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The geometrical theory of constraints applied to thedynamics of vakonomic mechanical systems:The vakonomic bracket

Sonia Martı́nez,a)

Jorge Cortés,b)

and Manuel de Leónc)

Instituto de Matemá ticas y Fı́ sica Fundamental, Consejo Superior de InvestigacionesCientı́ ficas, Serrano 123, 28006 Madrid, Spain

Received 2 August 1999; accepted for publication 14 September 1999

A vakonomic mechanical system can be alternatively described by an extended

Lagrangian using the Lagrange multipliers as new variables. Since this extended

Lagrangian is singular, the constraint algorithm can be applied and a Dirac bracket

giving the evolution of the observables can be constructed. © 2000 American

Institute of Physics. S0022-24880002802-4

I. INTRODUCTION

There are two different approaches to Lagrangian systems subjected to nonholonomic con-straints. The first one is based on the d’Alembert principle1– 5 and the corresponding equations of

motion are termed nonholonomic. The second approach is purely variational and was proposed by

Kozloz.6 Arnold, Kozlov, and Neishtadt1 coined the name of vakonomic mechanics of variational

axiomatic kind to refer to that sort of mechanics. Interesting comparisons between both ap-

proaches can be found in Refs. 3, 7, and 8.

Both topics have received a lot of attention in recent years in the context of geometric

mechanics. Nonholonomic mechanics has been studied from a Hamiltonian point of view,9–11

from a Lagrangian one,12–17 and even from a Poisson one.18–20 Several papers are devoted to

highlighting the equivalence among these viewpoints.21–23 Indeed, nonholonomic mechanics has

many applications to engineering robotics, control of satellites, etc., since it seems appropriate to

model the dynamical behavior of phenomena like rolling, etc. see Ref. 2, and references therein.

On the other hand, vakonomic mechanics is applied to study problems of optimal control theory

being related to sub-Riemannian geometry,24,25 economic growth theory,26 motion of microor-

ganisms at low Reynolds number,27 etc. A geometric unified approach was recently developed in

Ref. 28.

The aim of this paper is to study the equations of motion of vakonomic mechanical systems in

the framework of singular Lagrangian theories. As is well known, a vakonomic system given by

a Lagrangian function L L(q A, q̇ A) and constraints i(q A, q̇ A)0, can be equivalently described

by the extended Lagrangian LL(q A, i, q̇ A,̇ i) L(q A, q̇ A) i i see Ref. 1. This new La-

grangian is obviously singular, and its dynamics can be studied using Dirac’s machinery of

constraints.29 A first step in this direction is due to Cariñena and Rañada,30 where they considered

a global constraint function and treated the problem in the Lagrangian formalism.

Our program here is to apply the geometric version of the Dirac–Bergmann constraint algo-

rithm due to Gotay and Nester31–33 to the extended Lagrangian L. For that purpose, we first

enlarge the original space of velocities Q to PQRm

, and then we apply Gotay–Nester’sprocedure to L. We assume that L is a natural Lagrangian, that is, LT U where T is the kinetic

energy derived from a Riemannian metric on Q, and U is the potential energy. In addition, the

constraints are supposed to be linear in the velocities. With these assumptions, we find that the

algorithm stabilizes at the second step or, in other words, there are only secondary constraints.

aElectronic mail: [email protected] mail: [email protected] mail: [email protected]

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Moreover, all the constraints are second class in according with Dirac’s terminology. This last fact

implies that the final constraint submanifold M 2 is symplectic with respect to the canonical

symplectic structure on T *P and the symplectic structure induced there provides a Poisson

bracket that is just the same induced by the ambient Dirac bracket.29,34 A first result is that this

procedure ‘‘reduces’’ the phase space from T *P to M 2 .Furthermore, the final constraint submanifold is diffeomorphic with M ¯ Rm, where M ¯ is the

image in T *Q by the Legendre transformation of M . An interesting consequence of this identifi-

cation is the possibility of defining a Poisson bracket on functions on M ¯ which produces a

function on M 2 since we have to take account of the Lagrange multipliers. We are then impelled

to call this bracket the vakonomic bracket, in distinction with the so-called nonholonomic bracket

in nonholonomic mechanics.19,20,21,23,35 Indeed, the vakonomic bracket gives the evolution of the

observables of the vakonomic system.

If we consider a more general kind of constraints or Lagrangian not necessarily regular

situations which are more common in applications, the process is of course very much involved,

since tertiary and higher order constraints will appear. We leave this problem for further research.

The paper is organized as follows. In Sec. II, we review the two kinds of mechanics, non-

holonomic and vakonomic mechanics, from a unified variational approach. The constraint algo-

rithm in its geometric version is described in Sec. III and applied to vakonomic mechanics in Secs.IV and V. In Sec. VI, we study the second-order differential problem and in Sec. VII, we classify

the constraints according to Dirac. In Sec. VIII, we discuss what happens if the constraints are not

globally defined on TQ.

II. VARIATIONAL METHODS IN MECHANICS

In this section we shall give a brief account of the variational principles involved in the

derivation of the equations of motion in classical mechanics. For a more extended discussion see,

for instance, Refs. 3, 8, 28 and 36.

Let Q be an n-dimensional configuration manifold, and L :TQ→R an autonomous Lagrangian

function. If (q A) are coordinates on Q, we denote by ( q A, q̇ A) the natural bundle coordinates on

TQ such that the tangent bundle projection Q : TQ →Q reads as Q(q

A

, q̇

A

)

(q

A

).Given two points x , yQ we define the manifold of twice differentiable curves joining x and

y as

C 2 x , y c : 0,1→Q / c is C 2, c 0 x and c 1 y .

Let c be a curve in C 2( x, y ). As is well known, the tangent space of C 2( x , y ) at c is given by

T cC 2 x, y X : 0,1→TQ / X is C 1, X t T c t Q , X 0 0 and X 1 0.

We will assume here that L is subjected to nonholonomic linear constraints given by a submani-

fold M of TQ. Alternatively, the submanifold M can be viewed as the total space of a vector

subbundle of TQ, or, equivalently, as a distribution on Q which will be denoted by the same letter.

Therefore, if the annihilator M ° of M is locally spanned by m independent one-forms

1 , . . . , m, where i iA dq A, we have that the constraint functions 1 , . . . , m are just theevaluation functions of this basis, that is, i(vq)vq , i(q) for all vqT qQ , 1im . Now,we introduce the submanifold of C 2( x, y ) which consists of those curves which are compatible

with the constraint submanifold M ,

C ˜ 2 x , y c̃ C 2 x , y / c8 t M c̃ t , t 0,1.

Given a curve c̃ C ˜ 2( x, y ), the constraints allow us to consider a special vector subspace of

T c̃ C 2( x , y),

V c̃ X T c̃ C 2 x , y / i X 0, 1i m,

2091J. Math. Phys., Vol. 41, No. 4, April 2000 The geometrical theory of constraints . . .



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which are the allowed variations. Then, if X X A( / q A), we deduce that X V c̃ if and only if

iA X A0, 1im , 1

along the curve c̃ .

Next, define a functional J by

J :C 2 x, y →R

c0

1

L ċ t dt .

A direct computation using integration by parts shows that see Ref. 8

d J c X 0

1 L q A

d

dt L q̇ A

X Adt for cC 2( x , y ) and X T cC

2( x, y ).

A. Unconstrained systems

In this case, M TQ . The Hamilton principle states that a curve cC 2( x, y ) is a motion of the

Lagrangian system defined by L if and only if c is a critical point of J ; that is, iff d J (c)( X )

0 for all X T cC 2( x, y ), or

0

1 L q A

d

dt L q̇ A

X Adt 0, X A.This condition is equivalent to the Euler–Lagrange equations

d

dt L

q̇ A L

q A0, 1 An.

B. Nonholonomic mechanics

In this case, a curve c̃ C ˜ 2( x , y) is a motion if and only if it satisfies d J ( c̃ )( X )0, for all

X V c̃ , that is,

0

1 L q A

d

dt L q̇ A

X Adt 0,for all X A satisfying Eq. 1.

