a historical review on nonholomic mechanics.pdf

8/14/2019 A historical review on nonholomic mechanics.pdf

1/34

RACSAM (2012) 106:191224DOI 10.1007/s13398-011-0046-2

SURVEY

A historical review on nonholomic mechanics

Manuel de Len

Received: 4 May 2011 / Accepted: 17 June 2011 / Published online: 16 July 2011 Springer-Verlag 2011

Abstract The aim of this paper is to present a short historical/scientific review onnonholonomic mechanics, with special emphasis on the latest developments. Indeed, theuse of differential geometric tools has permitted in the last 25 years a fast and unsuspectedadvance in the theory, particularly in a better understanding of symmetries and reduction,HamiltonJacobi theory and integrability characterizations, and the construction of suitablegeometric integrators. The last part of the paper is devoted to discuss the latest results in

HamiltonJacobi theory for nonholonomic dynamics using our own approach.

Keywords Nonholonomic mechanics Symmetries and reduction HamiltonJacobi theory Geometric integrators

Mathematics Subject Classification (2000) 37J60 70F25 70H20 70H33

1 Introduction

Nonholonomic mechanics, that is, mechanical systems governed by a lagrangian functionsubject to kinematic constraints, is an old topic in mechanics that has attracted a lot of atten-tion due to its applications to engineering sciences[17,22] (and also to microswimming inBiology [76]).

Nonholonomic mechanics has experimented a dramatic change in the last 25 years, duemainly to the introduction in its study of geometric techniques. Indeed, it was in the beginningof the eighties when nonholonomic mechanics was fully incorporated into the realm of theso-called Geometric Mechanics.

The first objective of this paper is to give a brief historical overview on nonholonomic

mechanics, based in our personal experiences in the approach to the subject. So we start

M. de Len (B)Instituto de Ciencias Matemticas, Consejo Superior de Investigaciones Cientficas,c/ Nicols Cabrera, 13-15, Madrid 28049, Spaine-mail: [email protected]


2/34

192 M. de Len

reviewing some historical developments due to Hertz, Poincar, and the Russian schoolof mechanics, continuing with the seminal work by Koiller [75] (see also Vershik andFaddeev [140], and Weber[93,118,148]) in the beginning of the 1980s which introducesthe modern language of differential geometry in the description of nonholonomic mechani-

cal systems.The content of this paper does not include new results but we reviewed some of the lastachievements; the term of historical review should be understood as an indication of ourpurpose to include some historical references (indeed, linked to the own experience of theauthor), but the paper is not a history of the matter. Therefore we discuss some relevant topicsin nonholonomic mechanics, like symmetries and reduction, the development of a convenientHamiltonJacobi theory and new geometric integrators.

At the end, wediscover that these results requirea more sophisticated tool, the so-called Liealgebroids (or even a more general notion, say skewsymmetric algebroid, where the Jacobiidentity is not required). Indeed, as it was remarked by Alan Weinstein [149], mechanics

should be developed in this new geometric setting. Let us recall that Lie algebroids are theinfinitesimal objects associated to Lie groupoids (in the same sense that tangent bundles areassociated to manifolds or a Lie algebra is associated to a Lie group); the use of Lie groupoidsand algebroids is allowing new insights in the theory.

The paper is structured as follows. In Sect. 2 we recall the main notions in Lagrangian andHamiltonian mechanics as well as derive the nonholonomic equations of motion. Section 3is devoted to give a brief historical review. In Sect. 4we consider examples of nonholonomicsystems as well as nonholonomic toys; we also discuss recent results in different topics: non-holonomic field theories, piecewise nonholonomic systems or higher order nonholonomic

constraints. Section5 we introduce the new geometric setting to deal with nonholonomicmechanics based in Lie groupoids and Lie algebroids. Finally, in Sect.6we shall give a firstapplication of this new setting to study HamiltonJacobi theory of nonholonomic mechanicalsystems.

2 Lagrangian and hamiltonian mechanics. Nonholonomic mechanics

2.1 Lagrangian mechanics

A mechanical system is governed by a Lagrangian function L which is just the differencebetween the kinetic energy Tand the potential energy Vof the system. Usually, there aresome relations among the coordinates, the so-called holonomic constraints, which are usedto introduce generalized coordinates (qA)in the configuration manifold Q, where Arunsfrom 1 ton =dim Q (nis the number of degrees of freedom).

Therefore, the Lagrangian function is L = L(qA, qA), where (qA)are the generalizedvelocities. The lagrangian is L = T V, where T(qA, qA) = 12 gA B (q)q

AqB , g being a

Riemannian metric on Q, andV = V(qA)a function on Q.The Hamiltons principle produces the EulerLagrange equations

d

dt

LqA

L

qA =0, 1 An . (1)

For simple mechanical systems, Eq. (1) are just the translation of the well-known Newtonsecond law.

A geometric version of Eq. (1)can be obtained as follows (see [92]).


3/34

A historical review on nonholomic mechanics 193

The lagrangian is a function L : T Q R, where here the tangent bundle T Qrepresents the space of velocities. Consider the (1,1)-tensor field Sand the Liouville vectorfielddefined locally on the tangent bundle T QofQ:

S=

qA dqA, = qA

qA .

We construct the Poincar-Cartan 1 and 2-forms

L = S(d L), L = dL ,

whereS denotes the adjoint operator ofS.The energy is given by

EL =(L) L ,

so that we recover the classical expressions

L =d qA dpA, EL = q

A pAL ,

where pA = L

qAdenotes the generalized momenta.

We say that Lis regular if the 2-formL is symplectic, which in coordinates turns to beequivalent to the regularity of the Hessian matrix ofLwith respect to the velocities

WA B = 2L

qAqB In this case, the equation

iXL =d EL (2)

has a unique solution, X =L , called the EulerLagrange vector field; L is a second orderdifferential equation (SODE) that means that its integral curves are tangent lifts of their pro-

jections on Q(these projections are called the solutions ofL ). The solutions ofL are justthe solutions of Eq. (1).

If L : T T Q TT Q is the musical isomorphism, L (v) = iv L , then we haveL (L )= d EL .

The Legendre transformation F L : T Q T Q is a fibred mapping (that is,Q F L = Q , where Q : T Q Q and Q : T Q Q denote the canonicalprojections of the tangent and cotangent bundle ofQ, respectively) defined by

F L(qA, qA)= (qA,pA).

A direct computation shows that Lis regular if and only ifF Lis a local diffeomorphism.In what follows, we will assume that F L is in fact a global diffeomorphism (in other

words, Lis hyperregular) which is the case when Lis a lagrangian of mechanical type, sayL = T V, where

Tis the kinetic energy defined by a Riemannian metric g on Q, V : Q Ris a potential energy.

Indeed, in this case we have WA B = gA B , wheregA B = g( qA

, qB

).


4/34

194 M. de Len

2.2 Hamiltonian mechanics

The hamiltonian counterpart is developed on the cotangent bundle T Q of Q. Denote byQ =d q

A d pAthe canonical symplectic form, where (qA,pA)are the canonical coordi-

nates onT

Q. The Hamiltonian energy is just H = ELF L1

and the Hamiltonian vectorfield is the solution of the symplectic equation

iXHQ =d H.

The integral curves(qA(t),pA(t))ofXHsatisfies the Hamilton equations

dqA

dt=

H

pA,

d pA

dt=

H

qA (3)

If f, gare two functions defined on the phase space we define the Poisson bracket

{f, g} = Q (Xf ,Xg)= fqA

gpA

fpA

gqA

The Poisson bracket has the following properties:

bilinear skewsymmetric Leibniz rule

{f, gh} = {f, g}h+ g{f, h}

satisfies the Jacobi identity

{f, {g, h} } + {g, {h, f} } + {h, {f, g}} =0

Since F LQ = L , we deduce that L and XHare F L-related, and consequently F Ltransforms the EulerLagrange equations(1) into the Hamilton equations (3).

An important fact is that the Lie bracket of vector fields onQ is related with the canonicalsymplectic structure on T Q as follows. Given a vector field Xon Q, that is, a section ofT Q Q, we define a linear function X : T Q Rusing the natural pairing betweenvectors and covectors; then we have

[X, Y] = {X, Y}

Notice also that the bracket on T Qis equivalent to have a tensor field of type (2,0) onT Qgiven by:

Q (d f, dg)= {f, g}

Qis an example of Poisson tensor or Poisson structure.

2.3 Nonholonomic systems

A nonholonomic mechanical system consists of

1. a lagrangian functionL = L(qA, qA),2. subject to nonholonomic constraints i (qA, qA)= 0.

These are kinematic constraints (constraints involving the velocities).If i (qA, qA) = iA(q)q

A (respectively, i (qA, qA) = iA(q)qA + bi (q)) is linear

(respectively, affine) in the velocities the constraints are called linear (respectively, affine).Otherwise, they are called nonlinear.


5/34


Invoking the D Alembert principle for linear and affine constraints (or the Chetaevprinciple, for nonlinear constraints) we derive the nonholonomic equations of motion

d

dt L

qA

L

qA

= ii

qA

, 1 A n

i (qA, qA)= 0 (4)wherei =i (qA, qA)are Lagrange multipliers to be determined.

2.4 A paradigmatic nonholonomic system: the rolling disk

To illustrate the general theory, let us consider a disk rolling without sliding on a horizontalplane, which can be considered as a paradigmatic example of a nonholonomic mechanicalsystem.

Let(x,y)be the coordinates of the point of contact with the floor, the angle measuredfrom a chosen point of the rim to the point of contact (rotation angle), is the angle betweenthe tangent to the disk at the point of contact and the xaxis (heading angle), and is theangle of inclination of the disk.

The configuration manifold is then Q = R2 S1 S1 S1.The lagrangian is given by L = T Vwhere

T = 1

2m( x2 + y2 + R22 +R22 sin2 ) m R(cos ( xsin ycos )

+sin ( xcos + ysin )) +1

2

I1(22 cos2 ) +

1

2

I2(+ sin )2

and

V =mg Rcos

Here,m is the mass of the disk,Ris the radius, and I1and I2are the principal momenta ofinertia. The rolling without sliding condition means that the following constraints have to befullfilled along the motion

1 = x (Rcos ) =0, 2 = y (Rsin ) =0.

The important fact is that all the configurations are available, but not all the velocities.

3 Some historical remarks

3.1 Nonholonomic mechanics before Hertz

Nonholonomic systems are, roughly speaking, mechanical systems with constraints on theirvelocities that are not derivable from position constraints.

