geometric mechanics, lagrangian reduction, and ...way, symplectic reduced spaces. in the reduction...

Geometric Mechanics, Lagrangian Reduction, and

Nonholonomic Systems

Hernan CendraDepartamento de Matematica

Universidad Nacional del Sur, Av. Alem 12548000 Bahia Blanca, Argentina

[email protected]

Jerrold E. MarsdenControl and Dynamical Systems 107-81

California Institute of TechnologyPasadena, CA 91125

[email protected]

Tudor S. RatiuDepartement de Mathematiques

Ecole Polytechnique Federale de LausanneCH 1015 Lausanne. Switzerland

[email protected]

September, 1999; this version: January 30, 2001

Mathematics Unlimited–2001 and Beyond

Edited by B. Engquist and W. Schmid, Springer-Verlag, 221–273.

Abstract

This paper surveys selected recent progress in geometric mechanics, focussing onLagrangian reduction and gives some new applications to nonholonomic systems, thatis, mechanical systems with constraints typified by rolling without slipping.

Reduction theory for mechanical systems with symmetry has its roots in the clas-sical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincare andothers. The modern vision of mechanics includes, besides the traditional mechanicsof particles and rigid bodies, field theories such as electromagnetism, fluid mechanics,plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories,including gravity.

Symmetries in mechanics ranges from obvious translational and rotational symme-tries to less obvious particle relabeling symmetries in fluids and plasmas, to subtlesymmetries underlying integrable systems. Reduction theory concerns the removal ofsymmetries and utilizing their associated conservation laws. Reduction theory hasbeen extremely useful in a wide variety of areas, from a deeper understanding of manyphysical theories, including new variational and Poisson structures, stability theory,integrable systems, as well as geometric phases. Much effort has gone into the de-velopment of the symplectic and Poisson view of reduction theory, but recently theLagrangian view, emphasizing the reduction of variational principles has also matured.

1

CONTENTS 2

While there has been much activity in the geometry of nonholonomic systems, thetask of providing an intrinsic geometric formulation of the reduction theory for non-holonomic systems from the point of view of Lagrangian reduction has been somewhatincomplete. One of the purposes of this paper is to finish this task. In particular, weshow how to write the reduced Lagrange d’Alembert equations, and in particular, itsvertical part, the momentum equation, intrinsically using covariant derivatives. Theresulting equations are called the Lagrange-d’Alembert-Poincare equations.

Contents

1 An Introduction to Reduction Theory 2

2 Geometric Mechanics and Nonholonomic Systems 8

3 The Lagrange-d’Alembert Principle with Symmetry 15

4 The Local Momentum and Horizontal Equation 28

5 The Snakeboard 31

6 Miscellany and Future Directions 36

1 An Introduction to Reduction Theory

The Purpose of this Paper. This paper outlines some features of general reductiontheory as well as the geometry of nonholonomic mechanical systems. In addition to thissurvey nature, there are some new results. Our previous work on the geometric theory ofLagrangian reduction provides a convenient context that is herein generalized to nonholo-nomic systems with symmetry. This provides an intrinsic geometric setting for many of theresults that were previously understood primarily in coordinates. This solidification andextension of the basic theory should have several interesting consequences, some of whichare spelled out in the final section of the paper. Two important references for this workare Cendra, Marsden and Ratiu [2000], hereafter denoted CMR and Bloch, Krishnaprasad,Marsden and Murray [1996], hereafter denoted BKMM.

A Brief History of Reduction Theory. Reduction theory has its origins in the classicalwork of Euler, Lagrange, Hamilton, Jacobi, Routh and Poincare and is one of the funda-mental tools in the study of mechanical systems with symmetry. Several classical casesof reduction (using conservation laws and symmetry to create smaller dimensional phasespaces), such as Routh’s elimination of cyclic variables and Jacobi’s elimination of the node,were developed by these founding masters. The ways in which reduction theory has beengeneralized and applied since that time has been rather impressive.

Routh (circa 1850) pioneered reduction theory for Abelian groups, that is, for La-grangians with cyclic variables. By 1890, Lie had discovered many of the basic structuresin symplectic and Poisson geometry and their link with symmetry. Meanwhile, Poincarein 1901 discovered the generalization of the Euler equations for rigid body mechanics andfluids to general Lie algebras. From then to about 1965, the subject lay, to some extent,dormant, in a state of gestation.

The modern era of reduction theory began with the fundamental papers of Arnold [1966]and Smale [1970]. Arnold focussed on systems on Lie algebras and their duals, as in the


works of Lie and Poincare, while Smale focussed on the Abelian case giving, in effect, amodern version of Routh reduction.

The description of many physical systems such as rigid bodies and fluids requires non-canonical Poisson brackets and constrained variational principles of the sort implicitly stud-ied by Lie and Poincare. It is quite astounding how much rediscovery and duplication ofthe basic work of Routh, Lie and Poincare occurred in the first two thirds of the twentiethcentury.

A basic noncanonical Poisson bracket on g∗, the dual of a Lie algebra g, is called, followingMarsden and Weinstein [1983], the Lie-Poisson bracket. These structures were known toLie around 1890, although Lie seemingly did not recognize their importance in mechanics.The symplectic leaves in these structures, namely the coadjoint orbit symplectic structures,although implicit in Lie’s work, were discovered by Kirillov, Kostant, and Souriau in the1960’s. At first it may seem remarkable that the dual of any Lie algebra carries, in a naturalway, a Poisson structure. However, this structure is nothing but the natural quotient Poissonstructure induced from the canonical bracket structure on T ∗G by the quotient operationg∗ = (T ∗G)/G. This fact is explained in, for example, Marsden and Ratiu [1999].

To synthesize the Lie algebra reduction methods of Arnold [1966] with the techniques ofSmale [1970] on the reduction of cotangent bundles by Abelian groups, the work of Mars-den and Weinstein [1974] developed reduction theory in the general context of symplecticmanifolds and equivariant momentum maps; related results, but with a different motivationand construction (not stressing equivariance properties of momentum maps) were found byMeyer [1973].

The construction is now standard: let (P,Ω) be a symplectic manifold and suppose thata Lie group G acts freely and properly on P by symplectic maps. The free and properassumption is to avoid singularities in the reduction procedure as is discussed later. Assumethat this action has an equivariant momentum map J : P → g∗. Then the symplecticreduced space J−1(µ)/Gµ = Pµ is a symplectic manifold in a natural way; the inducedsymplectic form Ωµ is determined uniquely by π∗

µΩµ = i∗µΩ where πµ : J−1(µ) → Pµ

is the projection and iµ : J−1(µ) → P is the inclusion. If the momentum map is notequivariant, Souriau [1970] discovered how to centrally extend the group (or algebra) tomake it equivariant. Using either this construction or a modification of the original method,one can carry out nonequivariant reduction as well.

The paper of Marsden and Weinstein [1974] showed that coadjoint orbits are, in a naturalway, symplectic reduced spaces. In the reduction construction, one chooses P = T ∗G, withG acting by (say left) translation, the corresponding space Pµ is identified with the coadjointorbit Oµ through µ together with its coadjoint orbit symplectic structure. Likewise, the Lie-Poisson bracket on g∗ is inherited from the canonical Poisson structure on T ∗G by Poissonreduction, that is, by simply identifying g∗ with the quotient (T ∗G)/G. It is not clear whofirst explicitly observed this, but it is implicit in many works such as Lie [1890], Kirillov[1962, 1976], Guillemin and Sternberg [1980], and Marsden and Weinstein [1982, 1983], andis explicit in the papers Marsden, Weinstein, Ratiu, Schmid and Spencer [1982] and Holmesand Marsden [1983].

Kazhdan, Kostant and Sternberg [1978] showed that Pµ is symplectically diffeomorphicto an orbit reduced space Pµ

∼= J−1(Oµ)/G and from this it follows that Pµ are the symplec-tic leaves in P/G. This paper was also one of the first to notice deep links between reductionand integrable systems, a subject continued by by many authors, but the group theoreticexplanation of the integrability of the Kowalewski top in work of [1989] is particularlynoteworthy.

The way in which the Poisson structure on Pµ is related to that on P/G was clarified ina generalization of Poisson reduction due to Marsden and Ratiu [1986], a technique that hasalso proven useful in integrable systems (see, e.g., Pedroni [1995] and Vanhaecke [1996]).