As before, we deduce that c̃ is a motion if and only if

L q A

d

dt L q̇ A

X A0, 2for all X A satisfying Eq. 1, which is just the statement of d’Alembert’s principle. Therefore, c̃ is

a motion for the nonholonomic system if and only if

d

dt L q̇ A

L q A

i iA , 1 An , 3

for some Lagrange multipliers 1, . . . , m.

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C. Vakonomic mechanics

In vakonomic mechanics, a curve c̃ C ˜ 2( x , y) is a motion if and only if d J ( c̃ )( X ˜ )0, for all

X ˜ T c̃ C ˜ 2( x , y), i.e., the motions are the extremals of the restriction of the functional to the curves

satisfying the constraints.

Now, using the Lagrange multipliers theorem in an infinite dimensional context, we deduce

see Refs. 1, 3, 8, and 36 that c̃ is an admissible regular motion if and only if there exist m

functions 1, . . . , m, i:0,1→R such that

d

dt L q̇ A

L q A

i iA q B

q̇ B iB

q A q̇ B d

i

dt iA , 1 A n. 4

An alternative approach to vakonomic mechanics is the following. From 4 we deduce that a

curve c̃ (q A( t )) in C ˜ 2( x, y ) is a solution of the vakonomic equations if and only if there exist

local functions 1, . . . , m on R such that c̄ (t )(q A(t ), i( t )) is an extremal for the extended

Lagrangian

L:T QRm→R, L L i i ,

i.e., it satisfies the Euler–Lagrange equations

d

dt L q̇ A

L q A

0, 1 An ,

d

dt L ̇ i

L i

i q A, q̇ A0, 1im

see Refs. 1, 3, 8, and 36 for details.

III. THE CONSTRAINT ALGORITHM

First of all, let us recall the geometric formulation for Lagrangian mechanics see Ref. 37.

Let S / q̇ Adq A be the canonical almost tangent structure on TQ and q̇ A( / q̇ A) the

Liouville vector field on TQ. From the Lagrangian L, we construct the Poincaré –Cartan two-form

LdS *(dL) and the energy E L( L) L .

Then, the equations of motion can be equivalently written as

i x LdE L . 5

Indeed, if the Lagrangian L is regular, i.e., its Hessian matrix Hess( L)( 2 L / q̇ A q̇ B) is not

singular, then L is symplectic, and 5 has a unique solution L which is a second-order differ-

ential equation SODE. The solutions of L are just the ones of the Euler–Lagrange equations. If

L is not regular, then 5 has no solution in general, and even if a solution exists, it will not be

unique or a SODE.

In order to treat with this kind of system, Gotay and Nester31–33 developed a constraintalgorithm a geometrization of the Dirac– Bergmann algorithm, applicable in the general frame-

work of presymplectic manifolds as is described in the following. A presymplectic system is a

triple, M, , , that consists of a smooth manifold M, a closed two-form with constant rank,

and a closed one-form .

We are interested in searching the possible solutions of

i x . 6

Let :T M→T *M be the map defined by ( X )i X . If is not symplectic, then is not

surjective and, consequently, 6 has no global solution on M in general.




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Consider the points of M where 6 has a solution and assume that this set is a submanifold

M2 of M1M this will be our case, since we are assuming that has constant rank . It could

still happen that the solutions on M2 are not tangent to M2 . In consequence, we take a submani-

fold M3 of M2 where the solutions are tangent to M2 . Continuing with this process repeatedly,

we generate a sequence of submanifolds

¯Mi¯M2¯M1M,

in such a way that if the algorithm stabilizes for some k , i.e., Mk Mk 1M f , then there exists

a vector field on M f such that

i / M f .

Notice that if we finish the process at the step k 1, it will mean that there is a global solution

on the whole of M.

Alternatively, the above submanifolds can be obtained as follows:

Mi xM / x z 0, zT xMi1

,

where

T xMi1 zT xM / x v , z 0, vT xMi1.

We call M2 the secondary constraint submanifold, M3 the tertiary constraint submanifold, and in

general Mi will be the i-ary constraint submanifold. If the algorithm stabilizes, then M f will be

the final constraint submanifold. Accordingly, the local functions defining these submanifolds

will be termed secondary constraints, ternary constraints, and so on.

IV. THE LAGRANGIAN FORMALISM

Let Q be an n-dimensional manifold representing the configuration space of a mechanical

system described by a Lagrangian function L:TQ→R and subjected to linear nonholonomic

constraints given by a submanifold M of TQ.

We shall assume that the Lagrangian is of natural type, that is LT U , where T is the

kinetic energy of a Riemannian metric g on Q, and U :Q→R is a potential energy.

In bundle coordinates L reads as

L q A, q̇ A12 g AB q q̇

Aq̇ BU q .

As we have seen earlier, the constraint submanifold M is locally defined as the zero set of m

independent linear nonholonomic constraints i(q A, q̇ A) iA (q)q̇

A.

For the sake of simplicity, we shall assume that the constraints i are globally defined on the

whole TQ. Later, we shall consider the general case.

Consider the product manifold PQRm with local coordinates ( q A, i). As we have seen in

Sec. II, the equations of motion corresponding to the vakonomic problem given by L and M can be

formulated in terms of the extended lagrangian L:T P→R, L L i i .

In what follows, we will identify TP with T QT Rm, and denote by 1 :TQT Rm→TQ and

2 :TQT Rm→T Rm the canonical projections of TQT Rm onto TQ and T Rm, respectively.

The Poincaré–Cartan two-form L associated to L is

L g AC q B q̇ C i iA

q B dq A∧dq B iA dq A∧d ig BA dq A∧dq̇ B.




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Notice that L is not symplectic because of the singular character of L. Indeed, L / ̇i0.

However, it still has constant rank as shows its Hessian matrix

Hess L 2L

q̇ A q̇ B

2L

̇ i q̇ B

2L

q̇ A ̇ j

2L

̇ i ̇ j Hess L 0

0 0 .

Therefore, we have

rank Lrank Hess L)rank Hess L )rank L 2n .

We deduce that the triple ( T P , L ,dE L) is a presymplectic system, with E L(L)L the

energy of L.

In this presymplectic framework the equations of motion are written as

i X LdE L . 7

Next, we will apply Gotay and Nester’s algorithm described in Sec. III to find a solution of 7.

Put P 1T P , then

P 2 xP 1 / dE L , Z x 0, Z T x P 1,

where

T x P 1 Z T x P 1 / L Z ,W 0,W T x P1 Z T x P1 / L Z 0.

Thus, to obtain P 2 we need first to calculate kerL .

A direct computation shows that

i

̇ i L0.

Moreover, we also have

i Z i L0,

where

Z i

ig BC iC

q̇ B , 1 im .

Therefore, since the vector fields / ̇ i, Z i are linearly independent and rank L2n , wededuce that they generate ker L , that is,

ker Lspan Z i , ̇ i

. Remark IV.1: It is not difficult to see that

dimker L)2 dim V T P ker L,




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where V (T P) is the vertical bundle over P. Therefore, L is a singular Lagrangian of Type II

according to the classification in Refs. 38 and 39.

Notice that E L( 1)*( E L), where E L is the energy corresponding to the Lagrangian L. In

what follows, we will write E L instead of ( 1)*( E L), for brevity.

Now, in order to compute the constraint functions which define P 2 , we calculate(dE L) x( / ̇

i) and (dE L) x( Z i), 1im ,

dE L ̇ i

E L ̇ i

0,

dE L Z i Z i E L ig BC iC

q̇ B L

q̇ A q̇ A L

g BC iC L

q̇ Bg BC iC

L

q̇ Bg BC iC g AB q̇

A iA q̇

A,

which are the original constraints.Thus, we have

P 2 xP 1 / i 1 x 0,1im.