Nonholonomic systems arise, for instance, in mechanical systems that have rolling contact(for example, the rolling of wheels without slipping) or certain kinds of sliding contact (suchas the sliding of skates). They are a remarkable generalization of classical Lagrangian andHamiltonian systems in which one allows position constraints only.

The oldest publication that addresses the dynamics of a rolling rigid body is Euler in1734 [47], in which small oscillations of a rigid body moving without slipping on a horizon-tal plane were studied.


6/34

196 M. de Len

Later, the dynamics of a rigid body rolling on a surface was studied in Routh (1860) [119],Slesser (1861) [128] Vierkandt (1892) [145], and Walker (1896) [147].

Historically, certain errors of the well-known mathematicians Neumann and Lindelof [95]were caused by an incorrect application of the Lagrange equations in the presence of non-

integrable constraints in the description of the problem of a body rolling without sliding onthe horizontal plane.

The Lindelof mistake was detected by Chaplygin (see [33], vol. 1, pages 5175), and

attracted the attention of many scholars of that time, like Appell, Bobylev, Chaplygin,Cenov, Hamel, Hertz, Maggi, Voronec, Zukovskii, It should be noticed that early work byFerrers, Korteweg, Neumann was ignored.

According to[19] the derivation of the equations of motion of a nonholonomic systemin the form of the EulerLagrange equations corrected by some additional terms to takeinto account the constraints (but without Lagrange multipliers), was outlined by Ferrers in1872 [51]. The formal derivation of this form of equations was performed by Voronetz in1901 [146]. In the case in which some of the configuration variables are cyclic, such equations(now called Chaplygin equations) were obtained by Chaplygin in 1895 (see [32,33]).


7/34


In any case, the general understanding of inapplicability of Lagrange equations andvariational principles to the nonholonomic mechanics is due to Hertz, who expressed itin his fundamental work[62] that deals mostly with his conception of hidden cyclic param-eters (coordinates, masses), as opposed to the conventional notion of interaction as a result

of force application. Furthermore, Hertz coined the term nonholonomic system in 1894.

Hertz wanted to construct the Foundations of Mechanics disposing entirely with the notion

of force, replacing it by equivalent velocity constraints. His basic principle, which yields pre-cisely the LagrangedAlembert equations, states that the geometric curvature of the path isalways a minimum, subjected to the constraints.

One of the first discoveries by Hertz was that the usual integral variational principlessuch as the principle of least action or Hamiltons principle do not hold for non-holonomicsystems. However, Otto Holder pointed out that if the variations in the variational principlesare chosen in the right way, the principles remained valid. Instead of assuming, as Hertz haddone, that the varied motion should satisfy the constraints, Holder assumed that the variationssatisfy the constraints. If the system is non-holonomic the varied motion will not satisfy theconstraints, Holders variational principle is not about an ordinary variational problem, but

it gives the correct trajectories.It is interesting to remark the reaction by Poincar to Hertz developments. In his review of

Hertzs book, Poincar says [117] (we borrow the next paragraphs from Borisov and Mamaev[21]):

Hertz terms a system holonomic when the following holds: if the systems constraints

do not allow a direct transition from one position to another infinitely close position,

then they either do not allow indirect transitions between these positions. Only rigid

constraints exist in such systems. It is evident that our sphere is not a holonomic system.

So, it sometimes happens that the principle of least action cannot be applied to non-holonomic systems. Indeed, one can proceed from position A to position B taking the

path that we have just discussed, or, undoubtedly, one of many other paths.

Among these, there is, evidently, one path corresponding to the least action.

Hence, it should have been possible for the sphere to follow this path from A to B. But

this is not so: whatever the initial conditions of motion may be, the sphere will never

pass from A to B. In fact, if the sphere does pass from position A to position A?, it


8/34

198 M. de Len

does not always follow the path that corresponds to the least action. The principle of

least action holds no more.

Hertz says,In this case, a sphere obeying this principle would seem to be a living

creature, which deliberately pursues a certain goal, while a sphere following the laws

of Nature would look as an inanimate monotonously rolling mass But such con-

straints do not exist in Nature. So-called rol ling without sliding is actually rolling with

slight sliding. This phenomenon belongs to the class of irreversible phenomena, such

as friction; these phenomena are still poorly investigated, and we have not yet learned

to apply to them the true principles of Mechanics.Our reply is,Rolling without sliding does not contradict either the law of energy con-

servation, or any other law of physics known to us. This phenomenon can be realized

in the observable world within the accuracy that would allow its application to con-

struction of the most accurate integration machines (planimeters, harmonic analyzers,

etc.). We have no right to exclude it from consideration as impossible. As for our prob-

lems, they still remain regardless of whether such rolling is realized exactly or only

approximately. To accept the principle, it is necessary to require that its application to a

problem with almost exact source data would yield the results, as close to the exactness

as the source data were. Besides, other (rigid) constraints can also be realized in Nature

only approximately. But nobody is going to exclude them from consideration

3.2 A milestone: Neimark and Fuffaevs book

An important milestone in the development of nonholonomic mechanics has been the bookby Neimark and Fufaev [112].

Let us recall some facts about this book. The idea of the book started in 1949/1950, aroundthe A.A. Andronovs seminar, and the work started in 1951 by Neimark. In the next year thework was discontinuous because the interest of the authors in the subject decreased. Muchlater, the authors were again attracted by the study of nonholonomic dynamics and the bookwas finally written. The book is plenty of ideas, examples and applications.

Another important step was the paper by Vershik and Faddeev [140]. Indeed,

This paper containsin our opinionthe introduction of Modern Differential Geometryin the study of nonholonomic systems.

It contains many ideas and techniques later rediscovered by other authors.


9/34


The reduction procedure for nonholonomic systems with symmetries has been discussedby Jair Koiller[75], where the author revises the work by Chaplygin and extends his resultsto the case of non Abelian groups. Indeed, Koiller considers mechanical systems whoseconfiguration space is the total space of a principal bundle, the constraints given by the

horizontal space of a principal connection.Indeed, symmetries are very relevant in mechanics, and, in particuular in nonholonomicsystems, as the following quotation by Fomenko [53] shows:

One of the interesting ocurrences of symmetry in mechanics is the rolling of a solid

body without slipping along a two dimensional surface (possibly of a complex pro-

file). The results of this process are studied by the mechanics of nonholonomic sys-

tems Recently, deep and interesting connections of this subject with Lie groups were

discovered

Another important step (inspired in Koillers paper), again in the realm of the reductionof nonholonomic systems with symmetry, is the paper by Bloch et al.[18]. A relevant issuehere is that the existence of a symmetry does not imply necessarily a conserved quantity.Therefore we have to discuss the compatibility of the constraints with the symmetries, thatis, how the symmetries are placed with respect to the constraints. In the quoted paper, theauthors classify the constraints and introduce a nonholonomic momentum mapping.

Let us describe some alternative approaches:

A symplectic setting (also inspired in Koiller) is due to Bates, Snyaticki and collabora-tors [13] (see also[11,12,40]). The idea is to construct a symplectic subbundle on where

the constraints vanish and then the symplectic mechanism produces the dynamics. Cantrijn, Sarlet and Saunders [124,125] have described the dynamics in terms of jetbundles (see also[57,85,108]).

Marle[98,99] has developed a hamiltonian point of view (see also Dazord [42]). Ehlers, Koiller, Montgomery and Rios [46] were inspired in the original treatment by

Cartan. An approach using implicit differential equations can be found in Ibort et al. [67] (see

also Grcia and Martn [60]).

Our own approach was also inspired in Koiller, Bloch et al, and particularly in an almost

ignored paper by Cariena and Raada [27]. This approach has permitted: a better understanding of the introduction of the nonholonomic bracket [24,26,68] a method of reduction [25] which integrates those previously developed by Bloch et al.

[18] and Bates et al. [13] the study of existence of invariant measures [23].

4 Geometric nonholonomic mechanics

4.1 Nonholonomic examples and devices

Some examples and devices have played an important role in the development of nonholo-nomic mechanics. We shall a list of some of them:

Classical examples. Any class of rolling body. A large variety of examples can be foundin Neimark and Fuffaev book [112]. It should be noticed that, related with these prob-


10/34

200 M. de Len

lems, there is a relevant research connected with differential and algebraic geometry:Agrachev[2], Fatima Leites and co-workers [63], Jurdjevic[71,72],

Another important source of examples, coming from the study of rigid bodies, is the studyof nonholonomic systems on Lie groups: Veselov and Veselova [143,144], Fedorov and

co-workers[48,49], In recent times, the so-callednonholonomic toyshave deserved a lot of attention fromthe scholars: the wowblestone, ratleback or celtic stone (see the papers by Bondi [20]and Tokieda and Moffatt [109]; the skateboard (see Kuleshov [82], Kremnev andKuleshov [81]); the snakeboard (see Ferraro, Kobilarov, de Len, Martn de Diego andMarrero [86,74]).


11/34


4.2 Nonhonomic and vakonomic dynamics

One of the more interesting historical events in this issue was the paper of Korteweg [ 80].Up to that point (and even persisting until recently) there was some confusion in theliterature between nonholonomic mechanical systems and variational nonholonomic systems(also called vakonomic systems). The latter are appropriate for optimal control problems.

One of the purposes of Kortewegs paper was to straighten out this confusion, and in doingso, he pointed out a number of errors in papers up to that point.

The difference between nonholonomic and vakonomic dynamics relies in the differentprinciple applied in both cases:

nonholonomic dynamics is derived using the d Alembert principle. vakonomic dynamics is obtained using a variational principle looking for extremals curves

among those satisfying the constraints.