Stability Theory. Reduction theory for mechanical systems with symmetry has provento be a powerful tool enabling advances in stability theory (from the Arnold method tothe energy-momentum method of Simo, Lewis and Marsden [1991]) as well as in bifurcationtheory of mechanical systems, geometric phases via reconstruction — the inverse of reduction— as well as uses in control theory from stabilization results to a deeper understanding oflocomotion. For a general introduction to some of these ideas and for further references, seeMarsden and Ratiu [1999].

Tangent and Cotangent Bundle Reduction. The simplest case of cotangent bundlereduction is reduction at zero in which case one chooses P = T ∗Q and then the reducedspace at µ = 0 is given by P0 = T ∗(Q/G), the latter with the canonical symplectic form.Another basic case is when G is Abelian. Here, (T ∗Q)µ

∼= T ∗(Q/G) but the latter has asymplectic structure modified by magnetic terms; that is, by the curvature of a mechanicalconnection.

The Abelian version of cotangent bundle reduction was developed by Smale [1970] andSatzer [1977] and was generalized to the nonabelian case in Abraham and Marsden [1978].Kummer [1981] interpreted Abraham and Marsden’s results in terms of a connection, nowcalled the mechanical connection. The geometry of this situation was used to great effectin, for example, Guichardet [1984], Iwai [1987, 1990], and Montgomery [1984, 1990, 1991].Routh reduction may be viewed as the Lagrangian analogue of cotangent bundle reduction.

Tangent and cotangent bundle reduction evolved into what we now term as the “bundlepicture” or the “gauge theory of mechanics”. This point of view was first developed byMontgomery, Marsden and Ratiu [1984] and Montgomery [1984, 1986]. That work wasmotivated and influenced by the work of Sternberg [1977] and Weinstein [1978] on a Yang-Mills construction that is in turn motivated by Wong’s equations, that is, the equationsfor a particle moving in a Yang-Mills field. The main result of the bundle picture gives astructure to the quotient spaces (T ∗Q)/G and (TQ)/G when G acts by the cotangent andtangent lifted actions. This point of view of the bundle picture will be central to the mainbody of this paper.

Lagrangian versus Hamiltonian Reduction. In symplectic and Poisson reduction onefocusses on how to pass the symplectic two form and the Poisson bracket structure as wellas any associated Hamiltonian dynamics to a quotient space for the action of a symmetrygroup. In Lagrangian reduction theory, which proceeds in a logically independent way, oneemphasizes how the variational structure passes to a quotient space. Of course, the twomethodologies are related by the Legendre transform, although not always in a straightfor-ward way. We shall begin our brief summary of Lagrangian reduction theory with two ofthe most important classical cases, namely Routh reduction and Euler-Poincare reduction.

Routh Reduction. Routh reduction for Lagrangian systems is classically associated withsystems having cyclic variables (this is almost synonymous with having an Abelian symmetrygroup); modern accounts can be found in Arnold, Kozlov and Neishtadt [1988] and inMarsden and Ratiu [1999], §8.9. A key feature of Routh reduction is that when one dropsthe Euler-Lagrange equations to the quotient space associated with the symmetry, and whenthe momentum map is constrained to a specified value (i.e., when the cyclic variables andtheir velocities are eliminated using the given value of the momentum), then the resultingequations are in Euler-Lagrange form not with respect to the Lagrangian itself, but withrespect to the Routhian. In his classical work, Routh [1877] applied these ideas to stabilitytheory, a precursor to the energy-momentum method for stability mentioned above. Ofcourse, Routh’s stability method is still widely used in mechanics.


Euler-Poincare Reduction. Besides Routh reduction, another fundamental case of La-grangian reduction is that of Euler–Poincare reduction, which occurs for the case in whichthe configuration space is a Lie group G. One thinks of this case as primarily intended forsystems governed by Euler equations, such as those of a rigid body or a fluid. This casehas its origins in the work of Lagrange [1788] and Poincare [1901a]. Both of them clearlyhad some idea of the reduction process. Poincare realized that both the equations of fluidmechanics and the free rigid body and heavy top equations could all be described in Liealgebraic terms in a beautiful way.

The classical Euler-Poincare equations are as follows. Let ξa be coordinates for the Liealgebra g of a Lie group G and let Ca

bd be the associated structure constants. Let L : TG → R

be a given left invariant Lagrangian and let l : g = (TG)/G → R be the correspondingreduced Lagrangian. Then the Euler-Poincare equations for a curve ξ(t) ∈ g are

d

dt

∂l

∂ξb=

∂l

∂ξaCa

dbξd,

where a summation is implied over repeated indices. These equations are equivalent to theEuler-Lagrange equations for L for a curve g(t) ∈ G, where g(t)−1g(t) = ξ(t). This is oneof the most basic formulations of Lagrangian reduction. A nice way to see this equivalenceis to make use of the technique of reduction of variational principles.

The history of the Euler-Poincare equations is complicated, despite their simplicity andthe literature has been full of repeated rediscovery. For example, apparently unaware of thework of Poincare, it was only in Newcomb [1962], Arnold [1966], and Bretherton [1970] wasthe reduction procedure and the reduced variational principle established for rigid body andfluid equations. Both the intrinsic (coordinate free) formulation of Routh reduction as wellas the general formulation of Euler–Poincare reduction in terms of variational principles weregiven in Marsden and Scheurle [1993a, b]. The Euler–Poincare case was further developedin Bloch, Krishnaprasad, Marsden and Ratiu [1996]. An exposition of Lagrangian reductionfor both the Routh and Euler–Poincare cases can be found in Marsden and Ratiu [1999].

Lagrange-Poincare Reduction. Marsden and Scheurle [1993a, b] showed how to gen-eralize the Routh theory to the nonabelian case as well as realizing how to get the Euler-Poincare equations for matrix groups by the important technique of reducing variationalprinciples. This approach was motivated by related work of Cendra and Marsden [1987]and Cendra, Ibort and Marsden [1987]. The Euler–Poincare variational structure was ex-tende to general Lie groups in Bloch, Krishnaprasad, Marsden and Ratiu [1996]. CMRcarried out a Lagrangian reduction theory that extends the Euler–Poincare case to arbi-trary configuration manifolds. This work was in the context of the Lagrangian analogue ofPoisson reduction in the sense that no momentum map constraint is imposed.

One of the things that makes the Lagrangian side of the reduction story interesting isthe lack of a general category that is the Lagrangian analogue of Poisson manifolds. Such acategory, that of Lagrange-Poincare bundles is developed in CMR, with the tangent bundleof a configuration manifold and a Lie algebra as its most basic examples. That work alsodevelops the Lagrangian analogue of reduction for central extensions and, as in the case ofsymplectic reduction by stages (see Marsden, Misiolek, Perlmutter and Ratiu [1998, 2000]),cocycles and curvatures enter in this context in a natural way.

The Lagrangian analogue of the bundle picture is the bundle (TQ)/G, which, as shownlater, is a vector bundle over Q/G; this bundle was studied in CMR. In particular, theequations and variational principles are developed on this space. For Q = G this reducesto Euler-Poincare reduction and for G Abelian, it reduces to the classical Routh procedure.Given a G-invariant Lagrangian L on TQ, it induces a Lagrangian l on (TQ)/G. The


resulting equations inherited on this space are the Lagrange–Poincare equations (or thereduced Euler-Lagrange equations).

The Lagrange-Poincare equations in coordinates are given as follows. Let ξa becoordinates for the Lie algebra g of G and let Ca

bd be the associated structure constants. Forgiven local coordinates xα in Q/G, let Aa

α(x) be the local coordinate expression of a chosenprincipal connection on the bundle Q → Q/G and let Ba

β,α the associated curvature, whosedefinition we review later. Let pa denote the momentum conjugate to ξa; that is, pa = ∂l

∂ξa .Then

dpb

dt= pa

(Ca

dbξd − Ca

dbAdαxα

)∂l

∂xα− d

dt

∂l

∂xα=

∂l

∂ξa

(Ba

βαxβ + Cadbξ

dAbα

),

where a summation is implied over repeated indices. We shall review the intrinsic geometryassociated with these equations later. An interesting point is that if one is willing to workonly with the quotient space TQ/G rather than its global realization as T (Q/G) ⊕ g, thena connection is not needed and the local form of the equations is given by the Hamelequations (see Hamel [1904]), namely the equations that are formally obtained by takinga trivial connection:

dpb

dt− paCa

dbξd = 0

∂l

∂xα− d

dt

∂l

∂xα= 0,

Methods of Lagrangian reduction have proven very useful in a number of areas, as withHamiltonian reduction. In particular, these methods are useful in optimal control problems;for example, Koon and Marsden [1997a] extended the falling cat theorem of Montgomery[1990] to the case of nonholonomic systems.