Next, we shall compute T P 2 . Take X a vector field tangent to P 2 , that is, if

X X 1 A

q A X 2

i

i X 3

A

q̇ A X 4

i

̇ i,

we have

X i X 1 Aq̇ B

iB q A

X 3 A iA0, i . 8

The matrix ( iA ) has rank m, so we can assume that the submatrix ( i j), 1i , j m is invertible,

with inverse matrix ( ji ). Equation 8 can be written as

X 3 j i j X 3

a ia X 1 A

q̇ B iB

q A ,

where 1i , jm and m1an . Now, multiplying by ( ji ) we obtain that

X 3 j ji X 1

Aq̇ B

iB

q A ji X 3

a ia .

Consequently, we deduce that T P 2 is spanned by the vector fields

i

,

̇ i,

q Aq̇ B

iB

q A ji

q̇ j ,

q̇a ji ia

q̇ j .

Next, we want to compute T P 2 . Consider a vector field Y ,

Y Y 1 A

q AY 2

i

iY 3

A

q̇ AY 4

i

̇ i,




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such that Y T P 2 . After some calculations, we obtain that

Y 1 A0,

Y 3

Ag EA

iE Y

2

i .

Then

dE L Y g AB q̇ BY 3

Ag AB q̇

Bg EA iE Y 2i q̇ E iE Y 2

i iY 2

i0, 9

on P 2 and, therefore, P 3P 2 . This means that the algorithm stabilizes at P 2 , and P 2 is the final

constraint submanifold.

Our aim in the rest of this section is to get explicit expressions for the solutions of Eq. 7. For

that purpose, take an arbitrary vector field on TP locally written as

A A

q AB i

iC A

q̇ ADi

̇ i,

and assume that it satisfies

i LdE L .

A straightforward computation shows that

i LA B g BC q A g AC

q B q̇ C i iB

q A

iA

q B B i iAC Bg AB dq A

A A iA d iA Ag AB dq̇

B,

dE L12 g BC

q A q̇ C q̇ B

U

q Adq Ag AB q̇ Bdq̇ A.

Comparing the coefficients of d q̇ B and d i we deduce that

A Bg ABq̇ Bg BA , A

A iA0,

which implies A Aq̇ A, 1 An , and

iA q̇ A0, 1im . 10

Comparing now the coefficients of d q A, we find that B i and C B are related as follows:

B i iAC Bg AB

iq̇ D iD q A

iA

q D 1

2

g DC

q A

g AC

q D q̇ C q̇ D U

q A ,

or, equivalently,

C Bg AB q̇ D i iD q A

iA

q D 1

2

gCD

q A

g AC

q D q̇ C g AB U

q Ag AB iA B

i. 11

Moreover, since has to be tangent to P 2 , we get

C B jBq̇ Aq̇ B

jB

q A 0. 12

Introducing the expression for C B obtained in 11 into 12, we have




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The obstacle for the above-mentioned splitting to be ‘‘clean,’’ that is, X being independent of

and Z being independent of (q, q̇), is the coupling of the coordinates (q , q̇) and in the

vakonomic equations, a fact that can also be seen in the explicit expressions for B iB i(q , , q̇)

and C BC B(q , , q̇) see 14 and 15. A look to these local expressions shows that if the crossed

terms iB / q

A iA / q

B

vanish, then we will be able to project ‘‘cleanly’’ onto a vectorfield X independent of parameters. Of course, this is just the case when the constraints are

holonomic.8

On the other hand, this can also be done for some mechanical systems subjected to nonholo-

nomic constraints: for example, whenever we can get an expression for the Lagrange multipliers

( i(t )) along solutions ( q A(t ), q̇ A(t )). This is the case of the vertical rolling disk see Example

VII.4. In fact, we have that X 0( t ) L, M , where ( 0i (t )) is a special curve of Lagrange multi-

pliers and L, M is the nonholonomic vector field along M . Consequently, the solutions of the

nonholonomic problem may be regarded as a subset of the vakonomic ones.8,24 As a by-product of

the application of the Gotay and Nester algorithm, we have found a geometric characterization of

this fact. However, it will not be true in general as pointed out in Ref. 8 and the question of when

this can be done is still unanswered.

V. THE HAMILTONIAN FORMALISM

In this section, we will discuss the vakonomic system within the framework of the cotangent

bundle T *P . First of all, note that the Lagrangian L is almost regular, so we are just in the

assumptions of Gotay and Nester.31,32

Our interest in developing this formulation is to classify the constraints appeared in the

process following Dirac’s criterion and, then, to define a Dirac bracket giving the evolution of

dynamical variables.

Consider the Legendre transformation of L,

FL:T P→T *P .

As is well known, the Legendre mapping is a fibered mapping over P, i.e., PFL P , where P : T *P→ P is the canonical projection. In local coordinates the Legendre transformation reads

as

FL q A, i, q̇ A,̇ i q A, i, L q̇ A qA, q̇A

i i q̇ A

qA, q̇A

,0 .Therefore, if (q A, i, p̂ A , p̂ i) are bundle coordinates in T *P we have

p̂ Ag AB q̇ B i iA , p̂ i0,

along the image of FL.

Next we will prove that L is almost regular according to the definition in Refs. 31 and 32.

Proposition V.1: The following statements are true(i) FL(T P) M 1 is a submanifold of T *P .

(ii) F L is a submersion on its image and its fibers are connected submanifolds of TP. There-

fore, L is almost regular .

Proof: The Jacobian matrix of FL is

I n 0 K 0

0 I m K ¯ 0

0 0 Hess L 0

0 0 0 0

,




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where K q̇C ( g BC / q A) i( iB / q

A) and K ¯ ( iA ). Then, rank FL2nm at every x

T P , and from the rank theorem we deduce that M 1 is a submanifold of T *P . Moreover, with

this differentiable structure the mapping FL:T P→ M 1 is a submersion.

Next, we will prove that FL1( y )span( / i) P( y), for all y M 1 . In this case, the fibers

of FL would be connected. Indeed, let x1 , x2FL1

( y). Then both are in the same fiber of TP,i.e., P( x 1) P( x 2), and from the definition of FL we deduce that F L( 1( x 1))F L( 1( x2)).

Therefore 1( x 1) 1( x2) since F L is a diffeomorphism. Consequently, x 1 and x 2 differ only in

their components ̇ i. Thus, we have completed the proof.

Notice that M 1 is locally defined by the equations p̂ i0 for all i. Denote by 1 j 1* P ,

where Pdq A∧d p̂ Ad

i∧d p̂ i is the canonical symplectic form on T *P and j 1 : M 1→T *P is

the canonical inclusion. Then

1dq A∧d p̂ A

is a closed two-form on M 1 with constant rank 2ndim M 1 .

Since L is almost regular, the energy E L is constant along the fibers of FL and it induces a

well-defined function h 1 : M 1→R by the relation h 1FL E L . In fact,

h 1 q A, i, p̂ A,0

12 g

AB p̂ Ai iA p̂ B

j jB U q .

Thus, the system ( M 1 , 1 ,dh1) is presymplectic and we can apply to it the constraint algorithm.

It should be noticed that Gotay and Nester’s equivalence theorem see Refs. 31 and 32 implies

that this algorithm will stabilize at a submanifold M 2 of M 1 so that the following diagram

P1T P →FL

T *P

i 1↑ FL1 ↑ j 1

P2

M 1

FL2 ↑ j 2

M 2

is conmutative. Here, i 1 and j 2 are the canonical inclusions, and FLk FL Pk are submersions on

their images M k for k 1,2.

The primary constraints are those defining M 1 , that is, p̂ i0. In order to calculate the sec-

ondary constraints which in turn define M 2 , we first compute

ker 1 y T y M 1 T y M 1 / 1 y , 0, T y M 1.

In terms of the induced coordinate system on M 1 , the tangent space of M 1 at y is locally generatedby

q A y

, i

y

, p̂ A

y .

If

1 A

q A

y

2i i

y

3 A

p̂ A

y

T y M 1 ,




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1 A

q A

y

2i i

y

3 A

p̂ A

y

T y M 1 ,

then we have

1 y , dq A∧d p̂ A y , 1

A 3 A 3

A 1 A0, 1

A , 3 A .

Thus 1 A 3

A0, which implies that

T y M 1span i y .