Given two pointsx,y Qwe define the manifold of twice piecewise differentiable curveswhich connectxand yas

C2(x,y)= {c: [0, 1] Q | c is C2, c(0)= x, c(1)= y}

We know that

TcC2(x,y)= {X : [0, 1] T Q | Xis C1,X(t) Tc(t) Q,X(0)= 0,X(1)= 0}

Assume that the lagrangianL : T Q R is subjected to constraints given by a subman-ifold MofT Q. Locally, Mis defined by the equations

i (qA, qA)= 0

DefineC

2(x,y)= {c C2(x,y) |c(t) M for all t [0, 1]}

Given a curve c C2(x,y)we define a vector subspace

Vc = {X TcC2(x,y) | S(di )(X)= 0, i,

for all vector fields X T Qalongc which projects ontoX}


12/34

202 M. de Len

Therefore, ifX= XA qA

, we deduce that X Vcif and only if

XA i

qA =0

Define the functional

J : C2(x,y ) R

c

10

L(c(t)) dt

We have

dJ(c)(X)=

1

0 L

qA

d

dt L

qA XA dt

for all X TcC2(x,y ), c C2(x,y).Therefore we have three different options:

Unconstrained systems. A curve c is a motion of the lagrangian system defined by L ifand only ifc a critical point ofJ, that is,

dJ(c)(X)= 0

for allc C2(x,y); this is equivalent to satisfy the following equations of motion

L

qA

d

dt

L

qA

= 0

X Tc C2(x,y).

Nonholonomic mechanicsA curve c C2(x,y)is a motion if and only if

dJ(c)(X)= 0

for all X Vc; this is equivalent to the equations of motion

L

qA d

dt LqA= i iqAplus the constraints.

Remark The derivation of the equations of motion for nonlinear constraints and even theexistence of such constraints have been deserved much attention from the researchers. Werefer here to[14,41,43,67,121] for detailed discussions, including the famous Atwoodmachine (see also [78,116]).

Vakonomic mechanicsA curve c C2(x,y)is a motion if and only if

dJ(c)(X)= 0

for all c Tc C2(x,y ), or in other words, cis a critical point ofJrestricted to C2(x,y ).

This is equivalent to satisfy the following equations

L

qA

d

dt

L

qA

= i

d

dt

i

qA

i

qA

+

di

dt

i

qA


13/34


We can also prove that cis a motion if and only if the curve (c(t), (t))is a motion ofthe unconstrained lagrangian

L= L i i .

The terminology vakonomic (mechanics of variational axiomatic kind) was coined byKozlov ([4]).Vakonomic dynamics will not be discussed here in detail, but we refer to [4,810,59,5456,106,107] for more details. For the relation to Sub-Riemannian geometry we referto [110].

Comparison between both approaches have been discussed in[36,59,94], where our ownapproach based in a convenient setting for both dynamics using an extension of the Skinnerand Rusk formalism (see [126,127]). The method developed is sofisticated but it gives auniversal and computational way to compare both dynamics.

4.3 A geometrical setting

If we realize that the bundle of 1-formsS((T M)0)is locally generated by the local 1-forms{S(di )}, we can rewrite Eq.(4)as follows

iXL d EL S((T M)0)

X T M.

(5)

We assume theadmissibility condition:

dim(T M)0

=dimS

((T M)0

)

which is equivalent to say that the matrixi

qA

has maximal rankm .(For linear constraints the above conditions means that the set of 1-forms {i =

iA(q)dqA} is linearly independent and, indeed, a local cobasis of the distribution M).

We also assume thecompatibility condition:

F T M= {0}

where Fis the distribution on T Q(along M) such that

F0 = S((T M)0)

and F denotes theL -complement ofF.Notice that F = Zi whereL (Zi )= S(di ), thereforeL (F)= F0.Consider a possible solution of the equation

iXL d EL =i S(di );

thenX =L+ iZi . If we impose the condition to the dynamics be tangent to the constraintsubmanifold we obtain

0= X( j )= L (j ) + iZ

i ( j ) (6)

DenoteCi j = Zi ( j ). Notice that if the matrix(Ci j )is regular (the compatibility condi-tion), then we can compute the Lagrange multipliers solving the linear equation(6) at each


14/34

204 M. de Len

point ofM. In this case we can obtain the nonholonomic dynamics Xnh which is the uniquesolution of Eq.(5).

A simple calculation gives

Ci j = i

qAWA B

j

qB

where(WA B )is the inverse matrix of(WA B ), and shows that if(WA B )is definite (positiveor negative) then(Ci j )is inversible.

As a consequence, if the lagrangian function L is of mechanical type then the nonholo-nomic system is always admissible and compatible.

Remark We can consider more general nonholonomic systems without these regularity con-ditions, even with singular lagrangians (see [89]).

Assume that the nonholonomic system is compatible and admissible, then we have a directsum decomposition

Tx(T Q )= TxMF

x

for allx M. In terms of vector bundles we have a Whitney sum decomposition

T T Q |M = T M F

with two complementary projections P : T T Q |M = T Mand Q: T T Q |M = F such thatXnh = P(L ).

Remark To be more precise, the resultXnh = P(L ) holds if the constraint are homogenous,that is, is tangent to the constraint submanifold, |M T M. This is the case for linearconstraints.

Assuming the regularity of the Lagrangian, we have that the Lagrangian andHamiltonian formulations are locally equivalent. If we suppose, in addition, that the Lagrang-ian Lis hyperregular, then the Legendre transformation

F L : T Q T Q, (qA, qA)(qA,pA =L/qA)

is a global diffeomorphism. The constraint functions onT

Qbecomei

=i

F L1

, i.e.

i (qA,pA)= i

qA,

H

pA

,

where the Hamiltonian H : T Q R is defined by H = EL F L1. Since locally

F L1(qA,pA)= (qA,

H

pA), then

H = pAqA L (qA, qA),

where qA is expressed in terms ofq A and pAusing F L1.The equations of motion for the nonholonomic system on T Q can now be written as

follows

qA = H

pA

pA = H

qA i

i

pBHB A

(7)


15/34


together with the constraint equations

i (q,p)= 0

where HA B are the components of the inverse of the matrix

(HA B )= (2H/pApB )

It should be noticed thati

pBHB A

(q,p)=

i

qA F L1

(q, p).

The symplectic 2-formL is related, via the Legendre map, with the canonical symplec-tic formQ on T Q. Let Mdenote the image of the constraint submanifold Munder the

Legendre transformation, and let Fbe the distribution onT

Qalong M, whose annihilatoris given by

F0 = F L(S((T M)0)).

Observe that F0 is locally generated by the m independent 1-forms

i = i

pAHA B dq

B , 1 i m .

The nonholonomic Hamilton equations for the nonholonomic system can be then rewrittenin intrinsic form as

(iXQ d H)| M F0

X| M T M

(8)

The compatibility condition is now written as F T M = {0}, where denotes thesymplectic complement with respect to Q . Equivalently, the matrix

(Ci j )=

i

pA

HA B j

pB (9)

is regular. On the Lagrangian side, the compatibility condition is locally written as

det(Ci j )= det

i

qAWA B

j

qB

=0 , (10)

whereWA B are the entries of the Hessian matrix

2L

qAqB

1A,Bn

.

The compatibility condition is not too restrictive, since it is trivially verified by the usualsystems of mechanical type (lagrangian = kinetic energypotential energy), where the H

A Brepresent the components of a positive definite Riemannian metric. The compatibility con-dition guarantees the existence of a unique solution of the constrained equations of motion(8)which, henceforth, will be denoted by Xnh on the Hamiltonian side and Xnh on theLagrangian side. Moreover, ifXH is the Hamiltonian vector field ofH(iXHQ =d H) then

i = Ci jXH( j ). (11)


16/34

206 M. de Len

4.4 Reduction

The question of reducing ordinary or partial differential equations which are invariant underthe action of a Lie group has attracted considerable attention in recent years. To reduce means

to obtain equations with fewer coordinates or, when possible, to obtain a globally defineddifferential operator on a quotient manifold.Let us consider a nonholonomic lagrangian system with symmetry, that is, we have a

regular lagrangian function L : T Q R, a constraint submanifold MofT Q and a Liegroup G acting on Q such that L and Mare G-invariant with respect to the lifted actiononT Q.

For any element g we denote byQand T Q the induced vector fields on Qand T Q,respectively. Notice that T Q is tangent to Msince theG -invariance.

We also assume that the different actions ofG are free and proper, so that we have well-defined quotient manifolds:

Q Q = Q/G

T Q T Q = T Q /G

M M = M/G

The energy EL also projects to a function EL on T Q .In the sequel we denote by Vthe subbundle ofT T Q whose fibers are the tangent spaces

to theG -orbits:

Vx =Tx(Gx)

for allx M.Notice that

Vx TxM

due to the G -invariance ofM.In the case of unconstrained systems that admits a Lie group of symmetries, Noethers

theorem states that the invariance of the lagrangian implies a momentum conservation law.Indeed, let

J : T Q g

be the momentum mapping defined by

J(x), = L (x), T Q

The function Jis defined by

J(x)= J(x),

LetL be the solution of the equation

iL L =d EL

Therefore, we have

L (J)= d J(L )= (iT Q L )(L )= (iL L )(T Q )= T Q (EL )

In consequence, the invariance ofL implies the invariance ofELso that Jis a conservedquantity.


17/34


But in the case of nonholonomic mechanics, we will have

X(J)= (iXL )(M)= M(EL ) + (M)

where Fo.

Therefore, theG -invariance only implies

X(J)= (M)

IfM VF, then Jis a conserved quantity. Suchis called ahorizontal symmetry.In general, the situation is more involved, and we distinguish three cases:

1. The pure kinematic case

VxFx = {0}

for allx M(no horizontal symmetries at all) andTxM= Vx+ (Fx TxM)

for allx M.2. The case of horizontal symmetries:

VxFx = Vx

for allx M, that is, Vx Fx, for all x M.3. The general case:

{0} Vx Fx Vx

The pure kinematic case is really important, since it contains as a particular case theCaplygin systems:

1. Q Q/Gis a principal bundle with structure group G ;2. L :T Q R is a G -invariant lagrangian;3. the constraints are given by the horizontal subspaces of a connection onQ Q/G.

IfT Q = H Vthen we can prove that the nonholonomic mechanical system can be

reduced to an unconstrained system on Q/G with a reduced lagrangian L

on T(Q/G)subject to an external force of gyroscopic type.

4.5 Geometric integration for nonholonomic mechanical systems

The development of the so-called geometric integrators has been a milestone in the historyof numerical approaches to mechanics (we refer to [61,102,103,123] for more thoroughexpositions; see also [101,105]).

Let us say that given a hamiltonian system on a symplectic manifold (P, ) andhamiltonian energyH, an algorithm (say, a collection of mapsFh : P P, ha parameter)

is said to be:

a symplectic integratorif each Fh : P P is a symplectic map (that is, it leaves invariant).

anenergy integratorifHFh = H. amomentum integratorif J Fh = Jwhere J : P g is the momentum map for

the action of a Lie group G on ( P, ).