Lagrange-Routh Reduction. If one imposes the constraint of constancy of the momen-tum map on the Lagrangian side, then Lagrange-Poincare reduction becomes Lagrange-Routh reduction. It is the nonabelian analogue of the classical Routh procedure mentionedabove. This theory in the nonabelian case began with Marsden and Scheurle [1993a]. Deeperinsight into the constraints one might impose on the variations was obtained by Jalnapurkarand Marsden [2000]. Putting this theory into an intrinsic bundle context, relaxing the con-straints needed on the variations and obtaining an intrinsic version of the Lagrange-Routhequations is the subject of Marsden, Ratiu and Scheurle [2000].

Semidirect Product Reduction. In the simplest case of a semidirect product, one hasa Lie group G that acts on a vector space V (and hence on its dual V ∗) and then one formsthe semidirect product S = G V , generalizing the semidirect product structure of theEuclidean group SE(3) = SO(3) R

3.Consider the isotropy group Ga0 for some a0 ∈ V ∗. The semidirect product reduction

theorem states that each of the symplectic reduced spaces for the action of Ga0 on T ∗Gis symplectically diffeomorphic to a coadjoint orbit in (g V )∗, the dual of the Lie algebraof the semi-direct product . This semidirect product theory was developed in a series ofpapers: Guillemin and Sternberg [1980], Ratiu [1980a, 1981, 1982a], and Marsden, Ratiuand Weinstein [1984a, b].

This construction is used in applications where one has “advected quantities” (such asthe direction of gravity in the heavy top, density in compressible flow and the magnetic field


in MHD). This theory has many other interesting applications, such as to the the dynamicsof underwater vehicles (Leonard and Marsden [1997]). This semidirect product theory wasa direct precursor to the development of symplectic reduction by stages described below.

Lagrangian analogues of the semidirect product theory, focussing on variational struc-tures, were developed in Holm, Marsden and Ratiu [1998b] with applications to many fluidmechanical problems of interest. The point of view was to extend the Euler–Poincare the-ory to the case of systems such as the heavy top and compressible flows in which there areadvected parameters. This methodology was applied to the interesting (and also degener-ate) case of the Maxwell-Vlasov equations by Cendra, Holm, Hoyle and Marsden [1998] andCendra, Holm, Marsden and Ratiu [1998] showed how it fits into the general framework ofLagrangian reduction.

Reduction by Stages and Group Extensions. The semidirect product reduction the-orem can be viewed using reduction by stages: one reduces T ∗S by the action of the semidi-rect product group S = G V in two stages, first by the action of V at a point a0 andthen by the action of Ga0 . Semidirect product reduction by stages for actions of semidirectproducts on general symplectic manifolds was developed and applied to underwater vehicledynamics in Leonard and Marsden [1997]. Motivated partly by semidirect product reduc-tion, Marsden, Misiolek, Perlmutter and Ratiu [1998, 2000] gave a significant generalizationof semidirect product theory in which one has a group M with a normal subgroup N ⊂ M(so M is a group extension of N) and M acts on a symplectic manifold P . One wants toreduce P in two stages, first by N and then by M/N . On the Poisson level this is easy:P/M ∼= (P/N)/(M/N) but on the symplectic level it is quite subtle.

Cotangent bundle reduction by stages is especially interesting for group extensions. Anexample of such a group, besides semidirect products, is the Bott-Virasoro group, wherethe Gelfand-Fuchs cocycle may be interpreted as the curvature of a mechanical connection.The work of CMR briefly described above, contains a Lagrangian analogue of reduction forgroup extensions and reduction by stages.

Singular Reduction. Singular reduction starts with the observation of Smale [1970] thatz ∈ P is a regular point of J iff z has no continuous isotropy. Motivated by this, Arms,Marsden and Moncrief [1981] showed that the level sets J−1(0) of an equivariant momentummap J have quadratic singularities at points with continuous symmetry. While such a resultis easy for compact group actions on finite dimensional manifolds, the main examples ofArms, Marsden and Moncrief [1981] were, in fact, infinite dimensional—both the phasespace and the group. Otto [1987] has shown that if G is a compact Lie group, J−1(0)/Gis an orbifold. The singular structure of J−1(µ) is closely related to important convexityproperties of the momentum map that have been studied by Guillemin, Sternberg, Atiyah,Kirwan, and many others starting in the early 1980’s. We shall not review that literaturehere.

The detailed singular stratified structure of J−1(0)/G for compact Lie groups acting onfinite dimensional manifolds was developed in Sjamaar and Lerman [1991] and extended forproper Lie group actions to J−1(Oµ)/G by Bates and Lerman [1997], if Oµ is locally closedin g∗. Ortega [1998] and Ortega and Ratiu [2001] redid the entire singular reduction theoryfor proper Lie group actions starting with the point reduced spaces J−1(µ)/Gµ and alsoconnected it to the more algebraic approach to reduction theory of Arms, Cushman andGotay [1991]. Specific examples of singular reduction and further references may be foundin Cushman and Bates [1997]. This theory is still under development.

The Method of Invariants. The method of invariants seeks to parameterize quotientspaces by group invariant functions. It has a rich history going back to Hilbert’s invariant


theory begun in the late 1800’s. It has been of great use in bifurcation with symmetry(see Golubitsky, Stewart and Schaeffer [1988] for instance). In mechanics, the method wasdeveloped by Kummer, Cushman, Rod and coworkers in the 1980’s. We will not attemptto give a literature survey here, other than to refer to the papers of Kummer [1990], Kirk,Marsden and Silber [1996], Alber, Luther, Marsden and Robbins [1998] and the book ofCushman and Bates [1997] for more details and references.

2 Geometric Mechanics and Nonholonomic Systems

Problems in nonholonomic mechanics are typified by those involving velocity dependentconstraints such as problems in robotics, wheeled vehicular dynamics and motion genera-tion. These problems involve important engineering issues such as path planning, dynamicstability, and control. When constraints are expressed in terms of distributions, then thenotion of “nonholonomic” can be precisely and easily stated in terms of integrability via theFrobenius theorem. We shall recall this below.

Nonholonomic systems come in two varieties. First, there are those with dynamic non-holonomic constraints, i.e., constraints preserved by the basic Euler-Lagrange or Hamiltonequations, such as angular momentum, or more generally momentum maps. Of course,these “constraints” are not externally imposed on the system, but rather are consequencesof the equations of motion, and so it is sometimes convenient to treat them as conservationlaws rather than constraints per se. Second, kinematic nonholonomic constraints are thoseimposed by the kinematics, such as rolling constraints, which are constraints linear in thevelocity.

Despite the long history of nonholonomic mechanics, the establishment of productivelinks with corresponding problems for systems with configuration-space constraints (i.e.,holonomic systems) has taken some time to develop. Much more attention has been givento the development of the theory for systems with holonomic constraints, such as reductiontheory, stability theory, geometric integrators, averaging theory, etc. The main purpose ofthe body of this paper is to develop an intrinsic reduction theory for nonholonomic systemswith symmetry using the bundle point of view mentioned in the preceding section.

The Lagrange-d’Alembert principle. For systems with rolling constraints or moregenerally nonholonomic systems, one finds the equations of motion and properties of thesolutions (such as the fate of conservation laws) using the Lagrange-d’Alembert princi-ple . These systems are not literally variational but the basic mechanics still comes downto F = ma.

Consider a configuration space Q and a distribution D (below this is assumed to be a(smooth) subbundle, so it has constant rank) that describes the kinematic constraints; Dis a collection of linear subspaces: Dq ⊂ TqQ, for q ∈ Q. We say that q(t) ∈ Q satisfiesthe constraints if q(t) ∈ Dq(t). This distribution is, in general, nonintegrable in the senseof Frobenius’ theorem; i.e., the constraints can be nonholonomic. Anholonomy is measuredby the curvature of D.