Then dh1( / i)( h1 /

i) i provides the new constraints

i iA g AB p̂ B

j j B , 1im .

Consequently, M 2 is defined by the constraints p̂ i( y )0 and i( y)0,1im .

One can directly check that M 2FL( P 2). As we already know, M 2 is the final constraint

submanifold, that is, M 2 M f with the usual notations. Observe that we can introduce local

coordinates in M 2 as follows. Since i0, for all i, we have

i D i j jA g AB p̂ B , 1im .

Thus, we can take local coordinates ( q A, p̂ A) in M 2 . More precisely, the mapping

q A, p̂ A q A, D i j jA g

AB p̂ B , p̂ A,0

defines M 2 as a submanifold of T *P .

We summarize the above results in the following diagram:

P 1

T P

T QR

m

→

FL

T *P

i1↑ FL1

↑ j 1

P 2 i0 M 1 p̂ i0

FL2

↑ j 2

M 2 p̂ i0, i0.

Remark V.2: Observe that 2 j 2* P is in fact a symplectic form on M 2 since

rank 22ndim M 2 .

Then, we have that ( M 2 , 2 ,h 2) is a symplectic Hamiltonian system, where h 2 denotes the

restriction of h 1 to M 2 . In local coordinates,

h212 g

AB p̂ B p̂ A Dik kC iA g

CD p̂ DU .

Let us denote by M ¯ F L( M ) the submanifold of T *Q obtained by means of the Legendre

transformation associated to L. Indeed, M ¯ is defined by the linear constraints iA g AB p B , where

(q A, p A) stand for the bundle coordinates in T *Q . Notice that M ¯ is a vector subbundle of T *Q

since FL is a vector bundle isomorphism over Q.




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To end this section, we will investigate the relation between M 2 and M ¯ , and will compare 2with Q , the canonical symplectic form on T *Q .

Let : M T Rm→ P 2 be the global diffeomorphism between M T Rm and P 2 , which is

induced from the canonical diffeomorphism TQT Rm→T (QRm). By means of , we define

the global mapping

: M ¯ Rm→ M 2

ȳ , FL F L1 ȳ , ,0.

In local coordinates we have

q A, p A , i q A, i, p A

i iA ,0.

Proposition V.3: is a diffeomorphism.

Proof: Indeed, it is differentiable and its inverse is

M 2→ M ¯ Rm,

q A, p̂ A q A, p̂ A

i iA , i,

where i D i j jA g AB p̂ B . Obviously,

1 is differentiable, too.

Via one obtains that

* 2 Qd i iA ∧dq

A.

VI. THE SODE PROBLEM

In this section we will discuss the problem of finding a vector field ˜ satisfying the equations

i ˜ LdE L S ,

S ̃ S ,

on some submanifold S of P 2 . That is, we are looking for a solution satisfying the SODE

condition, since our problem is variational and it requires second-order equations.

First of all, let us recall that points in the same fiber of FL2 only differ one from each other

in their components ̇ i. Indeed, if y 0 is a point in M 2 with local coordinates (q0 A , 0

i , p̂ 0 A,0) then

we have

FL21

y 0 q 0 A , 0

i ,g0 AB

p̂ 0 B0i 0iB , ̇

i / ̇ iRP 2 .

This fact implies that, if

D0iq̇ A

q AB i q , , q̇

iC A q , , q̇

q̇ AD0

i

̇ i

is an arbitrary solution of Eq. 17, then it is projectable by FL onto a vector field ̄ tangent to M 2defined by

̄ y FL*

D0i x , xFL1 y ,

since B i and C A do not depend on ̇ i.

Moreover, since ¯ is such that ( i¯ 1dh1) M 2, we deduce




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i¯ 2dh2 ,

and ¯ is the Hamiltonian vector field associated to h 2 , i.e., ̄ h2. For each y M 2 , with local

coordinates (q A, i, p̂ A,0) we have

̄ y FL*

D0i x

g AB p̂ Bi iB q A

y

B i x i

y

g AD q B q̇ D i iA

q B

y

g BC p̂ C i iC B

i iA x C Bg AB x p̂ A y

g AB p̂ Bi iB q A

y

B i x i

y

L

q A p̂ A

y

,

where x is an arbitrary point in FL1( y).

Now, we define the mapping s: M 2→ P2 by putting

s y s q A, i, p̂ A,0 q A, i,g AB p̂ B

i iB ,B i x , y M 2 , xFL

1 y ,

where i D i j jA g AB p̂ B . It is not difficult to see that s is well defined and that it does not depend

on the choice of the local coordinates on M 2 . In fact, one can define s by taking the value of D0i

at x and then project the result by the canonical projection from TP onto P see Refs. 31 and 33.

Moreover, we have that s ( y )FL21( y), for each y M 2 so s is a differentiable section of FL2 .

Then, S s( M 2)P 2 is a submanifold of P 2 , and hence of TP as well. Observe that on this

submanifold, D satisfies the SODE condition: indeed, we have

S D S B i̇ i ̇ i

S

0.

However, in general, one cannot ensure that D is tangent to S .

This problem is solved by transporting the vector field ̄ from M 2 to S by using the global

diffeomorphism s: M 2→S , that is, we define

̃ s*

̄ .

Therefore, ̃ will verify the SODE condition because of the form of s and, in addition, the

equation

i˜ LdE L S .

Next, we will obtain a local expression for ̃ . Let x be a point in S ; since s is injective, there

is a unique point y M 2 such that s( y) x. Then,

̃ x s* y

̄ y .

As we know from the above discussion, q̇ x Ag AB ( p̂ B

i iB ) y and ̇ xiB x

i , so that we have




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˜ xq̇ x A

q A

x

̇ xi i

x

q̇ x Aq̇ x D g CD g BC q A

x

q̇ x A x

i g x BC iC

q A

x

̇ ig BC iC x

g x BA g EA

q D q̇ E i

iA

q D xq̇ x D ̇ i iA x C Dg AD x

q̇ B x q̇ x A B i

q A

x

̇ x j B i

j

x

L

q A B i p̂ A

x

̇ i

x

.

This expression can be simplified as follows:

˜ xq̇ x A

q A

x

̇ xi i

x

C x B

q̇ B

x

q̇ x A B i q A

x

̇ x j B i

j

x

p̂ ˙ A B i p̂ A

x

̇ i

x

q̇ x

A

q A x

˙

x

i

i x

C x B

q̇ B x

B ˙

x

i

̇ i ,taking into account that

q̇ Ag CD q̇ D g BC

q A q̇ Dq̇ E g BA

g EA

q D q̇ Dq̇ E

q D g BC gCE 0.

Remark VI.1: We have obtained a vector field ¯ on M 2 , and a vector field ˜ on S , both vector

fields solving the dynamics of the singular Lagrangian L. It should be noticed that, since the

equations of motion for L are the same as the equations of motion for the vakonomic problem, we

have obtained a sort of reduction of the latter problem. Indeed, the integral curves of ¯ or

equivalently, of ˜ ) give the vakonomic dynamics. But M 2 or, if we want, S has dimension 2 n

and we have started with a state system TP with dimension 2n2m .

Recall that we have proved ¯ h2. In addition, the vector field ˜ on S is also a Hamiltonian

vector field. In fact, ˜ is the Hamiltonian vector field corresponding to the restriction of E L and

with respect to the restriction of L to S . Both Hamiltonian vector fields are related by the

symplectomorphism s.

VII. CLASSIFICATION OF THE CONSTRAINTS ACCORDING TO DIRAC

The application of the Dirac–Bergmann–Gotay–Nester algorithm has produced the following

constraints:

i the primary constraints, p̂ j

0, 1 jm ,ii and the secondary constraints, j0, 1 jm ,

which together define the final constraint submanifold M 2 .

In according with Dirac’s terminology,29 the constraints can be classified into first class and

second class constraints. Let us recall that a constraint is said to be first class if its brackets with

all the other constraints vanish; otherwise, it is said to be second class.

