner-Rusk equations (17) become

dyA

dx =

H nhp A

and d pA

dx =

H nhy A

+ y A

,

together with the constraint equations y = (x , yA , ya ). By using the ex-pression for H nh as well as (20), we nally obtain that the Skinner-Rusk equa-tions imply the standard nonholonomic eld equations:

ddx

Ly A

Ly A

= y A

,

together with the constraints.


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We now focus on affine constraints, and employ a similar convention for thebundle D of constraint forms as in the vakonomic case. In this case, the thirdequation of (19) splits into two sets of equations,

C a = Ly a

B a and C = Ly

+ .

One can combine these two expressions to eliminate the Lagrange multipliers.In the resulting expression, one can then substitute expression (21) to elim-inate C A , and expression (22) to express D in terms of D a . After a longcomputation, we nally obtain

D b 2L

y a y b =

2L

x y a yb

2L

y by a

2L

y y a+ L

y a

+ Ly

ybB ay b

+ B ay

+ B ax

y a

.(23)

5 Comparison between both approaches

Denition 2 Let X be a manifold and consider two brations C , D : C, D X . Consider a smooth map f : C D and let h be a connection on C , and k a connection on D . These connections are then said to be f -related if

T f h p = k f ( p) T f for all p C.

Consider now the vakonomic and nonholonomic manifolds W 3 and W 3. Thereexists an obvious surjective submersion f : W 3 W 3, given in coordinates byf (x , yA , y a ; p ) = ( x , yA , y a ) (see [9,5]). The map f can be given an intrinsicmeaning by using the Legendre transformation.

In order to study the relation between W 3 and W 3, and hence the relationbetween vakonomic and nonholonomic classical eld theory, we make use of the following observation of Krupkov a [9] and Cortes et al. [5]: if h and kwere f -related connections, then any integral section of h would project down(under f ) to an integral section of k . The original theorem concerned integralcurves of vector elds, but using denition 2 also covers integral sections of connections.

Let h be a vakonomic connection (with connection coefficients as determinedin section 3) and k be a nonholonomic connection (with coefficients as insection 4). By considering the set of points S 1 of W 3 where h and k are f -related, we obtain a rst characterization of the equivalence between h and k .Let us assume that S 1 is not empty, otherwise both connections are entirely

unrelated. A comparison of both sets of connection coefficients then shows thefollowing:


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Proposition 3 S 1 is locally determined by the vanishing of the following set of functions on W 3:

a = Ly

p (Rab y

b + R

a ).

Proof. The local expression for S 1 follows by considering the following con-tracted difference:

a = 2Ly a y b

D b Db ,

where D is the set of vakonomic connection coefficients (13), and D is the setof nonholonomic coefficients (23).

The submanifold S 1 can be seen as the rst stage in a certain constraint algorithm (see [9,5]), the result of which is a nal submanifold S (whichmight be empty) where the vakonomic and nonholonomic dynamics are equiv-alent. A general discussion of this constraint algorithm would not differ signi-cantly from the treatment of Krupkova and Cortes et al. and is hence omitted.We only wish to point out that, if the constraints are holonomic, and henceR ab = Ra = 0, then S 1 is the whole of W 3 and vakonomic and nonholo-nomic dynamics are everywhere equivalent, by which it is conrmed that thevakonomic and nonholonomic description give the same results for holonomicconstraints.

References

[1] E. Bibbona, L. Fatibene, M. Francaviglia, Gauge-natural parameterizedvariational problems, vakonomic eld theories and relativistic hydrodynamicsof a charged uid, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 15731608.

[2] E. Bibbona, L. Fatibene, M. Francaviglia, Chetaev versus vakonomicprescriptions in constrained eld theories with parametrized variationalcalculus, J. Math. Phys. 48 (3) (2007) 032903.

[3] E. Binz, M. de Leon, D. Martn de Diego, D. Socolescu, NonholonomicConstraints in Classical Field Theories, Rep. Math. Phys. 49 (2002) 151166.

[4] J. F. Cari nena, M. Crampin, L. A. Ibort, On the multisymplectic formalism forrst order eld theories, Diff. Geom. Appl. 1 (4) (1991) 345374.

[5] J. Cortes, M. de Leon, D. Martn de Diego, S. Martnez, Geometric descriptionof vakonomic and nonholonomic dynamics. Comparison of solutions, SIAM J.Control Optim. 41 (5) (2002) 13891412.

[6] M. de Leon, J. C. Marrero, D. Martn de Diego, A new geometric setting for

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[7] A. Echeverra-Enrquez, C. L opez, J. Marn-Solano, M. C. Mu noz-Lecanda,N. Rom an-Roy, Lagrangian-hamiltonian unied formalism for eld theory, J.Math. Phys. 45 (1) (2004) 360380.

[8] P. L. Garca, A. Garca, C. Rodrigo, Cartan forms for rst order constrainedvariational problems, J. Geom. Phys. 56 (4) (2006) 571610.

[9] O. Krupkov a, The Geometry of Ordinary Variational Equations, vol. 1678 of Lecture Notes in Mathematics, Springer-Verlag, 1997.

[10] O. Krupkov a, Partial differential equations with differential constraints, J. Diff.Eq. 220 (2) (2005) 354395.

[11] O. Krupkov a, P. Voln y, Euler-Lagrange and Hamilton equations fornonholonomic systems in eld theory, J. Phys. A: Math. Gen. 38 (40) (2005)87158745.

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[14] D. J. Saunders, F. Cantrijn, W. Sarlet, Regularity aspects and Hamiltonisationof non-holonomic systems, J. Phys. A: Math. Gen. 32 (39) (1999) 68696890.

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The Skinner-Rusk Approach for Vakonomic