18/34

208 M. de Len

An algorithm having one or more of these properties is called a mechanical integrator.It is also a well-known fact that one algorithm preserving at the same time the symplecticstructure, the energy and the momentum, gives the exact solution.

TheseintegratorsarecloselyrelatedtothosebasedontheMoserandVeselovdiscretization

technique [111,141,142].The extension of these geometric integrators to the case of nonholonomic systems is farto be reached but it is an active field of research.

We want to describe in a brief way two recent approaches (later we will discuss a moresophisticated approach based on Lie groupoids).

Corts and Martnez approach [35,39]:The starting point to construct integrators for mechanical systems with non-holonomicconstraints is to develop a discrete version of the LagrangedAlembert principle (DLA).Thus we can construct numerical integrators that capture the essential qualitative fea-tures of this kind of systems. Indeed. these non-holonomic integrators derived from theDLA principle preserve the symplectic structure along trajectories. Moreover, for non-holonomic systems with symmetry, non-holonomic integrators possess a discrete versionof the non-holonomic momentum equation. For horizontal symmetries (see the abovesection), the associated momenta are preserved exactly by the discrete flow. For the casewhen no non-holonomic momentum map exists, the continuous flow is reduced to anunconstrained systems with a non-conservative force.The authors also prove that their approach works for the discrete reduced space, so thatone has a generalized variational integrator in the sense of Kane et al. [73]. As one could

expect, when the non-conservative forces vanish, the proposed integrator is indeed avariational integrator.The developed nonholonomic integrator is tested against a RungeKutta method.

de Len et al. approach[90,91]:In this approach, the nonholonomic integrators are based on the method of generatingfunctions (recall that generating functions form the basis for many symplectic integra-tion procedures). The modification to the holonomic scheme includes the effects in thecanonical transformation of the constraint forces. This idea is based on understanding theeffects of the constraint forces on the rate of change of the symplectic form. The numericalscheme seems to work quite well in the examples given in the paper, even better than inCorts and Martnez integrator.

4.6 Piecewise nonholonomic systems

As we have previously discussed, rolling a disk or a ball on a plane are standard examplesof nonholonomic mechanical systems, and these problems are well understood; even, someprogress have been made for any smooth deformation of the surfaces. But for non-smoothlydeformed surfaces, much less is known.

Think for instance on a polyhedral approximationc of a smooth surfaces; it is clear that

nonholonomy is conserved. But current definitions of nonholonomic constraints (like thosethat we have discussed in the previous sections) are referred to systems described by ordinarydifferential equations, and are thus inapplicable to a plohedral system. In these systems westudy the set of positions and orientations that a polyhedral part can reach by rolling on aplane through sequences of adjacent faces. Such more general systems with nonholonomicfeatures may be used to represent some very general classes of systems and devices of greatpractical relevance.


19/34


An interesting paper is[120] by Andy Ruina. We find interesting to reproduce here thecontent of the Summary of the paper:

There the author considers mechanical systems with intermittent contact that are smoothand holonomic except at the instants of transition. Overall such systems can be nonholo-

nomic in that the accessible configuration space can have larger dimension than the instanta-neous motions allowed by the constraints. The known examples of such mechanical systemsare also dissipative. By virtue of their nonholonomy and dissipation such systems are notHamiltonian. Thus there is no reason to expect them to adhere to the Hamiltonian prop-erty that exponential stability of steady motions is impossible. Since nonholonomy andenergy dissipation are simultaneously present in these systems, it is usually not clear whethertheir sometimes observed exponential stability should be attributed solely to dissipation, tononholonomy, or to both. However, it is shown here on the basis of one simple example, thatthe observed exponential stability of such systems can follow solely from the nonholonomicnature of intermittent contact and not from dissipation. In particular, it is shown that a dis-

crete sister model of the Chaplygin sleigh, a rigid body on the plane constrained by oneskate, inherits the stability eigenvalues of the smooth system even as the dissipation tendsto zero. Thus it seems that discrete nonholonomy can contribute to exponential stability ofmechanical systems.

(see also[15,34,97]).In our opinion, research in this issue should be fostered.

4.7 Non classical nonholonomic constraints

A field of great importance which has not been included in the general theory of nonholo-nomic systems are the various systems in which the rolling constraints are not classical.

For instance, the kinematics constraints that arise during the rolling of an elastic pneu-matic tire or a railroad wheel are very different from the classical idealization of the rollingof a rigid body without sliding. Different and inspiring models can be found in the book byNeimarl and Fuffaev[112].

This kind of systems have inspired a suitable geometrical setting, developed in a series ofpapers: Cendra, de Len, Ibort and Martn de Diego [31], or the so-calledgeneralized non-holonomic systemsmainly studied by the Argentinian school (Sergio Grillo, Jorge Solomn,Hernn Cendra and Paula Balseiro [6,7,29,30]).

4.8 Nonholonomic field theory

A natural extension of nonholonomic mechanical systems is that to classical field theories.A classical field theory is governed by a Lagrangian function depending on the space-

time coordinates (x), the fields (yA), and its derivatives with respect to the space-timecoordinates, that is, L = L(x,yA,zA ).

In several papers we have considered classical field theories subjected to nonholonomicconstraints, that is, our system is given by:

A lagrangian

L = L(x,yA,zA )

subject to constraints of the form

i (x,yA,zA )= 0


20/34

210 M. de Len

A first result (Marsden et al. [104]), where the authors have considered incompressiblefluids. But in this case, the constraints are not truly nonholonomic (by integration on thespace coordinates, the constraints disappear).

In [16,139] we have studied this kind of systems, obtaining the nonholonomic field

equations using geometric tools. The weakness of the theory is the lack of examples,which should be a priority in the future research (some recent work has been developedby Vankerschaver[135138]).

4.9 Unilateral constraints

A interesting case is when the mechanical system is subjected to time-dependent two-sidednonholonomic as well as one-sided (impulsive) constraints. One simple example to illustratethis kind of systems is a sphere rolling without sliding on a horizontal plane and hitting arough wall.

A regular Lagrangian system is considered, with configuration space Q. The system issubjected to nonholonomic constraints which are modelled by a submanifold with boundaryof the tangent bundle T Q , where the boundary induces a one-sided nonholonomic con-straint. In addition, at points of the boundary, some impulsive reaction forces are assumed toact. Along the constraint submanifold, a co-distribution is defined, called the Chetaev bun-dle. This bundle, which in general will not be of constant rank, represents the reaction forcesinduced by the constraints and the impulsive forces. It is shown that when two additionalconditions (an admissibility and a compatibility condition) are satisfied, the system admits awell defined dynamics. The latter is recovered from the description of the constrained systemin terms of a submanifold of a fibred product bundle over TQ. From the equations definingthis submanifold one can then derive the equations of motion as well as the expressions forthe instantaneous jumps in the momenta due to the impulsive reaction forces. In the case ofpermanent constraints, suitable projectors are constructed which, among others, allow oneto evaluate these jumps in the momenta. This, in particular, leads to a geometric formulationof Carnots theorem, which establishes a relationship between the momenta before and afterthe action of the impulsive forces.

For more details we refer to [38,6466,83] (see also[122,150]).

5 A new setting for nonholonomic mechanics

5.1 Weinstein s program

In 1992 Alan Weinstein [149] proposed an ambitious program to develop mechanics on amore general setting, Lie algebroids. Let us recall the notion of a Lie algebroid:

Definition 1 A Lie algebroid structure in the vector bundle D : D Q is a R-linearbracketBD : (D ) (D ) (D )and a morphism of vector bundlesD : D T Q ,the anchor map, such that

1. BD is skew-symmetric,

BD (, )= BD ( , ) , for, (D );

2. BD satisfies the Jacobi identity

BD (BD (1, 2), 3) +BD (BD (2, 3), 1) + BD (BD (3, 1), 2)= 0;


21/34


3. If we denoteD : (D ) X(Q)the morphism ofC(Q)-modules induced by theanchor map, then

BD (, f )= f BD (, ) + D ()(f) ,

for, (D)and f C(Q).

The reasons behind this project were that Lie algebroids are natural extensions of tangentbundles and Lie algebras, which are the natural arena to develop mechanics. Indeed, for atangent bundle the Lie algebroid structure is given by the usual Lie bracket of vector fieldswith the identity as the anchor, and for a Lie algebra, the bracket is just the own of the algebra,being the anchor the zero mapping (the Lie algebra is considered as a trivial vector bundleover its zero vector). In addition, ifL : T Q Ris a lagrangian function and G is a Liegroup of symmetries, thenT Q/G Qis a Lie algebroid (the so-called Atiyah algebroid)and L reduces to a lagrangian l : T Q /G R, which is a true lagrangian in this new

universal language.In the rest of this paper we will develop a more general setting using the so-called skew-

symmetric algebroids (see[5,86], where the condition of integrability is not necessary. Thisnew setting allows to obtain unsuspected results for a HamiltonJacobi theory in nonholo-nomic systems.

5.2 Nonholonomic integrators and Lie groupoids

Another interesting research line is the construction of numerical algorithms that producesufficiently accurate, affordable, and robust numerical approximations of the nonholonomic

dynamics. Indeed, the construction of geometric integrators for nonholonomic dynamics isvery recent as an open problem [96]:

The problem for the more general class of non-holonomic constraints is still open, asis the question of the correct analogue of symplectic integration for non-holonomicallyconstrained Lagrangian systems

The main guiding idea for geometrically simulate nonholonomic systems comes fromHlders variational principle which is not a standard variational principle [4], but admits anadequate discretization. This is, roughly speaking, the procedure introduced by Corts and

Martnez [39] and followed by other authors[48,52,96] extending, moreover, the results tononholonomic systems defined on Lie groups (see also [91] for a different approach usinggenerating functions). From the geometric perspective it is possible to see all these situationsas particular cases of nonholonomic systems on Lie groupoids [70,100]. This idea follows theprogram proposed by Weinstein [149] for the study of discrete mechanics on Lie groupoids.

RememberthataLie groupoidover a differentiable manifoldQis a differentiable manifoldGtogether with the following structural maps:

two submersions, thesource map : G Qand target map : G Q. The mapsand define the set ofcomposable pairs

G2 = {(g, h) G G | (g)= (h)};

amultiplication map m : G2 G, (g, h)gh . an identity section : Q G of and , such that for all g G,((g))g = g =

g((g));

an inversion map i : G G, g g1, such that for all g G, gg1 = ((g)),g1g=((g)).