Here the bracket is the canonical one provided by the canonical symplectic form P on T *P ,

f ̄ , ḡ f ̄

q A ḡ

p̂ A

f ̄

i ḡ

p̂ i

f ̄

p̂ i

ḡ

i

f ̄

p̂ A

ḡ

q A ,




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for all pair of functions f ̄ , ḡ :T *P→R.

We construct the matrix C (C ), with C , , where 1 2m and p̂ for1 m and m if m1 2m . Then we have

C p̂ i , p̂ j p̂ i , j

i , p̂ j i , j 0 Di j

D i j N i j ,with

N i j i , j p̂ C g AB jA iD g

CD

q B iA

jD gCD

q B g AB k iA D k j q B jA

D ki

q B .

A straightforward computation shows that the matrix C is invertible with inverse

C 1 C D1 ND1 D1

D1 0 .

Therefore, all the constraints are second class.Thus, the Dirac bracket is

f ̄ , ḡ D f ̄ , ḡ f ̄ , C , ḡ ,

for all pair of functions f ̄ and ḡ on T *P .

An important observation is the following. Since the constraints become Casimir functions

with respect to the Dirac bracket, then it can be restricted to M 2 . Indeed, for all pairs of functions

f ,gC ( M 2) the bracket f ̄ , ḡ D M 2 does not depend on the choice of the extensions f ̄ , ḡ to T *P .

Consequently, we will denote f ,g* f ̄ , ḡ D M 2.

As Dirac proved, the bracket , D provides the evolution of any observable, that is,

f ̄

˙ f ̄ , h̄ D ,

for some convenient extension h̄ of the projected Hamiltonian h 1C ( M 1). In particular,

f ,h 2* gives the evolution of f : M 2→R.As we have noticed in Sec. V, ( M 2 , 2) is a symplectic submanifold of T *P . Let us denote

by , M 2 the Poisson bracket induced by 2 . We are interested in knowing which is the relation

between both brackets, ,* and , M 2. This is solved in the following.

Proposition VII.1: The bracket ,* coincides with , M 2, that is, we have that

f ,g* f ,g M 2,

for all f ,gC ( M 2).

Proof: As ( M 2 , 2) is a symplectic submanifold of T *P , we have the following decomposi-tion:

T M 2 T *P TM 2T M 2

,

with associated projectors

P :T M 2 T *P →T M 2 ,

Q:T M 2 T *P →T M 2

.




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It is proved in Ref. 34 that our Dirac bracket is precisely

f ̄ , ḡ D P P X f ¯ ,P X ḡ

for f ¯

,g¯

C

(T *P). Let us denote by Y f the Hamiltonian vector field on M 2 associated with afunction f : M 2→R with respect to 2 . A careful computation shows that j 2*

Y f P ( X f ¯ ), where f ̄

is an extension to T *P of f C ( M 2). Consequently, we have

f ,g* P P X f ¯ ,P X ḡ P j 2*Y f , j 2*

Y g 2 Y f , Y g f ,g M 2.

If we denote by : M ¯ Rm→ M ¯ the canonical projection, we can define a Poisson bracket along

˜ 1 as follows:

f ,gvak f ˜ , g ˜ *,

which is a function defined on M 2 . Therefore, we have a bracket

,vak : C M ¯ C

M ¯ —— → C M 2

f ,g —— → f ,gvak ,

which is in fact a bracket along ˜ . This bracket ,vak enjoys similar properties to those of ordinary Poisson brackets.

Definition VII.2: The bracket ,vak on M ¯ along ˜ will be called the vakonomic bracket .

The vakonomic bracket produces a function on M 2 from two functions defined on M ¯ , since we

need to specify the corresponding Lagrange multipliers i in the equations by means of the

above-mentioned diffeomorphism between M 2 and M ¯ Rm.

A careful computation shows that, in local coordinates, the expression for the vakonomic

bracket is

f ,gvak f ˜ , g ˜ * f ˜

q A g ˜

p̂ A

f ˜

p̂ A

g ˜

q A

f ̄

i D ik N jl D

l j ḡ

j , 18

where f ̄ , ḡ C (T *P) are arbitrary extensions of f ˜ and g ˜ , respectively.

Moreover, if ̄ is the ‘‘reduced’’ vakonomic vector field on M 2 , then, for any f : M ¯ → R, we

have

f , H M ¯ vak f ˜ , H M ¯ ˜ *̄ f ˜ f ˙ ,

where H :T *Q→R is the Hamiltonian defined by E L , that is, H F L E L .

Remark VII.3: It should be noticed that M 2 has a vector bundle structure over M ¯ with rank m.

Indeed, it is a vector subbundle of pr 1 :T *P T *QR2m→T *Q , that is,




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In this way, a vakonomic motion ( q(t ),(t )) in M 2 can be viewed as a motion in the total space

of that vector bundle, with base components q ( t ) in M ¯ and fiber components (t ) in Rm. Roughly

speaking, the Lagrange multipliers can be considered as a sort of internal variables in addition to

position variables.

Example VII.4: The vertical rolling disk . Let us consider the following problem for a disk of radius R and unit mass m1 which rolls on a horizontal plane.

The configuration space for this system can be identified with QR2S 1S 1. B y ( x, y )

R2 we denote the coordinates of the point of contact of the disk with the plane and ( , )

S 1S 1 give, respectively, the angle between the disk and the x axis, and the angle of rotation

between a fixed diameter in the disk and the y axis.

Given q 0 ,q 1Q , i.e., initial and final position variables, we want to find the trajectories of

the disk connecting such points that minimize the energy expenditure. Of course, we want the disk

to roll without slipping. This situation can be seen as an optimal control problem. 36 A problem of

optimal control is described by the following data: a configuration space B giving the states

variables of the system, a fiber bundle : N → B whose fibers describe the control variables, a

vector field Y : N →TB along the projection , and a ‘‘Lagrangian’’ function L: N →R. Now the

solutions of the optimal control problem will be those paths : I → N such that has fixed end

points, which extremize the action

L t dt

and satisfy the differential equation

d

dt Y ,

which rules the evolution of the state variables.

It is easy to show that this is indeed a vakonomic problem on the manifold N . The constraint

submanifold M TN , given by the above-mentioned differential equation is

M vnTN / *vnY n .

In the problem under consideration, we identify BQ , N TQ , and :TQ→Q as the natural

projection Q . The Lagrangian L:TQ→R is given by

L12 ẋ

2 ẏ 2 I 1 ˙

2 I 1 ˙

2 ,

with I 1 , I 2 the moments of inertia notice that the potential energy is not included since it is

constant. The vector field along Q is

Y : TQ —— → TQ

x , y , , ,d 1 ,d 2 ,d 3 ,d 4 → x, y , , , R cos d 4 , R sin d 4 ,d 3 ,d 4.

Notice that Y is simply a tensor 1, 1 on the manifold Q.

In fact, in this framework, we are considering the velocities as the ‘‘control’’ variables.

Solving this optimal control problem is precisely the same as considering the vakonomic problem

associated with the vertical rolling disk for the extended Lagrangian L:T (QR2)→R,

L L .

where

ẋ sin ẏ cos ,




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ẋ cos ẏ sin R ˙

are the constraint functions determining M . Note that we have chosen a linear combination of the

usual constraints

¯ ẋ R ˙ cos ,

¯ ẏ R ˙ sin .

In Sec. VIII, we will discuss how this change of constraints affects the final result. In addition,

as is stated in Refs. 8 and 24, the vakonomic solutions for this problem are also solutions of the

nonholonomic problem if the initial conditions for the Lagrange multipliers are properly chosen.

We have that

Ldx∧dẋsin d x∧d cos d x∧d cos sin dx∧d d y∧dẏcos d y∧d

sin d y∧d sin cos dy∧d I 1 d ∧d ˙ I 2d ∧d ˙ R d ∧d ,

is the Poincaré-Cartan two-form in local coordinates.The final constraint submanifold is

P 2 x , y , , , , , ẋ , ẏ , ˙ , ˙ , ̇, ˙ T QR2 / 0, 0.

Let be a general solution of equation i LdE L and tangent to P 2 . In local coordinates, we

have

ẋ

x ẏ

y ˙

˙

B

B

C x

ẋC y

ẏC

˙C

˙D

̇D

˙.