A simple example of a nonholonomic system is the rolling disk (see Figure 2.1). Here,the constraints of rolling without slipping define the distribution D:

x = −ψR cos φ and y = −ψR sinφ.

These equations for the velocities may be interpreted as defining a linear subspace of thetangent space to the configuration space. These linear spaces then make up the constraintdistribution D.


Figure 2.1: The rolling disk.

The dynamics of a nonholonomic system is determined by a Lagrangian L : TQ → R,usually the kinetic minus the potential energy. The basic equations of motion are given byrequiring that q(t) satisfy the constraints and that

δ

∫ b

a

L(q, q) dt = 0,

for all variations satisfying δq(t) ∈ Dq(t), a ≤ t ≤ b.Consistent with the fact that each Dq is a linear subspace, we consider only homogeneous

velocity constraints. The extension to affine constraints is straightforward, as in BKMM.Standard arguments in the calculus of variations show that this “constrained variational

principle” is equivalent to the equations

−δL :=(

d

dt

∂L

∂qi− ∂L

∂qi

)δqi = 0, (2.1)

for all variations δq such that δq ∈ Dq at each point of the underlying curve q(t). Theseequations are equivalently written as

d

dt

∂L

∂qi− ∂L

∂qi= λi,

where λi is a set of Lagrange multipliers (i = 1, . . . , n), representing the force of constraint.Intrinsically, this multiplier λ is a section of the cotangent bundle over q(t) that annihilatesthe constraint distribution. The Lagrange multipliers are often determined by using thecondition that q(t) lies in the distribution.

The Nature of the Lagrange-d’Alembert Principle. The Lagrange-d’Alembert prin-ciple is not a variational principle in the usual sense because constraints are put on thevariations. If one imposes the constraints before taking the variations (in other words, onerestricts the class of curves that are being varied), then one gets different equations — thewrong ones.

Replacement of the Lagrange-d’Alembert principle by a variational approach gives theso-called “vakanomic mechanics”. See, for example, Arnold, Kozlov and Neishtadt [1988]


and Bryant and Griffiths [1983]. While such a “truly variational” theory is relevant andinteresting for optimal control problems, it is not relevant for the dynamical equationsof nonholonomic mechanics for the sort of examples with rolling constraints consideredabove. This has been recognized in the classical literature such as Whittaker [1907], andthe Engineering literature for a long time; see, for example, Rosenberg [1977] and referencestherein.

One has to be careful reading the literature since this can cause confusion. However, thisissue of whether or not the equations of nonholonomic mechanics are variational or not wasdiscussed extensively and “put to rest” already by Korteweg [1899]. These distinctions arealso clearly discussed in Bloch and Crouch [1999]. We also refer to Lewis and Murray [1995]for an experimental verification of the Lagrange-d’Alembert principle, to Jalnapurkar [1994]for its derivation from F = ma for systems such as rolling rigid bodies, and to Martinez,Cortes and Leon [2000] for some comments on the Poisson nature of “vakanomic” systems.Thus, anyone who doubts the validity of the Lagrange-d’Alembert principle should take upthe matter with both the experimental evidence and with Mr. I. Newton.

The Roller Racer. Another example is the roller racer—it is a wheeled vehicle withtwo segments connected by a rotational joint, as in Figure 2.2.

θ

x

z

y

(x, y)φ

Figure 2.2: The roller racer—Tennessee racer.

The configuration space is given by SE(2)×S1, whose elements give the overall positionand orientation of the vehicle plus the internal shape angle φ. The constraints are definedby the condition of rolling without slipping, as in the case of the falling penny.

The roller racer is interesting because it generates locomotion. If you climb aboardand wiggle the joint, the vehicle moves!

The Rattleback. This system, also called the wobblestone, is another much studiedexample, illustrating the lack of conservation of angular momentum. This is demonstratedby the reversal of its direction of rotation when spun. General theory provides a replacementfor the conservation law from “standard” mechanics, namely the momentum equation.

Special Features of Nonholonomic Mechanics. Some of the key features of nonholo-nomic systems that set them apart from “ordinary” mechanical systems are:

1. symmetry need not lead to conservation laws, but rather lead to an interesting mo-mentum equation,


Figure 2.3: The rattleback.

2. equilibria can be stable, with some variables being asymptotically stable,

3. energy is still conserved ,

4. Jacobi’s identity for Poisson brackets can fail .

To explore the structure of the Lagrange-d’Alembert equations in more detail, let ωa,a = 1, . . . , k, be a set of k independent one forms whose vanishing describes the constraints;i.e., the distribution D. One can introduce local coordinates qi = (rα, sa), where α =1, . . . n − k, in which ωa has the form

ωa(q) = dsa + Aaα(r, s)drα,

where the summation convention is in force. We locally write the distribution as

D = (r, s, r, s) ∈ TQ | sa + Aaαrα = 0.

The equations of motion (2.1) may be rewritten by noting that the allowed variationsδqi = (δrα, δsa) satisfy δsa + Aa

αδrα = 0. Substitution into (2.1) gives

(d

dt

∂L

∂rα− ∂L

∂rα

)= Aa

α

(d

dt

∂L

∂sa− ∂L

∂sa

). (2.2)

Equation (2.2) combined with the constraint equations

sa = −Aaαrα (2.3)

gives a complete description of the equations of motion of the system; this procedure maybe viewed as one way of eliminating the Lagrange multipliers. Using this notation, one findsthat λ = λaωa, where

λa =d

dt

∂L

∂sa− ∂L

∂sa.

Equations (2.2) can be written in the following way:

d

dt

∂Lc

∂rα− ∂Lc

∂rα+ Aa

α

∂Lc

∂sa= − ∂L

∂sbBb

αβ rβ , (2.4)

where

Lc(rα, sa, rα) = L(rα, sa, rα,−Aaα(r, s)rα).


is the coordinate expression of the constrained Lagrangian defined by Lc = L|D and where

Bbαβ =

∂Abα

∂rβ−

∂Abβ

∂rα+ Aa

α

∂Abβ

∂sa− Aa

β

∂Abα

∂sa. (2.5)

Letting dωb be the exterior derivative of ωb, a computation shows that

dωb(q, ·) = Bbαβ rαdrβ

and hence the equations of motion have the form

−δLc =(

d

dt

∂Lc

∂rα− ∂Lc

∂rα+ Aa

α

∂Lc

∂sa

)δrα = − ∂L

∂sbdωb(q, δr).

This form of the equations isolates the effects of the constraints, and shows, in particular,that in the case where the constraints are integrable (i.e., dωa = 0), the equations ofmotion are obtained by substituting the constraints into the Lagrangian and then settingthe variation of Lc to zero. However in the non-integrable case the constraints generateextra (curvature) terms, which must be taken into account.

Some Geometry. The above coordinate results can be put into an interesting and usefulintrinsic geometric framework. The intrinsically given information is the distribution andthe Lagrangian. Assume temporarily that there is a bundle structure πQ,R : Q → R for ourspace Q, where R is the base manifold and πQ,R is a submersion and the kernel of TqπQ,R

at any point q ∈ Q is called the vertical space Vq. One can always do this locally. AnEhresmann connection A is a vertical valued one form on Q such that

1. Aq : TqQ → Vq is a linear map and

2. A is a projection: A(vq) = vq for all vq ∈ Vq.

Hence, TqQ = Vq ⊕ Hq where Hq = kerAq is the horizontal space at q, sometimesdenoted horq. Thus, an Ehresmann connection gives us a way to split the tangent space toQ at each point into a horizontal and vertical part.

If the Ehresmann connection is chosen in such a way that the given constraint distributionD is the horizontal space of the connection; that is, Hq = Dq, then in the bundle coordinatesqi = (rα, sa), the map πQ,R is just projection onto the factor r and the connection A canbe represented locally by a vector valued differential form ωa:

A = ωa ∂

∂sa, ωa(q) = dsa + Aa

α(r, s)drα,

and the horizontal projection is the map

(rα, sa) → (rα,−Aaα(r, s)rα).