22/34

212 M. de Len

From this structure maps it is possible to define the left-translation by g Gand theright-translation by gas the diffeomorphisms

lg :1((g)) 1((g)); h l g (h)= gh,

rg :1((g))1((g)); h rg(h)= h g.

Take the vector bundle : AG Q, whose fiber at a point x Q is AxG = V(x) =K er(T(x)). Indeed AG is equipped with an structure of Lie algebroid. For any sectionX Sec (AG )it is possible to construct the corresponding left-invariant (resp., right-invari-

ant) vector field on G , that will be denoted byX (resp.,

X) (see[70] for details).

A generalized discrete nonholonomic (or constrained) Lagrangian system on G is deter-mined by:

aregular discrete Lagrangian L d :G R, a constraint distribution, Dc, which is a vector subbundle of the bundle AG Q of

admissible directions. We will denote by Dc : Dc Qthe vector bundle projection andbyiDc : Dc AG the canonical inclusion.

adiscrete constraint embedded submanifoldMcofG , such that dimMc =d i mDc.

In[70] it is shown, applying a discrete version of Hlder principle, the discrete nonholo-nomic equationsare given as the solutions of the following difference equation onG :

Xa (gk)(Ld)

Xa (gk+1)(Ld)= 0 (12)

where(gk, gk+1) G2 (Mc Mc)(with (g) = (h) = x) and where{Xa }is a localbasis of SecDc on an open subsetUofQsuch thatxU.

Equation(12)admits different applications and interpretations depending on the differ-ent groupoid we consider. In the case, G = Q Q we obtain the same equations thanin reference [39]. When G is a Lie group, we obtain from Equation (12) the so-calleddiscrete EulerPoincarSuslov equations (see [48]). Also we may work with Atiyah bundlesG =( Q Q )/Gobtaining discretization of nonholonomic reduced systems, including dis-crete Chaplygin systems, etc.

5.3 Generalized hamiltonian systems

The use of Lie algebroids or even more general concepts leads us to develop a new and very

general setting for hamiltonian mechanics [37,58,84].The idea is to extract the main geometric ingredients in ordinary hamiltonian mechanics,

that is, that developed on cotangent bundles.Let us recall these main ingredients:

The phase space is the cotangent bundle T Q, dual of the tangent bundle T Q of theconfiguration manifold Q;

The canonical symplectic 2-form onT Qand its contravariant counterpart, the canonicalPoisson 2-vectorT QonT Q;

The hamiltonian function defined onT Q.

If we extrapolate these items we can consider a more general setting:

D : D Ma vector bundle, andD : D Mits dual vector bundle. A linear bivectorD on D (not Jacobi identity is required). We denote by{ , }D the

corresponding almost-Poisson bracket. h : D Ra hamiltonian function.


23/34


Here linear means that the bracket of two linear functions is a linear function.

Proposition 1 SinceD is linear, we deduce that

(a) 1, 2 (D ) {1,2}D is a linear function on D,

(b) (D ), f C(M) { , f D }D is a basic function (that is, a function doesnot depend on the fibres) with respect to D,

(c) f, gC(M) {f D , g D }D =0

The linear bivectorD induces the following structure on D:

an almost Lie bracket on the space (D )of sections ofD

[, ]D : (D ) (D ) (D )

(1, 2) [1, 2]D

where [1, 2]D = {1,2}D . an anchor mapD : (D )X(M)

f C(M), (D) D ()(f) D = { , f D }DThis bracket has the following properties:

a) [, ]D is antisymmetricb) [1, f2]D = f[1, 2]D + D (1)(f)2

In general,[

,]D does not satisfy the Jacobi identity. In the case when it satisfies the Jacobiidentity we say that(D, [, ]D , D )is a Lie algebroid.

Next, using the almost Lie bracket we can introduce the almost differentialdD : (kD) (k+1D)as follows:

Given (kD)thendD (k+1D)and

dD (0, 1, . . . , k)=

ki =0

(1)i D (i )((0, . . . ,i , . . . , k))+

i


24/34

214 M. de Len

Under these hypothesis, the dynamics given by a hamiltonian h : D Ris just thevector field Xh =D (dh)on D.

Given local coordinates(x)in the base manifold Mand a local basis of sections ofD,{e}, we induce local coordinates (x,y )on D and the bivectorD is written as

D =

y

x +

12

Cy

y

y

The corresponding Hamiltonian vector field is

Xh =D (dh)

or, in coordinates,

Xh =

h

y

x

h

x + C

y

h

y

y

5.4 Mechanical systems with nonholonomic constraints

Let G: EQ E Rbe a bundle metric on a Lie algebroid (E, [, ], ).The class of systems that were considered is that of mechanical systems with nonholo-

nomic constraints determined by:

The Lagrangian function

L(a)=1

2

G(a, a) V( (a)), a E,

withVa function on M The nonholonomic constraints given by a subbundle DofE

Consider the orthogonal decomposition E = D D, and the associated orthogonalprojectors

P : E D

Q : E D

Take local coordinates (x

) in the base manifoldMand a local basis of sections ofE, {e },adapted to the nonholonomic problem (L ,D), in the sense that

(i) {e} is an orthonormal basis with respect to G(that is G(e, e )= )

(ii) {e} = {ea , eA} where D =span{ea },D =span{eA}.

Denoting by(x,y )= (x,y a ,yA)the induced coordinates on E, the constraint equa-tions determining Djust read yA =0. Therefore we can choose (x,y a )as a set of coordi-nates on D.

In these coordinates we have the inclusion

iD : D E

(x,y a ) (x,y a , 0)

and the dual map

i D : E D

(x,ya ,yA) (x,ya )


25/34


where(x,y)are the induced coordinates on E by the dual basis of{e}.Moreover, from the orthogonal decomposition we have that

P : E D

(x,y a ,y ) (x,y a )

and its dual map

P : D E

(x,ya ) (x,ya , 0)

In these coordinates, the nonholonomic system is given by

(i) The LagrangianL(x,y )= 12

(y)2 V(x),

(ii) The nonholonomic constraintsyA =0.

In this case, the Legendre transformation associated with L is the isomorphism F L :E E induced by the metric G. Therefore, locally, the Legendre transformation is

F L : E E

(x,y ) (x,y = y)

and we can define the nonholonomic Legendre transformation F L nh = i D F L iD :D D

F Lnh : D D

(x,y a ) (x,ya = ya )

Notice that (E, [ , ], ) is a Lie algebroid and E is then a linear Poisson structureon E.

If f1and f2are functions on M, and1and 2are sections ofE, then:

{f1 E , g1 E }E =0, {1, f1 E , }E =((1)) f1 E , {1, 2}E = [1, 2]In the induced coordinates(x,y), the Poisson bracket relations on E are

{x,x}E =0, {y,x, }E =

, {y,y }E =C

y

In other words

E =

y

x

+ 12

Cy

y

y

The nonholonomic bracket on D, {, }nh ,D , is defined by

{F, G}nh ,D = {F i

D , G iD}E P

for all F, G C(D)The induced bivectornh ,D is

nh ,D =a

ya

x

+ 12 Ccabyc

ya

yb

That is,

{x,x}nh ,D =0, {ya ,x}nh ,D =a, {ya ,yb}nh ,D =C

cabyc

nh ,D is a linear bivector on D, but in general, does not satisfy the Jacobi identity. So,we are in the case considered above.


26/34

216 M. de Len

5.5 Particular cases

1. E = T M. Then the linear Poisson structure on E = TMis the canonical symplecticstructure. Thus,Dis a distribution onMand {, }nh ,Dis the nonholonomic bracket stud-

iedbyVanderSchaftandMaschke[134], Koon and Marsden [77],Cantrijnetal.[24,26],and others. It is interesting to note that a previous version of the nonholonomic bracketappears in [44,45] in two papers presented by P.A.M Dirac.

2. E=g, wheregis a Lie algebra. Eis a Lie algebroid over a single point (the anchor mapis the zero map). In this case, the linear Poisson structure onE =g is the Lie-Poissonstructure. Thus, if D = his a vector subspace ofg, we obtain that the nonholonomicbracket (nonholonomic Lie-Poisson bracket) is given by

{F, G}nh ,D()=

,P

F

,

G

for h, and F, G C(h). In adapted coordinates

{ya ,yb}nh ,D = Ccabyc

3. The Atiyah algebroid associated with a principalG -bundle : Q Q/G

E= T Q /G

The linear Poisson structure on E = T Q/Gis characterized by the following condi-tion: the canonical projectionT Q T Q/Gis a Poisson epimorphism. In this way,

we obtain the so-calledHamiltonPoincar bracketonT

Q/G(see [114]).IfDa G -invariant distribution on QD/Gis a vector subbundle ofE= T Q /G.Thus, we obtain a reduced non-holonomic bracket {, }nh ,D/G , which is the non-holonomic HamiltonPoincar bracket on D/G.

6 A first application: the HamiltonJacobi theory for nonholonomic mechanical

systems

Let us first recall the standard formulation of the HamiltonJacobi problem (see[1,3]).

The issue is to find a function S(t, qA)(called theprincipal function) such that

S

t+ h

qA,

S

qA

= 0. (13)

If we put S(t, qA)= W(qA) t E, where Eis a constant, then Wsatisfies

h

qA,

W

qA

= E; (14)

Wis called the characteristic function.

Equations(13) and(14) are indistinctly referred as the HamiltonJacobi equation.In modern symplectic language, Mis the configuration manifold, andTMits cotangent

bundle equipped with the canonical symplectic form

M =d qA d pA

where(qA)are coordinates in Mand(qA,pA)are the induced ones in TM.


27/34


Leth : TM R a hamiltonian function andXhthe corresponding hamiltonian vectorfield:

iXh M =d h

As we know, the integral curves ofXh , (qA(t),pA(t)), satisfy the Hamilton equations:

dqA

dt=

h

pA,

d pA

dt=

h

qA

Next, letbe a closed 1-form on M, sayd= 0; (then, locally = d W)

Theorem 1 (HamiltonJacobi Theorem)The following conditions are equivalent:

(i) If : I M satisfies the equation

dqA

dt= h

pA

then is a solution of the Hamilton equations;

(ii) d(h )= 0

Define a vector field on M:

Xh =TMXh

TM

M

Xh T(TM)

TM

M

Xh T M

The following conditions are equivalent:

(i) If : I Msatisfies the equation

dqA

dt=

h

pA

then is a solution of the Hamilton equations;(i) If : I Mis an integral curve ofXh , then is an integral curve ofXh ;(i) Xh and Xh are-related, i.e.