The coefficients satisfy the following equations:

C xB sin B cos ˙ cos sin ,

C yB cos B sin ˙ sin cos ,

C R

I 1 ˙ ,

C R

I 2B ,

and the tangency conditions

C x sin C y cos R ˙ ˙0,

C x cos C y sin RC 0.

Therefore, we get

1 0

0 1 R2

I 2 B B ˙ R ˙ ˙ ,

which leads to




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B R ˙ ˙ ˙ ,

B a ˙ ,

where a (1( R 2 / I 2))

1. In turn, the expressions for the other coefficients of become

C x 1a ˙ cos R ˙ ˙ sin ,

C y 1a ˙ sin R ˙ ˙ cos ,

C R

I 1 ˙ ,

C Ra

I 2 ˙ .

Continuing with the described process, we have that the submanifold S is given by

S x , y , , ,, , ẋ, ẏ , ˙ , ˙ , ̇ , ˙ T QR2 / 0, 0,B ̇ ,B ˙ ,

and ˜ is

˜ ẋ

x ẏ

y ˙

˙

̇

˙

C x

ẋC y

ẏC

˙C

˙D

̇D

˙,

with

D ˙ 2

R 2

I 1 ˙ 2

R

I 1 ˙ ,

D aR ˙ ˙2a ˙ 2

aR

I 1 2 ˙ .

Observe that the equations for the Lagrange multipliers

̇ ˙ R ˙ ˙ ,

˙ a ˙ ,

can be integrated to give

A sin B cos ,

A cos B sin R ˙ ,

where A and B are constants which depend on the initial conditions 0, 0. This allows us to

project ˜ ( A, B) to a vector field X ( A, B) on M giving different vakonomic solutions for each choice

of A, B. In particular

X 0,0 ẋ

x ẏ

y ˙

˙

R ˙ ˙ sin

ẋ R ˙ ˙ cos

ẏ




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is just the nonholonomic vector field, L, M , corresponding to the vertical rolling disk see the

discussion at the end of Sec. IV.

Now, the Legendre transformation FL:T (QR2)→T *(QR2) is given by

FL x , y , , ,, , ẋ, ẏ , ˙

, ˙ , ˙

, ˙

x , y , , , , , p̂ x , p̂ y , p̂ , p̂ , p̂ , p̂ ,

where

p̂ x ẋ sin cos ,

p̂ y ẏ cos sin ,

p̂ I 1 ˙ ,

p̂ I 2 ˙ R ,

p̂ 0,

p̂ 0.

So the presymplectic system ( M 1 , 1 ,h1) becomes

M 1FL T QR2 R10,

1dx∧d p̂ xdy∧d p̂ yd ∧d p̂ d ∧ p̂ ,

h 11

2 p̂ x sin cos 2 p̂ y cos sin 2 1 I 1 p̂ 2

1

I 2 p̂ R

2 .Applying Gotay–Nester’s algorithm we get the secondary constraints

p̂ y cos p̂ x sin ,

a1 p̂ y sin p̂ x cos

R

I 2 p̂ ,

through which we obtain the symplectic Hamiltonian system ( M 2 , 2 ,h2)

M 2FL P 2 R8,

2dx∧d p̂ xdy∧d p̂ yd ∧d p̂ d ∧d p̂ ,

h 21

2 1a cos2 p̂ x

2 1a sin2 p̂ y

2

1

I 1 p̂

2

a

I 2 p̂

2

1a sin 2 p̂ x p̂ y2 Ra

I 2cos p̂ x p̂ 2

Ra

I 2sin p̂ y p̂ .

As we have said, the natural bracket associated with the two-form 2 allows us to construct the

vakonomic bracket. This is, for any f ,g: M ¯ → R we have

f ,gvak f ˜ , g ˜ M 2,

where ˜ : M 2→ M ¯ is




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˜ z x, y , , , 1a cos2 p̂ x 1a sin cos p̂ y Ra I 2 cos p̂ ,

1a sin cos p̂ x 1a sin2 p̂ y

Ra

I 2sin p̂ , p̂ ,a R cos p̂ x R sin p̂ y p̂

.

If H M ¯ is the restriction of H to M ¯ , since H M ¯ ˜ h2 we have

f , H M ¯ vak f ˜ , h 2*

f ˜

x 1a cos2 p̂ x 1a sin cos p̂ y 2 Ra I 2 cos p̂

f ˜

y 1a sin2 p̂ y 1a sin cos p̂ x 2 Ra I 2 sin p̂

f ˜

p̂

I 1

f ˜

a

I 2 p̂

Ra

I 2cos p̂ x

Ra

I 2sin p̂ y

1

2

f ˜

p̂ 1a sin2 p̂ x2 1a sin 2 p̂ y2 1a 2 cos 2 p̂ x p̂ y

2 Ra

I 2sin p̂ x p̂

2 Ra

I 2cos p̂ y p̂ .

VIII. CONSISTENCY OF THE LOCAL CONSTRUCTION

In the previous sections we have assumed that the constraint functions i were globally

defined on the whole of TQ. Under this assumption, we have defined the extended Lagrangian L

on TP and, by means of the constraint algorithm, we have obtained an equivalent description of

vakonomic dynamics in terms of the vector fields ˜ and ¯ , on S and M 2 , respectively. An

alternative description was provided by the bracket ,vak .In this section, we will discuss the validity of the above results when a change of constraints

or a change of local coordinates is performed. We accomplish the two tasks at the same time.

Suppose that V and V ¯ are two coordinate neighborhoods in the configuration manifold Q such that

V V ¯ , and denote by (q A) and (q̄ A) the corresponding coordinate functions. Let

i : TV →R, i iA q̇ A,

¯ j : TV ¯ →R, ¯

j ¯ j Bq B,

be two sets of constraints defining M TV and M TV ¯ , as in Sec. II. Notice that both sets of

constraints are obtained by taking two local basis i and ¯ i of the codistribution M ° on V and

V ¯

, respectively.Then, for each one, we have the extended Lagrangians

L:T V Rm→R, L L i iA q̇ A,

L̄ :T V ¯ Rm→R, L̄ L i ¯ i Aq A ,

and we can apply the constraint algorithm. In this way, we obtain the constraint submanifolds P 2

and P̄ 2 ,

P 2 TV M T RmP 1T V R

m,




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P̄ 2 TV ¯ M T Rm P̄ 1T V ¯ R

m.

Assume now that

i iA q dq

A

, ¯

i ¯

i A q¯

dq¯

A

.

Then, there exist differentiable functions

i j :V V ¯ →R2m,

̄ jk :V V ¯ →R2m,

which give the matrices of the change of basis at each point in V V ¯ ,

i j j ¯ i , ̄ j

k ¯ k j , i

j̄ jk i

k .

Consequently, we have

i j jA ¯ i B

q̄ B

q A ,

̄ jk ¯ k A jB

q B

q̄ A .

As a first result we deduce that

¯ i i j j .

Therefore, P 2 P̄ 2 can be glued to form a new submanifold of P 1 P̄ 1 , which is in turn a

submanifold of T (QRm).

Remark VIII.1: In spite of this, there is no way to extend L or L¯

to the whole of P1P¯

1 , sowe will have to consider the process for each neighborhood.

Next, define the transformation

̄ : P 1 P̄ 1→ P 1 P̄ 1

q A, i, q̇ A,̇ i q̄ A,̄ i j i, q̄ ˙ A,̄ i

j̇ i,

which permits us to relate the extended Lagrangians as

L̄ P1P¯

1̄ L̄ i

j ī j Li iLP1P

¯ 1.

This implies that on P 1P¯

1 we have

S * ̄ *d L̄ S * d ̄ *L̄ S * d L,

and therefore the Poincaré–Cartan two-forms verify

L̄ * L¯ ,

on P 1 P̄ 1 . Since the energy associated with both extensions is the same, E L , we deduce that if

D is a solution on P 2 for the constrained system defined by L, then ̄ *( D) is a solution for the

constrained system defined by L̄ . In other words, if D satifies the equation




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iD LdE L P2,

then we will have

i ¯

*D L¯

dE L¯

P¯

2.