The curvature of an Ehresmann connection A is the vertical valued two form defined byits action on two vector fields X and Y on Q as

B(X, Y ) = −A([horX, horY ])

where the bracket on the right hand side is the Jacobi-Lie bracket of vector fields and wherehorX denotes the horizontal projection of X calculated at each q ∈ Q. This definition


shows the sense in which the curvature measures the failure of the constraint distributionto be integrable.

In coordinates, one can evaluate the curvature B of the connection A by the followingformula:

B(X, Y ) = dωa(horX, horY )∂

∂sa,

so that the local expression for curvature is given by

B(X, Y )a = BaαβXαY β

where the coefficients Baαβ are given by (2.5).

The Lagrange d’Alembert equations may be written intrinsically as

δLc = 〈FL, B(q, δq)〉,

in which δq is a horizontal variation (i.e., it takes values in the horizontal space) and B isthe curvature regarded as a vertical valued two form, in addition to the constraint equationsA(q) · q = 0. Here 〈 , 〉 denotes the pairing between a vector and a dual vector and

δLc =⟨

δrα,∂Lc

∂rα− d

dt

∂Lc

∂rα− Aa

α

∂Lc

∂sa

⟩.

When there is a symmetry group G present, there is a natural bundle one can workwith and put a connection on, namely the bundle Q → Q/G. In the generality of thepreceding discussion, one can get away with just the distribution itself and can introducethe corresponding Ehresmann connection locally. In fact, the bundle structure Q → R isreally a “red herring”. The notion of curvature as a TqQ/Dq valued form makes good senseand is given locally by the same expressions as above. However, keeping in mind that weeventually want to deal with symmetries and in that case there is a natural bundle, theEhresmann assumption is nevertheless a reasonable bridge to the more interesting case withsymmetries.

Nonholonomic Reduced Equations. Reduction procedures were applied to the La-grange d’Alembert principle in BKMM. The general form of the resulting reduced equations,which we will also call the Lagrange-d’Alembert-Poincare equations is

g−1g = −A(r)r + B(r)p,

p = rT α(r)r + rT β(r)p + pT γ(r)pM(r)r = −C(r, r) + N(r, r, p) + τ

The first equation describes the motion in the group variables as the flow of a left-invariantvector field on G determined by the internal shape r, the internal velocity r, as well asthe nonholonomic momentum p, which is, roughly speaking, the component of momen-tum in the symmetry directions compatible with the constraints. The second equation is themomentum equation . Notice that the momentum equation has terms that are quadraticin r, linear in r and p and quadratic in p. The coefficients β(r) define a connection andthis term is called the transport part of the momentum equation. The curvature of thisconnection plays an important role in stability theory. The third equation describes themotion in the shape variables r. The term M(r) is the mass matrix of the system, C isthe Coriolis term and τ represent internal control forces. This framework has proven tobe useful for controllability, gait selection, and locomotion of systems like the snakeboard.

One of the goals of the present paper is to give the intrinsic geometry of the Lagrange-d’Alembert-Poincare equations, so we will return to this topic later.


A Brief History of Nonholonomic Mechanics. Of course the history of nonholonomicsystems is too vast to survey in detail here, but we shall give some highlights that are relevantto the goals of the present paper.

Some of the classical examples remain gems that have eventually led to considerableprogress. For example, Routh [1860] showed that a uniform sphere rolling on a surface ofrevolution is an integrable system (in the classical sense). This same example was studiedand the analysis completed in Zenkov [1995] using techniques that led to a nonholonomicversion of the energy-momentum method (Zenkov, Bloch and Marsden [1998]).

Another instructive example is the rolling disk (not necessarily vertical), which wastreated in Vierkandt [1892]; this classical paper shows that the solutions of the equationson what we would now call the reduced space are all periodic. For an exposition of thisexample from a more modern point of view, see, for example, the papers Getz and Marsden[1994], Hermans [1995], O’Reilly [1996], Cushman, Hermans and Kemppainen [1995] as wellas the work Cushman, Kemppainen, Sniatycki and Bates [1995]. A related, perhaps moreinteresting, example is a (much) simplified model of the bicycle studied in Getz and Marsden[1995] and Koon and Marsden [1997c].

The classical work of Chaplygin [1897a] studies the rolling of a solid of revolution ona horizontal plane. In this case, it is also true that the orbits are periodic on the reducedspace (this is proved by a technique of Birkhoff utilizing the reversible symmetry in Hermans[1995]). One should note that a limiting case of this result (when the body of revolutionlimits to a disk) is that of Vierkandt. Chaplygin [1897b, 1903] also studied the case of arolling sphere on a horizontal plane that additionally allowed for the possibility of sphereswith an inhomogeneous mass distribution.

The vertical rolling disk and the spherical ball rolling on a rotating table are examplesof systems with both dynamic and kinematic nonholonomic constraints. In either case, theangular momentum about the vertical axis is conserved; this property was exploited in, forexample, the papers Bloch, Reyhanoglu and McClamroch [1992], Bloch and Crouch [1994b],Brockett and Dai [1992], Yang, Krishnaprasad and Dayawansa [1993], and BKMM.

Another classical example is the wobblestone, studied in a variety of papers and bookssuch as Walker [1896], Crabtree [1909], Bondi [1986], and Zenkov, Bloch and Marsden[1998]. Walker establishes important stability properties of relative equilibria by a spectralanalysis; he shows, under rather general conditions (including the crucial one that the axesof principal curvature do not align with the inertia axes) that rotation in one directionis spectrally stable (and hence linearly and nonlinearly asymptotically stable). By timereversibility, rotation in the other direction is unstable. On the other hand, one can have arelative equilibrium with eigenvalues in both half planes, so that rotations in opposite sensesabout it can both be unstable, as Walker has shown. Presumably this is consistent with thefact that some wobblestones execute multiple reversals. However, the global geometry ofthis mechanism is still not fully understood analytically.

The Momentum Equation. One of the most interesting developments in this area hasbeen a nonholonomic version of the Noether theorem, leading to what has been calledthe momentum equation, the form of which was indicated above. This equation, derivedin BKMM (although it has some special cases in earlier works) gives an equation for theNoether quantity rather than a conservation law. This equation is critical in the context ofcontrol theory.

Another interesting example in this regard is the snakeboard (studied in some detail inOstrowski, Burdick, Lewis and Murray [1995], BKMM, Ostrowski, Desai and Kumar [1996]),which shares some of the features of these examples but which has a crucial difference as well.This example, like many of the others, has the symmetry group SE(2) of Euclidean motionsof the plane but, now, the corresponding momentum is not conserved. However, the equation


satisfied by the momentum associated with the symmetry is useful for understanding thedynamics of the problem and how group motion can be generated. The nonconservationof momentum occurs even with no forces applied (besides the forces of constraint) and isconsistent with the conservation of energy for these systems. In fact, nonconservation iscrucial to the generation of movement in a control-theoretic context.

Nonholonomic Reduction. As we have indicated already, one of the important tools ofgeometric mechanics is reduction theory (either Lagrangian or Hamiltonian), which providesa well-developed method for dealing with dynamic constraints. In this theory the dynamicconstraints and the symmetry group are used to lower the dimension of the system byconstructing an associated reduced system. Koiller [1992] was one of the first papers inwhich a systematic theory of nonholonomic reduction was begun. A Hamiltonian version ofthis theory was developed in Bates and Sniatycki [1993] while a Lagrangian version of thistheory was given in BKMM. Links between these theories and further developments weregiven in Koon and Marsden [1997b, c, 1998]. The Lagrangian point of view is a convenientcontext for applications to control theory.

Other Work on the Geometry of Nonholonomic Systems. There have been manyworks on the geometry of nonholonomic systems and reduction theory from many points ofview. We cannot survey them in a comprehensive way here, but we would like to mention thefollowing works: Chaplygin [1897a, 1897b, 1903, 1911, 1949, 1954], Cartan [1928], Neimarkand Fufaev [1972], Rosenberg [1977], Weber [1986], Vershik and Gershkovich [1994], Koiller[1992], Bloch and Crouch [1992], Yang, Krishnaprasad and Dayawansa [1993], van der Schaftand Maschke [1994], Marle [1995, 1998], Udwadia and Kalaba [1996], Cantrijn, de Leon,Marrero and Martin de Diego [1998], and Cantrijn, Cortes, de Leon and Martin de Diego[2000].