T(Xh )= Xh

Them we can reformulate the above theorem as follows:

Theorem 2 (HamiltonJacobi Theorem)Let be a closed1-form on M . Then the followingconditions are equivalent:

(i) Xh and Xh are-related;(ii) d(h )= 0


28/34

218 M. de Len

If

= A(q) dqA

then the HamiltonJacobi equation becomes

h(qA, A(qB ))= const.

and we recover the classical formulation when

A = W

qA

The advantage of this method is that, in spite of the difficulties to solve a partial dif-ferential equation instead of an ordinary differential one, in many cases it works, being anextremely useful tool, usually more than Hamiltons equations. Indeed, in these cases themethod provides an immediate way to integrate the equations of motion.

The modern interpretation relating the HamiltonJacobi procedure with the theory ofLagrangian submanifolds is an important source of new results and insights [1,3]. Let usremark that, recently, Cariena et al. [28] have developed a new approach to the geometricHamiltonJacobi theory.

A relevant difference with the unconstrained mechanical systems is that a nonholonomicsystem is not Hamiltonian in the sense that the phase space is just the constraint submani-fold and not the cotangent bundle of the configuration manifold; moreover, its dynamics isgiven by an almost Poisson bracket, that is, a bracket not satisfying the Jacobi identity. In[87,88], the authors proved that the nonholonomic dynamics can be obtained by projecting

the unconstrained dynamics; this was the point of view adopted in[69] in order to develop aHamiltonJacobi theory for nonholonomic systems.A natural question related with a possible notion of integrability is in what extent one could

construct a HamiltonJacobi theory for nonholonomic mechanics. Past attempts to obtain aHamiltonJacobi theory for nonholonomic systems were non-effective or very restrictive(see [44,130133] and also[115]), because, in many of them, they try to adapt the typicalproof of the HamiltonJacobi equations for systems without constraints (using Hamiltonsprinciple). Usually, the results are valid when the solutions of the nonholonomic problem arealso the solutions of the corresponding constrained variational problem (see[79,121,129]for a complete discussion).

We will present here an approach to this theory using a new setting in skewsymmetricalgebroids [86] (see also[113]).

LetD be a linear bivector on Dand : M D be a section ofD : D M.We define Xh = TD Xh It is easy to show that X

h (x) D (Dx), x M

Indeed, look the local expressions

Xh =

h

y

x =

h

ye

Theorem 3 (HamiltonJacobi Theorem)

Assume that d

D

= 0. Then the two following conditions are equivalent:(i) : I M integral curve of Xh integral curve of Xh

(ii) dD (h )= 0


29/34


Take local coordinates(x)in the base manifold M, a local basis of sections ofD, {e },and induced coordinates(x,y)on D. Then if

: (x)(x, (x)) (x, (x))

we have

dD (h )= 0

is locally written as

0 = dD (h )(e )x

= D (x)(e(x))(h )

= (x)

x(h )x

= (x) h

x(x, (x)) +

h

y(x, (x))

x(x)

,

Therefore, we obtain the HamiltonJacobi Equations:

(x)

h

x(x, (x)) +

h

y(x, (x))

x(x)

= 0

Acknowledgments We acknowledge the partial financial support of Ministerio de Innovacin y Ciencia,Project MTM2007-62478 and project Ingenio Mathematica(i-MATH) No. CSD2006-00032 (Consolider-

Ingenio 2010).

References

1. Abraham, R., Marsden, J.E.: Foundations of Mechanics, 2nd edn. Bejamin, New York (1978)2. Agrachev, A.A.: Rolling balls and octonions. Proc. Steklov Inst. Math.258(1), 1322 (2007)3. Arnold, V.I.: Mathematical Methods of Classical Mechanics. In: Graduate Texts in Mathematics,

vol. 60. Springer, Berlin (1978)4. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I: Dynamical Systems III. Springer, Berlin (1988)5. Balseiro, P., de Len, M.,Marrero, J.C., Martn de Diego, D.: The ubiquity of thesymplectic Hamiltonian

equations in mechanics. J. Geom. Mech.1(1), 134 (2009)6. Balseiro, P., Solomin, J.E.: Orthogonal projections and the dynamics of constrained mechanical systems.Q. Appl. Math.66(3):437446 (2008)

7. Balseiro, P., Solomin, J.E.: On generalized non-holonomic systems. Lett. Math. Phys. 84(1), 1530 (2009)8. Barbero-Lin, M., Muoz-Lecanda, M.: Geometric approach to Pontryagins maximum principle. Acta

Appl. Math.108(2), 429485 (2009)9. Barbero-Lin, M., Muz-Lecanda, M.: Constraint algorithm for extremals in optimal control prob-

lems. Int. J. Geom. Methods Mod. Phys. 6(7), 12211233 (2009)10. Barbero-Lin, M., Muoz-Lecanda, M.: Strict abnormal extremals in nonholonomic and kinematic

control systems. Discret. Contin. Dyn. Syst. Ser. S 3(1), 117 (2010)11. Bates, L.: Examples of singular nonholonomic reduction. Rep. Math. Phys.42, 231247 (1998)12. Bates, L., Graumann, H., MacDonnell, C.: Examples of gauge conservation laws in nonholonomic

systems. Rep. Math. Phys.37, 295308 (1996)13. Bates, L., Snyaticki, J.E.: Nonholonomic reduction. Rep. Math. Phys.32(1), 99115 (1992)14. Benenti, S.: Geometrical aspects of the dynamics of non-holonomic systems. Journees relativistes,

Univ. de Chambery (1987)15. Bicchi, A., Chitour, Y., Marigo, A.: Reachability and steering of rolling polyhedra: a case study in

discrete nonholonomy. IEEE Trans. Autom. Control49(5), 710726 (2004)16. Binz, E., de Len, M., Martn de Diego, D., Socolescu, D.: Nonholonomic constraints in classical field

theories. Rep. Math. Phys. 49(23), 151166 (2002)


30/34

220 M. de Len

17. Bloch, A.M.: Nonholonomic mechanics and control. In: Interdisciplinary Applied Mathematics, vol. 24.Systems and Control, pp. xx+483. Springer, New York (2003)

18. Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., Murray, R.M.: Nonholonomic mechanical systemswith symmetry. Arch. Ration. Mech. Anal.136, 2199 (1996)

19. Bloch, A.M., Marsden, J.E., Zenkov, D.V.: Nonholonomic dynamics. Notice AMS 52(3),

324333 (2005)20. Bondi, H.: The rigid body dynamics of unidirectional spin. Proc. R. Soc. Lond. A405, 265274 (1986)21. Borisov, A.V., Mamaev, I.S.: On the history of the development of the nonholonomic mechanics. Regul.

Chaotic Dyn.7(1), 4347 (2002)22. Bullo, F., Lewis, A.D.: Geometric control of mechanical systems. In: Modeling, analysis, and design

for simple mechanical control systems. Texts in Applied Mathematics, vol. 49, pp. xxiv+726. Springer,New York (2005)

23. Cantrijn, F., Corts, J., de Len, M., Martn de Diego, D.: On the geometry of generalized Chaplyginsystems. Math. Proc. Camb. Philos.132(2), 323351 (2002)

24. Cantrijn, F., de Len, M., Martn de Diego, D.: On almost Poisson structures in nonholonomic mechan-ics. Nonlinearity12, 721737 (1999)

25. Cantrijn, F., de Len, M., Marrero, J.C., Martn de Diego, D.: Reduction of constrained systems with

symmetries. J. Math. Phys.40, 725820 (1999)26. Cantrijn, F., de Len, M., Marrero, J.C., Martn de Diego, D.: On almost Poisson structures in nonhol-

onomic mechanics II: the time-dependent framework. Nonlinearity 13(4), 13791409 (2000)27. Cariena, J.F., Raada, M.F.: Lagrangian systems with constraints: a geoemtric approach to the method

of Lagrangian multipliers. J. Phys. A Math. Gen. 26, 13351351 (1993)28. Cariena, J.F., Grcia, X., Marmo, G., Martnez, E., Muoz-Lecanda, M., Rom-Roy, N.: Geometric

HamiltonJacobi theory for nonholonomic dynamical systems. Int. J. Geom. Methods Mod. Phys. 7(3),431454 (2010)

29. Cendra, H., Grillo, S.: Generalized nonholonomic mechanics, servomechanisms and related brackets.J. Math. Phys.47(2), 022902 (2006)

30. Cendra, H., Grillo, S.: Lagrangian systems with higher order constraints. J. Math. Phys.48(5), 052904(2007)

31. Cendra, H., de Len, M., Iborty, A., Martn de Diego, D.: A generalization of Chetaevs principle for aclass of higher order nonholonomic constraints. J. Math. Phys. 45(7), 27852801 (2004)

32. Chaplygin, S.A.: Analysis of the Dynamics of Nonholonomic Systems. Classical Natural Sciences,Moscow (1949)

33. Chaplygin, S.A.: Selected Works on Mechanics and Mathematics. State Publ. House, TechnicalTheoretical Literature, Moscow (1954)

34. Coleman, M., Holmes, P.: Motions and stability of a piecewise holonomicsystem: the discrete ChaplyginSleigh. Regul. Chaotic Dyn. 4(2), 123 (1999)

35. Corts, J.: Geometric, control and numerical aspects of nonholonomic systems. In: Lecture Notes inMathematics, vol. 1793. Springer, Berlin (2002)

36. Corts, J., de Len, M., Martn de Diego, D., Martnez, S.: Geometric description of vakonomic andnonholonomic dynamics Comparison of solutions. SIAM J. Control Optim. 41(5), 13891412 (2002)

37. Corts, J., Jorge de Len, M., Marrero, J.C., Martnez, E.: Nonholonomic Lagrangian systems on Liealgebroids. Discret. Contin. Dyn. Syst. 24(2), 213271 (2009)

38. Corts, J., de Len, M., Martn de Diego, D., Martnez, S.: Mechanical systems subjected to generalizednon-holonomic constraints. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457(2007), 651670 (2001)

39. Corts, J., Martnez, S.: Non-holonomic integrators. Nonlinearity14(5), 13651392 (2001)40. Cushman, R., Kemppainen, D., Sniatycki, J., Bates, L.: Geometry of nonholonomic constraints. Rep.