In terms of their integral curves, we have that an integral curve of a fixed vector field D0i of the

family of solutions D is transformed by ̄ into an integral curve of ¯

D¯ 0 j on P2 P̄ 2 , where

D̄ 0 j̄ ̄ i

jD0iq̇ Ȧ i( ̄ i

j / q A).

Indeed, if

t A t , i t , ˜ A t , ˜ i t

is an integral curve of D0i on P 2 P̄ 2 , then

¯ t

A t

q̄ B

q A ,

i t ̄ i

j t , ˜ A t

q̄ B

q A , ˜

i t ̄ i

j t

,

will be an integral curve of ¯ D¯ 0 j on P 2 P̄ 2 . It is very important to observe that, although

different, the projections of (t ) and ¯ (t ) to M coincide.

Remark VIII.2: If S respectively, S ¯ ) denotes as above the submanifold of P 2 respectively,

P̄ 2) where a SODE solution ˜ respectively, ˜

¯ ) exists, then

̄ *

˜ ˜ ¯ ̄ 19

holds on points in S S ¯ , that is, ˜ and ˜ ¯

are ̄ related on the overlapping. This can be seen as

follows. Recall that ˜ D0 jD with D0

i D

0 j (B i). Since B i does not depend on ̇ i, we have

that D0 j (B i) D j(B i) for all D j D and we can compute D0i choosing any member of the

family D . The same is true for the family ¯

D¯ . Then, taking D j and ¯

D¯ k such that ̄ * D j

¯ D¯ k ̄ , we can check that

D̄ 0i¯ D¯ k B ¯

i¯ D¯ k ̇iD j ̄ j

i ̇ j ̄ ji D0

jq̇ Ȧ j

̄ ji

q A ,

or, in other words, Eq. 19 holds.

Remark VIII.3: Given a ‘‘vakonomic motion,’’ c̃ (t )(q A(t )) , there are different curves in

P 2 P̄ 2 that project to ( c̃ (t ), c̃ ˙ (t )) M . Indeed, if we take ( q0 A , q̇ 0

A) M TV TV ¯ and ( 0i ,̇ 0

i )

as initial conditions for the Lagrange multipliers, we can consider the integral curve of ˜ starting

from (q0 A , 0i , q̇ 0 A , ̇0i ). Now, the curve ¯

¯ will be an integral curve of

˜ ¯

starting from(q 0

A ,̄ i j(q0

A) 0i , q̇ 0

A ,̄ i j(q 0

A) ̇ 0i ). Both curves project to the same solution of the vakonomic equa-

tions of motion. Therefore, in order to determine an unique curve on M T Rm whose projection

is ( c̃ (t ), c̃ ˙ (t )), we are forced to specify not only the initial conditions for the Lagrange multipliers,

but also the set of constraint functions such that (q 0 A ,0

i , q̇0 A ,̇ 0

i )P 2 .

We have seen what happens in the Lagrangian formalism when changing constraint functions.

Next, we accomplish the same task in the Hamiltonian context. As a consequence, we will give

later a relation of the above-mentioned integral curves with the solutions of vakonomic equations

of motion. By the Legendre transformations FL and FL̄ associated to L and L̄ , respectively, we

obtain the presymplectic systems ( M 1 , 1 ,h1) and ( M ¯ 1 , ¯ 1 , h̄ 1), where




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M 1FL P 1 , 1 j 1* , h 1FL E L , h 112 g

AB p̂ Ai iA p̂ B

j jB U ,

M ¯ 1FL̄ P̄ 1, ¯ 1 ̄ 1* , h̄ 1FL̄ E L¯ , h̄ 1

12 ḡ

AB p̄ ˆ Ai ¯ i A p̄ ˆ B

j ¯ jB U ,

with the obvious notations.Notice that M 1 M ¯ 1 can be provided of a differentiable structure such that it is a submanifold

of T *(V V ¯ )Rm. We also have that the restriction of the standard symplectic form of T *(Q

Rm) to M 1 M ¯ 1 is the natural extension of the two-forms 1 , ¯ 1 . However, there is no canoni-

cal extension to M 1 M ¯ 1 of the projected Hamiltonians h1 and h̄ 1 .

Define the transformations

̄ : M 1 M ¯ 1→ M 1 M ¯ 1

q A, i, p̂ A,0 q̄ A,̄ i j i, p̂ B q B

q̄ A,0 ,

such that the following diagram is commutative:

P 1 P̄ 1→FL

M 1 M ¯ 1

̄ ↓ ↓ ̄ 20

P 1 P̄ 1→FL̄

M 1 M ¯ 1

We have

̄ * ¯ 1

1

, h̄ 1̄ h

1.

Applying the algorithm to both presymplectic systems, we obtain the secondary constraint sub-

manifolds

M 2 y M 1 / i y 0, i iA g AB p̂ B

j jB ,

M ¯ 2 y M ¯ 1 / ¯ j y 0, ¯ j ¯ jA ḡ AB p̄ ˆ B

k ¯ k B.

Observe that

i y h 1 i y

h̄ 1̄ i

y

̄ ik y h̄ 1

k

¯ y

̄ ik ¯ k ̄ y ,

that is,

ji i ¯ j̄ .

As a consequence, the set M 2 M ¯ 2 does not define in general a submanifold of

M 1 M ¯ 1T *(( V V ¯ )Rm). However, we have a nice relation between both submanifolds, in-

deed,

̄ M 2 M ¯ 1 M ¯ 2 M 1 .




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It is important to observe that M 2 M ¯ 1 is an open submanifold of M 2 . Therefore, on restricting

the symplectic form 2 to M 2 M ¯ 1 , we do not lose its symplectic character.

Remark VIII.4: A careful computation shows that ̄ M 2 M ¯

2is just the identity. Consequently,

we have, for example, that

h̄ 2 M 2 M ¯

2 h 2 M 2 M

¯ 2.

In addition, using 20 and the relations:

FL D¯

M 2, FL̄ ¯ D¯

¯ M ¯ 2

, ̄ *

D¯

D¯ ̄ ,

we deduce that the vector fields ¯ M 2 and ¯

M ¯ 2fulfill along M 2 M ¯ 1

̄ *

¯ M 2¯

M ¯ 2̄ . 21

We see that the integral curves of M 2 and ¯

M ¯ 2 on M 2 M

¯ 2 are, in principle, different. However,

one can easily check that their projections onto M ¯ by

˜ : M 2→ M ¯ , q A, i, p̂ A,0 q

A, p̂ Ai iA ,

¯ ˜ : M ¯ 2→ M ¯ , q̄ A, i, p̄ ˆ A,0 q̄

A, p̄ ˆ Ai ¯ i A,

coincide, since

¯ ˜ ̄ M 2 M ¯

1 ˜ M 2 M

¯ 1.

We will now investigate the relation between the corresponding Dirac brackets, and more inter-

esting, about the induced brackets on the final constraint submanifolds M 2 and M ¯

2 ,

, D M 2 , , C , M 2,

, D M ¯ 2 , , ¯ C ¯ ¯ , M ¯ 2.

Recall that ̄ *( ¯ 1 ) M 1 M ¯

1( 1) M 1 M

¯ 1. This fact implies that ̄ *( ¯ 2 ) M ¯ 2 M 1( 2) M 2 M

¯ 1.

Consequently, we have for each pair of functions, f ,g: M ¯ 2→R that

f ,g*¯ ̄ M 2 M

¯ 1 f ̃ , g̃ *k , 22

where k : M 2 M ¯

1

M 2 is the canonical inclusion and f ˜

,g˜

: M 2→R

are extensions to M 2 of ̄ M 2 M

¯ 1 f M ¯ 2 M 1

,̄ M 2 M ¯

1g M ¯ 2 M 1

, respectively.

As a consequence, when defining the vakonomic brackets for functions f, g on M ¯ we have the

following two possibilities:

f ,gvak f ˜ , g ˜ M 2,

f ,gvak f ¯ ˜ , g ¯ ˜ M ¯ 2.