Applications of Reduction Theory. Reduction theory is important for many rea-sons. For example, it provides a context for understanding the theory of geometric (orHannay-Berry) phases (see the papers Krishnaprasad [1989], Marsden, Montgomery andRatiu [1990], Bloch, Krishnaprasad, Marsden and Sanchez de Alvarez [1992] and referencestherein). This is especially important for understanding locomotion generation in the non-holonomic context. An additional useful object that was introduced in BKMM in this regardis the nonholonomic connection, a nonholonomic analog of the mechanical connection. Wehave already mentioned the applications to the nonholonomic energy-momentum method.Lagrangian reduction theory has also had a significant impact on optimal control theory.See Vershik and Gershkovich [1994], Bloch and Crouch [1994a], Montgomery [1990, 1993],Koon and Marsden [1997a], and references therein. For a general overview of some of theapplications and for further references to locomotion and controllability issues, see Marsdenand Ostrowski [1998], Marsden [1999], and Bloch and Crouch [1998, 1999].

Goal of the Remainder of this Paper. In this paper we study geometric aspects ofbasic fundamental results in nonholonomic mechanics, with an emphasis on the role ofsymmetry. In particular, we provide a geometric formulation of the Lagrange-d’Alembert-Poincare equations.

3 The Lagrange-d’Alembert Principle with Symmetry

The Lagrange-d’Alembert Principle. As in the introductory sections, let Q be a man-ifold and let D be a distribution on Q, that is, D is a subbundle of TQ, and let L : TQ → R


be a given Lagrangian. We can state the Lagrange-d’Alembert principle as follows:

A curve q(t) ∈ Q, t ∈ [t0, t1], is an actual motion of the system if and only ifq(t) ∈ Dq(t) for all t and, besides, for any deformation q(t, λ) of q(t) such thatthe corresponding variation

δq(t) =∂q(t, λ)

∂λ

∣∣∣λ=0

satisfies δq(t) ∈ Dq(t) for all t, the following condition holds

δ

∫ t1

t0

L(q, q)dt = 0.

Here we use the standard notation in the calculus of variations, namely,

δ

∫ t1

t0

L(q, q)dt :=∂

∂λ

∣∣∣λ=0

∫ t1

t0

L (q(t, λ), q(t, λ)) dt.

Using the Lagrange-d’Alembert Principle, equations of motion can be derived and con-veniently written using an Ehresmann connection adapted to the given system in such away that D coincides with the horizontal distribution of the Ehresmann connection, as wasdescribed in the introductory sections.

Next, we shall describe the Euler-Lagrange operator and Euler-Lagrange equations as wedid in CMR. We shall begin by summarizing the definition of higher order tangent bundlesand connection-like structures defined on them.

The kth Order Tangent Bundle. First, we shall recall the definition of the kth-ordertangent bundle τ

(k)Q : T (k)Q → Q. For q ∈ Q, elements of T

(k)q Q are equivalence classes

of curves in Q, namely, two given curves qi(t), i = 1, 2, such that q1(t1) = q2(t2) = q areequivalent, by definition, if and only if in any local chart we have q

(l)1 (t1) = q

(l)2 (t2), for

l = 1, 2, . . . , k, where q(l) denotes the derivative of order l. The equivalence class of thecurve q(t) at q = q(t) will be denoted [q](k)

q . The projection

τ(k)Q : T (k)Q → Q is given by τ

(k)Q

([q](k)

q

)= q.

It is clear that T (0)Q = Q, T (1)Q = TQ, and that, for l < k, there is a well defined fiberbundle structure

τ(l,k)Q : T (k)Q → T (l)Q given by τ

(l,k)Q

([q](k)

q

)= [q](l)q .

The bundles T (k)Q for k > 1 are not vector bundles, except for k = 1. The bundle T (2)Qis often denoted Q, and is called the second order bundle (see, for example, Marsden,Patrick and Shkoller [1998], Marsden and Ratiu [1999] and references therein).

It is worth noting here that in field theory one uses the bundle R × Q → R, whosesections are curves in Q (the fields of classical mechanics). Then the k-jet bundle of thistrivial bundle over the time axis may be identified with the bundle R × T (k)Q → R × Q,where the first component of this map is the identity. From the point of view of jet bundlesassociated to maps between two manifolds, T (k)Q coincides with the fiber bundle Jk

0 (R, Q)formed by k-jets of curves from R to Q (based at 0 ∈ R), as defined, for example in Bourbaki[1983].


For any map f : M → N we have a naturally induced map

T (k)f : T (k)M → T (k)N given by T (k)f([q](k)

q

)= [f q](k)

f(q).

In particular, a group action ρ : G × Q → Q can be naturally lifted to a group action

ρ(k) : G × T (k)Q → T (k)Q given by ρ(k)g

([q](k)

q

)= [ρg q](k)

ρ(g,q .

We will often denote ρ(k)g

([q](k)

q

)= ρ(k)

(g, [q](k)

q

)= g[q](k)

q .Moreover, if π : Q → Q/G is a principal bundle with structure group G, then, with the

lifted action, T (k)Q is also a principal bundle with structure group G. Let M × N be theCartesian product of the manifolds M and N . Then, for any (m, n) ∈ M × N there is anatural identification T

(k)(m,n)((M × N) ≡ T

(k)m M × T

(k)n N , which induces an identification

T (k)(M × N) ≡ T (k)M × T (k)N .Let π : Q → Q/G be a principal bundle and consider the lifted principal bundle structure

T (k)Q with structure group G. The quotient T (k)Q/G can be easily shown to be a fiberbundle over the base Q/G. The bundle T (2)Q/G is the one that interests us most inthis paper, because the Lagrange-Poincare operator of a reduced Lagrangian is defined onT (2)Q/G. The class of the element [q](k)

q in the quotient T (k)Q/G will be denoted[[q](k)

q

]G

.

Since we have the projection π : Q → Q/G we obtain a bundle map.

T (k)πG(Q) : T (k)Q → T (k)(Q/G).

Moreover, it can be easily shown that this bundle map induces a well defined bundle map

T (k)Q/G → T (k)(Q/G) given by[[q](k)

q

]G→ T (k)π

([q](k)

q

).

Now assume that a principal connection A is given on the principal bundle Q → Q/G.Let [x](k)

x ∈ T (k) (Q/G) and also q ∈ Q such that π(q) = [q]G = x be given. Let x(t) be anycurve belonging to the class of [x](k)

x . Then there is a unique horizontal lift xhq of x(t). We

define the horizontal lift of [x](k)x at q by

[x](k),hx,q :=

[xh

q

](k)

q.

We must also remark that T (k)G carries a natural Lie group structure.1 If [g](k)g , and

[h](k)

hare classes of curves g and h in G, we define the product [g](k)

g [h](k)

has being the class

[gh](k)

ghat the point gh of the curve gh. The Lie algebra TeT

(k)G of T (k)G can be naturallyidentified, as a vector space, with (k + 1)g, which, therefore, carries a unique Lie algebrastructure such that this identification becomes a Lie algebra isomorphism. There is also anatural identification of T

(k)e G with kg.

Also, for k = 1, 2, . . . , T (k)Q is a principal bundle with structure group T (k)G in a naturalway. More precisely, if [g](k)

g ∈ T (k)G is the class of a curve g in G and [q](k)q ∈ T

(k)q Q is the

class of a curve q in Q we let [g](k)g [q](k)

q ∈ T(k)gq Q denote the class [gq](k)

gq of the curve gq atthe point gq. In particular, if ξ ∈ kg and [q](k)

q ∈ T (k)Q are given, there is a well definedelement ξ[q](k)

q ∈ T (k)Q.

1Recall that T (1)G = TG is the semidirect product G g and its Lie algebra is 2g which as a vectorspace is g⊕g and it carries the Lie algebra structure of the semidirect product g g where the second factoris regarded as the representation space of the adjoint action. We will not need, or study, the Lie groupstructure of T (k)G in this paper, although this would be interesting to do.