Math. Phys.36, 275286 (1995)41. Chetaev, N.G.: On Gauss principle. Izv. Fiz-Mat. Obsc. Kazan Univ.7, 6871 (1934)42. Dazord, P.: Mcanique Hamiltonienne en prsence de constraintes. Ill. J. Math.38, 148175 (1994)43. Delassus, E.: Sur les liaisons non lineaires. C.R. Acad. Sci. Paris153, 626628 (1911)44. Eden, R.J.: The Hamiltonian dynamics of non-holonomic systems. Proc. Roy. Soc. London. Ser.

A.205, 564583 (1951)

45. Eden, R.J.: The quantum mechanics of non-holonomic systems. Proc. Roy. Soc. Lond. Ser. A. 205,583595 (1951)

46. Ehlers, K., Koiller, J., Montgomery, R., Rios, P.M.: Nonholonomic systems via moving frames:Cartan equivalence and Chaplygin Hamiltonization. In: The breadth of symplectic and Poisson geometry,75120, Progr. Math., vol. 232. Birkhuser Boston, Boston, MA (2005)

47. Euler, L.: De minimis oscillationibus corporum tam rigidorum quam exililium, methodus nova etfacilis. Commentarii Academiae Scientiarum Imperialis Petropolitanae7, 99122 (1740)


31/34


48. Fedorov, Yu.N., Zenkov, D.V.: Discrete nonholonomic LL systems on Lie groups. Nonlinearity 18,22112241 (2005)

49. Fedorov, Y.N., Jovanovic, B.: Hamiltonization of the generalized Veselova LR system. Regul. ChaoticDyn.14(4-5), 495505 (2009)

50. Ferraro, S., Iglesias, D., Martn de Diego, D.: Momentum and energy preserving integrators for nonhol-

onomic dynamics. Nonlinearity21(8), 19111928 (2008)51. Ferrers, N.M.: Extension of Lagranges equations. Q. J. Pure Appl. Math.12(45), 15 (1872)52. Ferraro, S., Iglesias, D., Martn de Diego, D.: Numerical and geometric aspects of the nonholonomic

SHAKE and RATTLE methods, AIMS Proceedings (2011) (in press)53. Fomenko, A.T.: Visual and hidden symmetry in geometry. Symmetry : unifying human understanding

Part 1. Comput. Math. Appl.17(13), 301320 (1989)54. Garca, P.L., Garca, A., Rodrigo, C.: Cartan forms for first order constrained variational problems.

J. Geom. Phys.56(4), 571610 (2006)55. Garca, P.L., Rodrigo, C.: Cartan forms and second variation for constrained variational problems. In:

Geometry, integrability and quantization, pp. 140153. Softex, Sofia (2006)56. Garca, P.L., Fernndez, A., Rodrigo, C.: Variational integrators for discrete Lagrange problems.

J. Geom. Mech.2(4), 343374 (2010)

57. Giachetta, G.: Jet methods in nonholonomic mechanics. J. Math. Phys.33, 16521665 (1992)58. Grabowski, J., de Len, M., Marrero, J.C., Martn de Diego, D.: Nonholonomic constraints: a new

viewpoint. J. Math. Phys.50(1), 013520 (2009)59. Grcia, X., Marn-Solano, J., Muoz-Lecanda, M.C.: Some geometric aspects of variational calculus in

constrained systems. Rep. Math. Phys.51, 127148 (2003)60. Grcia, X., Martn, R.: Regularity and symmetries of nonholonomic systems. J. Phys. A Math. Gen.38,

10711087 (2005). doi:10.1088/0305-4470/38/5/00961. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration, Structure-Preserving Algorithms

for Ordinary Differential Equations.In: Springer Series in ComputationalMathematics, vol.31. Springer,Berlin (2002)

62. Hertz, H.: The principles of mechanics. In: Introduction by Robert S. Cohen, xlii+274 pp. DoverPublications, Inc., New York (1956) (preface by H. von Helmholtz. translation by D. E. Jones and

J. T. Walley)63. Kleinsteuber, K.H.M., Silva Leite, F.: Rolling Stiefel manifolds. Int. J. Syst. Sci.39(9), 881887 (2008)64. Ibort, L.A., de Len, M., Lacomba, E.A., Martn de Diego, D., Pitanga, P.: Mechanical systems subjected

to impulsive constraints. J. Phys. A30(16), 58355854 (1997)65. Ibort, L.A., de Len, M., Lacomba, E.A., Marrero, J.C., Martn de Diego, D., Pitanga, P.: Geometric

formulation of mechanical systems subjected to time-dependent one-sided constraints. J. Phys.A31(11), 26552674 (1998)

66. Ibort, L.A., de Len, M., Lacomba, E.A., Marrero, J.C., Martn de Diego, D., Pitanga, P.: Geometricformulation of Carnots theorem. J. Phys. A34(8), 16911712 (2001)

67. Ibort, L.A., de Len, M., Marmo, G., Martn de Diego, D.: Non-holonomic constrained systems asimplicit differential equations. Geometrical structures for physical theories, I (Vietri, 1996). Rend. Sem.Mat. Univ. Politec. Torino54(3), 295317 (1996)

68. Ibort, A., de Len, M., Marrero, J.C., Martn de Diego, D.: Dirac brackets in constrained dynamics.Forschritte der Physik47(5), 459492 (1999)

69. Iglesias-Ponte, D., de Len, M., Martn de Diego, D.: Towards a HamiltonJacobi theory for nonholo-nomic mechanical systems. J. Phys. A41(1), 015205 (2008)

70. Iglesias, D., Marrero, J.C., Martn de Diego, D., Martnez, E.: Discrete nonholonomic Lagrangian sys-tems on Lie groupoids. J. Nonlinear Sci.18(3), 221276 (2008)

71. Jurdjevic, V., Zimmerman, J.: Rolling problems on spaces of constant curvature. In: Lagrangian andHamiltonian methods for nonlinear control 2006, 221231. Lecture Notes in Control and InformationScience, vol. 366. Springer, Berlin (2007)

72. JurdjevicV., ZimmermanJ.:Rolling sphereproblems on spacesof constant curvature. Math. Proc. Camb.Philos. Soc.144(3), 729747 (2008)

73. Kane, C., Marsden, J.E., Ortiz, M., West, M.: Variational integrators and the Newmark algorithm for

conservative and dissipative mechanical systems. Int. J. Numer.Methods Eng. 49(10), 12951325 (2000)74. Kobilarov, M., Martn de Diego, D., Ferraro, S.: Simulating nonholonomic dynamics. Bol. Soc. Esp.

Mat. Apl. SeMA50, 6181 (2010)75. Koiller, J.:Reductionof some classicalnonholonomicmechanical systems with symmetry. Arch. Ration.

Mech. Anal.118, 113148 (1992)
http://dx.doi.org/10.1088/0305-4470/38/5/009http://dx.doi.org/10.1088/0305-4470/38/5/009http://dx.doi.org/10.1088/0305-4470/38/5/009


32/34

222 M. de Len

76. Koiller, J., Delgado,J.: On efficiency calculations for nonholonomic locomotion problems: an applicationto microswimming. Pacific Institute of Mathematical Sciences Workshop on Nonholonomic Constraintsin Dynamics (Calgary, AB, 1997). Rep. Math. Phys. 42(12), 165183 (1998)

77. Koon, W.S.,Marsden,J.E.:Poisson reductionof nonholonomic mechanical systems withsymmetry.Rep.Math Phys.42, 101134 (1998)

78. Koslov, V.V.: Realization of nonintegrable constraints in classical mechanics. Dokl. Akad. Nauk.SSSL273(3), 550554 (1983)

79. Kozlov, V.V.: On the integration theory of equations of nonholonomic mechanics. Regul. ChaoticDyn.7, 161176 (2002)

80. Korteweg, D.J.: Ueber eine ziemlich verbreitete unrichtige Behandlungsweise eines Problems der rol-lende Bewegung, die Theorie dieser Bewegung, und ins besondere kleine rollende Schwingungen umeine Gleichgewichtslage. Nieuw Archief Voor Wiskunde4, 130155 (1899)

81. Kremnev, A.V., Kuleshov, A.S.: Nonlinear dynamics and stability of the skateboard. Discret. Contin.Dyn. Syst. Ser. S3(1), 85103 (2010)

82. Kuleshov, A.S.: A mathematical model of the snakeboard (Russian). Mat. Model.18(5), 3748 (2006)83. Lacomba, E.A., Tulczyjew, W.M.: Geometric formulation of mechanical systems with one-sided

constraints. J. Phys. A23(13), 28012813 (1990)

84. de Len, M., Marrero, J.C., Martnez, E.: Lagrangian submanifolds and dynamics on Lie algebroids.J. Phys. A Math. Gen. 38, R241R308 (2005) (topical review)

85. de Len, M., Marrero, J.C., Martn de Diego, D.: Non-holonomic Lagrangian systems in jet manifolds.J. Phys. A30(4), 11671190 (1997)

86. Len, M., Marrero, J.C., Martn de Diego, D.: Linear almost Poisson structures and HamiltonJacobiequation. Applications to nonholonomic mechanics. J. Geom. Mech. 2(2), 159198 (2010)

87. de Len, M., Martn de Diego, D.: Solving non-holonomic Lagrangian dynamics in terms of almostproduct structures. Extracta Math.11, 325347 (1996)

88. de Len, M., Martn de Diego, D.: On the geometry of non-holonomic Lagrangian systems. J. Math.Phys.37, 33893414 (1996)

89. de Len, M., Martn de Diego, D.: A constraint algorithm for singular Lagrangians subjected to nonhol-onomic constraints. J. Math. Phys.38(6), 30553062 (1997)

90. de Len, M., Martn de Diego, D., Santamara-Merino, A.: Geometric numerical integration of nonhol-onomic systems and optimal control problems. Eur. J. Control 10(5), 515521 (2004)

91. de Len, M., Martn de Diego, D., Santamara-Merino, A.: Geometric integrators and nonholonomicmechanics. J. Math. Phys. 45(3), 10421064 (2004)

92. de Len, M., Rodrigues, P.R.: Methods of Differential Geometry in Analytical Mechanics. North-Holland, Amsterdam (1989)

93. de Len, M., Rodrigues, P.R.: Higher-order mechanical systems with constraints. Int. J. Theor.Phys.31(7), 13031313 (1992)

94. Lewis, A.D., Murray, R.M.: Variational principles for constrained systems: theory and experiment. Int.J. Non-Linear Mech.30(6), 793815 (1995) (including simulations and lab work)

95. Lindelof, E.: Acta Societatis Scientiarum Fennicae, vol. XX, issue no. 10 (1895)96. McLachlan, R., Scovel, C.: Open problems in symplectic integration. Fields Inst. Comm. 10,

151180 (1996)97. Marigo, A., Bicchi, A.: Rolling bodies with regular surface: controllability theory and applications. IEEE

Trans. Automat. Control45(9), 15861599 (2000)98. Marle, C.M.: Various approaches to conservative and nonconservative nonholonomic systems. Rep.