However, the relation ¯ ˜ ̄ M 2 M ¯

1 ˜ M 2 M

¯ 1

and 22 imply that




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f ,gvak k f ˜ , g ˜ M 2k f ¯ ˜ , g ¯ ˜ M ¯ 2̄

M 2 M

¯ 1 f ,gvak ̄ M 2 M

¯ 1,

which is coherent with the above-mentioned formula ̄ *

¯ M 2¯

M ¯ 2̄ .

Remark VIII.5: Therefore, although different, both brackets give the same valid information

about the evolution of a dynamical variable along ‘‘vakonomic curves’’ on M ¯ . In fact, given a

‘‘vakonomic’’ curve on M ¯ , c̄ (t )(q A( t ), p A(t ) ) , we take ( t )(q A(t ), i(t ), p̂ A(t ),0) on

M 2 M ¯ 1 and ̄ ( t )( q̄ A(t ), ̄ i

j i(t ), p̄ ˆ A(t ),0) on M ¯ 2 M 1 projecting onto it. Then, the evolu-

tion of f onto this curve on M ¯ will be

d

dt f q A t , p A t

d

dt f ¯ ˜ q̄ A t ,̄ i

j i t , p̄ ˆ A t ,0d

dt f ˜ q A t , i t , p̂ A t ,0,

that is,

f ˙̄ c̄ ¯

M ¯ 2 f ¯ ˜ ¯

¯ M 2

f ˜ f ˙ c̄ ,

or, equivalently,

f ˙̄ f , H M ¯ vak ̄ f , H M ¯ vak f ˙ .

Example VIII.6: The vakonomic particle. We consider the case of a particle of unit mass

moving through the space QR3 subjected to the global nonholonomic constraint ż y ẋ . In

order to illustrate the precedent discussion, we will take, instead of , the following constraints:

:TU →R, x, y , z, ẋ, ẏ , ż x ż y ẋ ,

:TV →R, x , y , z, ẋ, ẏ , ż z ż y ẋ ,

where

U x , y , z R3 / x0,

V x , y , z R3 / z0.

Here, the Lagrangian L is the kinetic energy L12 ( ẋ

2 ẏ 2 ż2), so the extended Lagrangians

are

L : T U R→R, L 12 ẋ

2 ẏ 2 ż 2 x ż xyẋ ,

L : T V R→R, L 12 ẋ

2 ẏ 2 ż2 zż zy ẋ .

Since ( x / z) in TU TV , the transformation ̄ is given by

̄ : T U V R → T U V R

x , y , z,, ẋ, ẏ , ż ,̇ x, y , z , x z

, ẋ, ẏ , ż , x

z ̇ .

The two-forms of Poincaré–Cartan are, respectively,

L dx∧dẋ xy dx∧d x d x∧dydy∧d ẏdz∧dż dz∧dx x dz∧d ,

L dx∧dẋ z y dx∧d z dx∧d y y dx∧dzdy∧dẏdz∧dż z d z∧d .




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Let , be the vector fields on P2 , P 2

satisfying

i L

dE L , i L dE L .

Then, the coefficients must fulfill the following equations:

C x ẏ yB

x ż, C x ẏ yB

z y ż,

C y xẋ , C y

zẋ ,

C x ẋB

x, C z y ẋB

z .

The tangency conditions ( )0, ( )0 are reduced to

C z ẏ ẋ yC x

0, C z

ẏ ẋ y C x

0.

It is easy to see now that in each case we obtain

B

ẋ

x

ẏ ẋ xy

x 1 y 2 ,

B

ż

z

ẏ ẋ zy

z 1 y 2 ,

so that we have

C x x ẏ

y ẏ ẋ xy

1 y 2 , C x

y ż

y ẏ ẋ z y

1 y 2 ,

C y

xẋ , C y

zẋ ,

C z

ẏ ẋ xy

1 y 2 , C z

ẏ ẋ zy

1 y 2 .

Consequently, we have determined the families D and D

. If we denote by S ,S the

submanifolds of P 2 , P 2

, respectively,

S yT U R3 / ̇ ẋ x

ẏ ẋ xy

x 1 y 2 ,

S

yT V R3 / ̇

ż

z

ẏ ẋ z y

z 1

y

2

,

we have proved that there is a vector field ˜ respectively, ˜ ) of D

respectively, D ) satis-

fying the SODE condition and tangent to S respectively, S ). These vector fields are determined

by

D ̇ ẋ

x

y ẏ

1 y 2 ẋ

x

2

ẏ1 y 2

2 y ẋ ẏ2

x 1 y 2 2

ẏ 2 y 21

1 y 22C x

ẏ x 1 y 2

xC y ẋ x 1 y 2

y

1 y 2 ,




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D ̇ ż

z

y ẏ

1 y 2 ż

z

2

ẏ y 1 y 2

2 y ẏ2 ẋ zy

1 y 22

ẏ 2

1 y2

2

z

C z C x

ẏ

z 1 y2

C y

ẋ

z 1 y2

y

1 y2

.

A straightforward but tedious computation shows that

̄ *

˜ ˜ ̄ .

We pass now to the Hamiltonian description of the problem. The Legendre transformations

are

FL : T U R —— → T * U R

x, y , z , , ẋ, ẏ , ż ,̇ x , y , z, , ẋ xy , ẏ , ż x ,0,

FL : T V R —— → T * V R

x , y , z, , ẋ, ẏ , ż, ̇ x, y , z , , ẋ z y , ẏ , ż z,0.

Therefore, we have that

M 1 FL T U R x0, p̂ 0R

7 / x0,

M 1 FL T V R z0, p̂ 0R

7 / z0,

with Poincaré–Cartan two-forms and Hamiltonian functions given by

dx∧d p̂ xd y∧d p̂ ydz∧d p̂ z ,

h 1

12 p̂ x x y

2 p̂ y

2 p̂ z x

2,

dx∧d p̂ xdy∧d p̂ ydz∧d p̂ z ,

h 1

12 p̂ x zy

2 p̂ y

2 p̂ z z

2 .

It is inmediate to see that h1 ̄ h 1

. The corresponding secondary constraints are

h1

x p̂ x x y y p̂ z x ,

h1

z p̂ x z y y p̂ z z ,

and, in fact, we verify that ( z / x) ̄ . The final constraint submanifolds in the Hamiltonian

side are

M 2 w M 1 / p̂ z y p̂ x x 1 y 2 R6 / x0,

M 2 w M 1 / p̂ z y p̂ x z 1 y 2 R6 / z0,

with two-forms and Hamiltonians




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dx∧d p̂ xd y∧d p̂ ydz∧d p̂ z ,

dx∧d p̂ xdy∧d p̂ ydz∧d p̂ z ,

h2 h 2 12 p̂ x

y p̂ z

2

1 y 2 p̂ y

2

.Note that and are not the same two-form, because they are defined on different manifolds,

that is, M 2 and M 2

, respectively.

To define the vakonomic brackets, we have

˜ : M 2

—— → M ¯

x, y , z , p̂ x , p̂ y , p̂ z x , y , z , p̂ x y p̂ z y p̂ x1 y 2 , p̂ y , p̂ z p̂ z y p̂ x

1 y 2 ,

˜ : M 2

—— → M ¯

x , y , z, p̂ x , p̂ y , p̂ z x, y , z , p̂ x y p̂ z y p̂ x

1 y 2 , p̂ y , p̂ z

p̂ z y p̂ x

1 y 2 .Given f ,g: M ¯ →R, we have on M 2

M 1

that

f ,gvak f ˜ ,g ˜ M

2 f ˜ , g ˜ M

2 ̄ f ,gvak

̄ .

ACKNOWLEDGMENTS

This work has been partially supported through Grant No. DGICYT Spain PB97-1257. S.

M. and J. C. wish to thank Spanish Ministerio de Educació n y Cultura for FPI and FPU grants,

respectively. The authors wish to thank D. Martı́n de Diego for helpful comments and suggestions.

We acknowledge the referee for his useful remarks.

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