Connection-like Structures on Higher Order Tangent Bundles. Throughout thispaper, following CMR, we shall denote by g the adjoint bundle (Q× g)/G where the actionof G on g is the adjoint action. The principal connection A on Q naturally induces an affineconnection ∇A on the vector bundle g whose covariant derivative is described below. Fora curve q(t) in Q and a curve ξ(t) in g we consider the curve [q(t), ξ(t)]G. The covariantderivative of this curve is given by the formula (see Lemma 2.3.4 of CMR)

D[q(t), ξ(t)]GDt

=[q(t),− [A (q(t), q(t)) , ξ(t)] + ξ(t)

]G

. (3.1)

For the curve [q(t), v(t)]G in g, where v(t) = A(q(t), q(t)) the above formula becomes

D[q(t), v(t)]GDt

= [q(t), v(t)]G .

Using (3.1), the second covariant derivative of [q(t), v(t)]G, is given by

D2[q(t), v(t)]GDt2

= [q(t),− [v(t), v(t)] + v(t)]G .

More generally, for each k = 1, 2, . . . , we can find, by induction, a curve vk(t) in g, havingan expression that involves v(t) and the derivatives v(l)(t), l = 1, 2, . . . , k − 1, such that

Dk−1[q(t), v(t)]GDtk−1

= [q(t), vk(t)]G .

More precisely, we have v1 = v and vk+1 = −[v, vk] + vk, for k = 1, 2, . . . . In particular,we obtain

v2(t) = v(t), v3(t) = − [v(t), v(t)] + v(t),

etc. In addition, we shall write, by definition, v0(t) = 0. Using the fact that v(t) =A (q(t), q(t)), we can also find expressions for vk(t) in coordinates in terms of q(l)(t), l =1, 2, . . . , k. We state the following lemma, which is readily proved.

Lemma 3.1. Let q(t) be a given curve in Q such that q(t) = q. For each k = 1, 2, . . . theformula

Ak

([q](k)

q

)= vk(t)

gives a well defined map Ak : T (k)Q → g. Therefore there is also a well defined mapAk : T (k)Q → kg, given, for each k = 1, 2, . . . , by Ak

([q](k)

q

)= ⊕k

l=1vl(t), where we have

written kg to stand for the vector space direct sum ⊕kl=1g of k copies of g.

Let g ∈ G and [q](k)q ∈ T

(k)q Q be given. Then we can easily prove that

Ak

(g[q](k)

q

)= Adg Ak

([q](k)

q

),

using induction, the definition of Ak, and taking into account the formulas

Adg vk =d

dtAdg vk

and Adg[v, vk] = [Adg v,Adg vk] .


For any curve q(t) in Q such that q(t) = q we have A(q, q) = gqg−1q , where gq is a curve

in G defined as follows. Let x(t) = π (q(t)). Let xhq (t) be the horizontal lift of x(t) such that

xhq (t) = q. Then the curve gq(t) is uniquely determined by the condition q(t) = gq(t)xh

q (t).The above remark and the inductive definition of vk inductively gives an expression for

vk(t) in terms of g(l)q (t), for l = 1, 2, . . . . For instance, for the case of matrix groups, we can

see directly that v1(t) = v(t) = gq(t), v2(t) = gq(t) − gq(t)2, etc. It is not difficult to seethat the expression of vk(t) is the sum of g

(k)q (t) plus terms involving only the lower order

derivatives g(l)q (t), l = 1, 2, . . . , k − 1. Using this, one can easily see that, given any element

ξ = (ξ1, . . . ξk) belonging to kg and any element q ∈ Q one can find a unique [q](k)q of the

form [gq](k)q where g(t) is a curve in G such that

g(t) = e and Ak

([q](k)

q

)= ξ.

In fact, since we obviously have gq(t) = g(t) it is enough to find g(t) such that the derivativesg(l)(t), l = 1, 2, . . . , k, satisfy the appropriate conditions as explained above. We shall callthis unique element ξq and the set of all such elements will be called kgq := gq ⊕ · · · ⊕ gq(k times). Moreover, it is not difficult to see that the restriction Ak : kgq → kg is adiffeomorphism, and therefore it naturally defines a unique vector space structure on kgqsuch that the restriction of Ak becomes a linear isomorphism, given by Ak(ξq) = ξ, for allξ ∈ kg.

By construction we see that there is a natural identification between kgq and T(k)q (Gq).

Let us define, for each k = 1, 2, . . . , the vector bundle kg as being the Whitney sum ofk copies of g. Define a map

T (k)Q → kg by [q](k)q →

[q, Ak

([q](k)

q

)]G

,

where the last term is defined by[q, Ak

([q](k)

q

)]G

= ⊕kl=1

[q, Al

([q](l)q

)]G

.

Using the preceding, it can be easily deduced that, given any curve q(t) in Q such thatq(t) = q, we have, at t = t,

[q, Ak

([q](k)

q

)]G

= ⊕kl=1

D(l−1)[q(t), v(t)]GDt(l−1)

∣∣∣∣t=t

.

The above is the essence of the proof of the following:

Proposition 3.2. The map αAk: T (k)Q/G → T (k)(Q/G) ×Q/G kg defined by

αAk

([[q](k)

q

]G

)= T (k)πG(Q)

([q](k)

q

)×Q/G

[q, Ak

([q](k)

q

)]G

is a well defined bundle isomorphism. The inverse of αA is given by

α−1Ak

([x](k)

x ×Q/G [q, ξ]G)

= ξ[x](k),hx,q .


The Euler-Lagrange Operator. Now we recall the definition of the Euler-Lagrangeoperator following the methods of CMR.

Theorem 3.3 (Euler-Lagrange). Let L : TQ → R be a given Lagrangian on a manifoldQ and let

S(L)(q) =∫ t1

t0

L(q, q) dt

be the action of L defined for curves q(t) in Q. Let q(t, λ) be a deformation of a curveq(t), let δq(t) be the corresponding variation and assume that δq(ti) = 0 for i = 0, 1. Thenthere is a unique bundle map EL(L) : T (2)Q → T ∗Q such that, for any deformation q(t, λ),keeping the endpoints fixed we have

dS(L)(q) · δq =∫ t1

t0

EL(L)(q, q, q) · δq,

where, as usual,

dS(L)(q) · δq =d

dλS(L) (q(t, λ))

∣∣∣∣λ=0

with δq(t) =∂q(t, λ)

∂λ

∣∣∣∣λ=0

.

The 1-form valued map EL(L) is called the Euler-Lagrange operator.

The Lagrange-d’Alembert Operator. Motivated by the preceding result, we now de-fine the Lagrange-d’Alembert operator and Lagrange-d’Alembert equation.

Definition 3.4. Let D ⊂ TQ be a subbundle of TQ. Then there is a natural projection ofvector bundles i∗D : T ∗Q → D∗, namely, the dual of the natural injection of vector bundlesiD : D → TQ. Let us define the subset T

(2)D Q ⊂ T (2)Q as being the subset of all [q](2)q such

that [q](1)q ∈ D and let i(2)D : T(2)D Q → T (2)Q be the natural inclusion.

The Lagrange-d’Alembert operator is the operator

LD(L) : T(2)D Q → D∗ defined by LD(L) = i∗D EL(L) i(2)D .

The Lagrange-d’Alembert equation is the equation LD(L) = 0.

The next theorem generalizes Theorem 3.3 in the sense that the latter consists of theparticular case in which D = TQ.

Theorem 3.5 (Lagrange-d’Alembert). Let L : TQ → R be a given Lagrangian on amanifold Q and let D be a subbundle of TQ. Let

S(L)(q) =∫ t1

t0

L(q, q) dt

be the action of L. Then the Lagrange-d’Alembert operator LD(L) : T(2)D Q → D∗ satisfies

dS(L)(q) · δq =∫ t1

t0

LD(L)(q, q, q) · δq =∫ t1

t0

EL(L)(q, q, q) · δq

where, as usual,

dS(L)(q) · δq =d

dλS(L) (q(t, λ))

∣∣∣∣λ=0

,

for all curves q(t) and variations δq(t) such that (q(t), q(t)) ∈ Dq(t), δq(t) ∈ Dq(t), for all t,and δq(ti) = 0, i = 1, 2.


The proof of this theorem is straightforward. It consists in applying the usual integrationby parts argument as in the proof of Theorem 3.3 (see for instance CMR), for the case ofcurves q(t) and variations δq(t) that satisfy the constraint D. We shall therefore omit thedetails.