Math Phys.42, 211229 (1998)99. Marle, C.M.: On symmetries and constants of motion in hamiltonian systems with nonholonomic con-

straints. Classical and quantum integrability (Warsaw, 2001), 223242, Banach Center Publ., PolishAcad. Sci., vol. 59, Warsaw (2003)

100. Marrero, J.C., Martn de Diego, D., Martnez, E.: Discrete Lagrangian and Hamiltonian mechanics onLie groupoids. Nonlinearity 19(6), 13131348 (2006). (Corrigendum: Nonlinearity 19 (2006), no. 12,30033004)

101. Marsden, J.E., Pekarsky, S., Shkoller, S.: Discrete Euler-Poincar and Lie-Poisson equations. Nonlin-

earity12, 16471662 (1999)102. Marsden, J.E.: Lectures on mechanics. London Mathematical Society Lecture Note Series, vol. 174,

pp. xii+254. Cambridge University Press, Cambridge (1992)103. Marsden, J.E., Patrick, G.W., Shadwick, W.F.: Integration algorithms and classical mechanics. In:

Proceedings of the conference held in Toronto, Ontario, October 1993. Edited by Fields Institute Com-munications, vol. 10, pp. xii+244. American Mathematical Society, Providence, RI (1996)


33/34


104. Marsden, J.E., Pekarsky, S., Shkoller, S., West, M.: Variational Methods, Multisymplectic Geometryand Continuum Mechanics. J. Geom. Phys.38, 253284 (2001)

105. Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numerica. 10, 357514(2001)

106. Martnez, S., Corts, J., de Len, M.: The geometrical theory of constraints applied to the dynamics of

vakonomic mechanical systems: the vakonomic bracket. J. Math. Phys. 41(4), 20902120 (2000)107. Martnez, S., Corts, J., de Len, M.: Symmetries in vakonomic dynamics: applications to optimal

control. J. Geom. Phys.38(34), 343365 (2001)108. Massa, E., Pagani, E.: Classical dynamics of non-holonomic systems: a geometric approach. Ann. Inst.

Henri Poincar: Physique Theorique55, 511544 (1991)109. Moffatt, H.K., Tokieda, T.: Celt reversals: a prototype of chiral dynamics. Proc. Royal Soc. Edinb.

138, 361368 (2008)110. Montgomery, R.: A tour of sub-Riemannian geometries their geodesics and applications. Math. Surveys

Monograph, vol. 91. American Mathematical Society, Providence (2002)111. Moser, J., Veselov, A.P.: Discrete versions of some classical integrable systems and factorization of

matrix polynomials. Comm. Math. Phys.139(2), 217243 (1991)112. Neimark, Ju.I., Fufaev, N.A.: Dynamicsof Nonholonomic Systems, Transactionsof Mathematical Mono-

graphs, vol. 33. AMS (1972) (Russian edition, 1967)113. Ohsawa, T., Bloch, A.M.: Nonholonomic HamiltonJacobi equation and integrability. J. Geom. Mech.

1(4), 461481 (2009)114. Ortega, J.P., Ratiu, T.S.: Momentum maps and Hamiltonian reduction Progress in Mathematics,

vol. 222. Birkhauser, Boston (2004)115. Pavon, M.: HamiltonJacobi equations for nonholonomic dynamics. J. Math. Phys.46, 032902 (2005)116. Pironneau, Y.: Sur les liaisons non holonomes non linaires deplacement virtuels travail nul, conditions

de Chetaev. In: Benenti, S., Francaviglia, M., Lichnerowicz, A. (eds.) Proceedings of the IUTAM-ISI-MMM Symposium on Modern Developments in Analytical Mechanics, Torino 1982, pp. 671686. ActaAcademiae Scientiarum Taurinensis (1983)

117. Poincar, H.:Les ides de Hertz sur la mcanique, Oeuvres VII, 231250. Gauthier-Villars, Paris (1952)118. Rodrigues, P.R.: On Lagrangian equations with generic constraints. An. Acad. Brasil. Cienc. 56(1),

1316 (1984)119. Routh, E.J.: Dynamics of a System of Rigid Bodies. Dover, New York (1960)120. Ruina, A.: Nonholonomic stability aspects of piecewise holonomic systems. Rep. Math. Phys.

42(12), 91100 (1998)121. Rumiantsev, V.V.: Hamiltonian principle for nonholonomic systems. Prikladnaia Matematika i

Mekhanika, vol. 42, pp. 387399, MayJune (1978) (in Russian)122. Rumyantsev, V.V.: Variational principles for systems with unilateral constraints. J. Appl. Math. Mech.

70, 808818 (2006)123. Sanz-Serna, J.M., Calvo, M.P.: Numerical Hamiltonian Problems. Chapman & Hall, London (1994)124. Sarlet, W., Cantrijn, F., Saunders, D.J. A geometrical framework for the study of non-holonomic

Lagrangian systems. J. Phys. A28(11), 32533268 (1995)125. Saunders, D.J., Sarlet, W., Cantrijn, F.: A geometrical framework for the study of non-holonomic

Lagrangian systems. II. J. Phys. A29(14), 42654274 (1996)126. Skinner, R., Rusk, R.: Generalized Hamiltonian dynamics. I. Formulation onT Q T Q . J. Math. Phys.

24((11), 25892594 (1983)127. Skinner, R., Rusk, R.: Generalized Hamiltonian dynamics. II. Gauge transformations. J. Math. Phys.

24(11), 25952601 (1983)128. Slesser, G.M.: Notes on rigid dynamics. Q. J. Math.4, 6577 (1861)129. Sumbatov, A.S.: Nonholonomic systems. Regul. Chaotic Dyn.7, 221238 (2002)130. Van Dooren, R.: The generalized HamiltonJacobi method for non-holonomic dynamical systems of

Chetaev s type. Z. Angew. Math. Mech.55, 407411 (1975)131. Van Dooren, R.: Motion of a rolling disc by a new generalized HamiltonJacobi method. J. Appl. Math.

Phys.27, 501505 (1976)132. Van Dooren, R.: Second form of the generalized HamiltonJacobi method for nonholonomic dynamical

systems. J. Appl. Math. Phys.29, 828834 (1978)133. Van Dooren, R.: On the generalized HamiltonJacobi method for nonholonomic dynamical systems.

Dienst Anal. Mech. Tw, Vub. 16 (1979)134. Vander Schaft, A.J., Maschke, B.M.: On the Hamiltonian formulation of nonholonomic mechanical

systems. Rep. Math. Phys.34(2), 225233 (1994)135. Vankerschaver, J.: The momentum map for nonholonomic field theories with symmetry. Int. J. Geom.

Methods Mod. Phys.2(6), 10291041 (2005)


34/34

224 M. de Len

136. Vankerschaver, J.: Classical field theories with external constraints. XIV Fall Workshop on Geometryand Physics. Publ. R. Soc. Mat. Esp., vol. 10, pp. 285291, R. Soc. Mat. Esp., Madrid (2006)

137. Vankerschaver, J.: A class of nonholonomic kinematic constraints in elasticity. J. Phys. A 40(14),38893913 (2007)

138. Vankerschaver, J., Martn de Diego, D.: Symmetry aspects of nonholonomic field theories. J. Phys. A

41(3), 035401 (2008)139. Vankerschaver, J., Cantrijn, F., de Len, M., Martn de Diego, D.: Geometric aspects of nonholonomic

field theories. Rep. Math. Phys.56(3), 387411 (2005)140. Vershik, A.M., Faddeev, L.D.: Differential geometry and lagrangian mechanics with constraints. Sov.

Phys. Doklady17(1), 3436 (1972)141. Veselov, A.P.: Integrable systems with discrete time, and difference operators (Russian). Funktsional.

Anal. i Prilozhen.22(2), 113, 96 (1988) (translation in Funct. Anal. Appl.22(2), 8393 (1988))142. Veselov, A.P: Integrable Lagrangian relations and factorization of matrix polynomials. (Russian). Funkt-

sional. Anal. i Prilozhen. 25(2), 3849, 96 (1991) (translation in Funct. Anal. Appl. 25(2), 112122(1991))

143. Veselov, A.P., Veselova, L.E.: Flows on Lie groups with a nonholonomic constraint and integrablenon-Hamiltoian systems. (Russian). Funktsional. Anal. i Prilozhen. 20(4), 6566 (1986)

144. Veselov, A.P., Veselova, L.E.: Integrable nonholonomic systems on Lie groups. (Russian) Mat. Zametki44(4), 604619, 701 (1988) (translation in Math. Notes44(56) (1988), 810819 (1989))

145. Vierkandt, A.: ber gleitende und rollende Bewegung. Monatshefte fr. Math.3(1), 97116. doi:10.1007/BF01692429

146. Voronetz, P.: On the Equations of Motion for Nonholonomic Systems. Mat. Sbornik XXII,659686 (1901)

147. Walker, G.T.: On a dynamical top. Q. J. Pure Appl. Math.28, 175184 (1896)148. Weber, R.W.: Hamiltonian systems with constraints and their meaning in mechanics. Arch. Ration.

Mech. Anal.91(4), 309335 (1986)149. Weinstein, A.: Lagrangian mechanics and groupoids. In: Mechanics day (Waterloo, ON, 1992). Fields

Institute Communications vol. 7, pp. 173239. American Mathematical Society, RI (1996)150. Zhuravlev, V.F., Fufayev, N.A.: Mechanics of Systems with Unilateral Constraints, Nauka, Moscow

(1993). Prikl. Mat. Mekh.,70(6), 902914 (2006)
http://dx.doi.org/10.1007/BF01692429http://dx.doi.org/10.1007/BF01692429http://dx.doi.org/10.1007/BF01692429http://dx.doi.org/10.1007/BF01692429

a historical review on nonholomic mechanics.pdf

Documents