Nonholonomic Systems with Symmetry. Now we shall assume that π : Q → Q/Gis a principal bundle with structure group G and we denote by V the vertical distribution,that is, Vq = Tq

(π−1([q]G)

), for each q ∈ Q, which is obviously an integrable distribution

whose integral manifolds are the group orbits π−1([q]G).Following BKMM, let us consider the following condition:

(A1) Dimension Assumption. For each q ∈ Q, we have TqQ = Dq + Vq.

It is easy to see that, under assumption (A1), the dimension of the space Sq = Dq ∩ Vq

does not depend on q ∈ Q, and moreover, the collection of spaces Sq, q ∈ Q, is a subbundleof D, of V, and of TQ.

Now let us now consider the following condition

(A2) Invariance of D. The distribution D is G-invariant.

It follows from (A1) and (A2) that S is a G-invariant distribution.

The Nonholonomic Connection. It is known and easy to prove that there is always aG-invariant metric on Q. See, for example, CMR, §6.3.

In many important physical examples there is a natural way of choosing an invariantmetric, representing, for instance, the inertia tensor of the system, see BKMM. Let usassume condition (A1). We can then define the principal connection form A : TQ → g suchthat the horizontal distribution HorA TQ satisfies the condition that, for each q, the spaceHorA TqQ coincides with the orthogonal complement Hq of the space Sq in Dq.

This connection is called the nonholonomic connection. For each q ∈ Q, let usdenote Uq the orthogonal complement of Sq in Vq. Then it is easy to see that U is a smoothdistribution and we have the Whitney sum decomposition

TQ = H⊕ S ⊕ U .

We obviously haveD = H⊕ S and V = S ⊕ U .

Under the additional assumption (A2), all three distributions H, S, and U are G-invariant, so we can write

TQ/G = H/G ⊕ S/G ⊕ U/G.

The Geometry of the Reduced Bundles. Recall from CMR (see also the paragraphConnection-like Structures on Higher Order Tangent Bundles) that there is avector bundle isomorphism

αA : TQ/G → T (Q/G) ⊕ g,

where g is the adjoint bundle of the principal bundle Q, defined as follows

αA[q, q]G = Tπ(q, q) ⊕ [q, A(q, q)]G.


Notice that the bundle T (Q/G) ⊕ g does not depend on the connection A; however thevector bundle isomorphism αA does depend on A. It is easy to see that

αA(H/G) = T (Q/G), and αA(V/G) = g.

Define the subbundles s and u of g by

s = αA(S/G) and u = αA(U/G),

respectively. Clearly, we have, g = s ⊕ u.

Necessary and Sufficient Condition for the Constraints to be Holonomic. Weremark that our conventions for the curvature B of the principal connection A are givenby Cartan’s structure equation dA(u, v) = B(u, v) + [A(u), A(v)]. Recall that we define theg valued 2-form B on the base Q/G by B([q]G)(X, Y ) = [q, B(Xh(q), Y h(q))]G, where Xh

and Y h are the horizontal lifts to Q of the vector fields X and Y on Q/G. Also recall thatwe denoted by ∇A the affine connection naturally induced on the vector bundle g by theprincipal connection A, defined in the paragraph Connection-like Structures on HigherOrder Tangent Bundles. Let Xi, i = 1, 2, be given invariant vector fields on Q. LetαA

([Xi]G

)= Xi⊕ ξi, i = 1, 2. A calculation given in CMR, shows that one has the following

formula for the Lie bracket on sections of Lagrange-Poincare bundles:

[X1 ⊕ ξ1, X2 ⊕ ξ2] = [X1, X2] ⊕ ∇AX1

ξ2 − ∇AX2

ξ1 − B(X1, X2) + [ξ1, ξ2],

where, by definition,

[X1 ⊕ ξ1, X2 ⊕ ξ2] = αA

([[Xi, Xi]

]G

).

Using this formula we can easily prove the following theorem.

Theorem 3.6. A given constraint distribution D satisfying assumptions (A1) and (A2) isholonomic if and only if for any given sections Xi⊕ ξi, i = 1, 2, of T (Q/G)⊕ s the followingconditions are satisfied:

(i) [ξ1, ξ2] ∈ s; (ii) ∇AX1

ξ2 ∈ s; (iii) B(X1, X2) ∈ s.

Lagrange-Poincare Operators. Now we shall recall the definition and basic propertiesof the Lagrange-Poincare operators introduced in CMR; we refer to this paper for the proofsof the statements below.

Theorem 3.7. Let L : TQ → R be an invariant Lagrangian on the principal bundle Q.Choose a principal connection A on Q and identify the bundles TQ/G and T (Q/G) ⊕ g

using the isomorphism αA and also the bundles T (2)Q/G and T (2)(Q/G)×Q/G 2g using theisomorphism αA2 . Thus an element [q, q]G of TQ/G can be written, equivalently, as anelement (x, x, v) of T (Q/G) ⊕ g. Let l : T (Q/G) ⊕ g → R be the reduced Lagrangian. Thenthere is a unique bundle map

LP(l) : T (2)(Q/G) ×Q/G 2g → T ∗(Q/G) ⊕ g∗

such that for any curve q ∈ Ω(Q; q0, q1) and any variation δq of q vanishing at the endpoints,the corresponding reduced curve [q, q]G = (x, x, v), where v = [q, A(q, q)]G, and covariantvariation δx ⊕ δAv, where

δAv(t) =Dη

Dt(t) + [v(t), η(t)] + B(δx(t), x(t)),

with η(t) = [q(t), η(t)]G and δx(t) = Tπ(δq(t)), satisfy

EL(L)(q(t), q(t), q(t)) · δq(t) = LP(l)(x(t), x(t), v(t)) · (δx(t) ⊕ η(t)).


Definition 3.8. The bundle map

LP(l) : T (2)(Q/G) ×Q/G 2g → T ∗(Q/G) ⊕ g∗

defined in the preceding theorem will be called the Lagrange-Poincare operator. Thedecomposition of the range space for LP(l) as a direct sum naturally induces a decompositionof the Lagrange-Poincare operator

LP(l) = Hor(LP)(l) ⊕ Ver(LP)(l)

which define the horizontal Lagrange-Poincare operator and the vertical Lagrange-Poincare operator. The Lagrange-Poincare equations are, by definition, the equationsLP(l) = 0. The horizontal Lagrange-Poincare equation and vertical Lagrange-Poincare equation are, respectively, the equations Hor(LP)(l) = 0 and Ver(LP)(l) = 0.

Reduced Covariant Derivatives. The question of calculating formulas for Hor(LP)(l)and Ver(LP)(l) rests on giving meaning to the partial derivatives ∂l

∂x , ∂l∂x and ∂l

∂v . Since g

and T (Q/G) are vector bundles, we may interpret the last two derivatives in a standard(fiber derivative) way as being elements of the dual bundles T ∗(Q/G) and g∗, for each choiceof (x, x, v) in T (Q/G) ⊕ g. In other words, for given (x0, x0, v0) and (x0, x

′, v′) we define

∂l

∂x(x0, x0, v0) · x′ =

d

ds

∣∣∣∣s=0

l(x0, x0 + sx′, v0)

and

∂l

∂v(x0, x0, v0) · v′ =

d

ds

∣∣∣∣s=0

l(x0, x0, v0 + sv′).

To define the derivative ∂l/∂x, one needs to chose a connection ∇ on the manifoldQ/G, as we will explain next. Let (x0, x0, v0) be a given element of T (Q/G) ⊕ g. For anygiven curve x(s) on Q/G, let (x(s), v(s)) be the horizontal lift of x(s) with respect to theconnection ∇A on g such that (x(0), v(0)) = (x0, v0) and let (x(s), u(s)) be the horizontal liftof x(s) with respect to the connection ∇ such that (x(0), u(0)) = (x0, x0). (Notice that ingeneral, (x(s), u(s)) is not the tangent vector (x(s), x(s)) to x(s).) Thus, (x(s), u(s), v(s))is a horizontal curve with respect to the connection C = ∇ ⊕ ∇A naturally defined onT (Q/G) ⊕ g in terms of the connection ∇ on T (Q/G) and the connection ∇A on g.

Definition 3.9. The covariant derivative of l with respect to x at (x0, x0, v0) in thedirection of (x(0), x(0)) is defined by

∂C l

∂x(x0, x0, v0) (x(0), x(0)) =

d

ds

∣∣∣∣s=0

l (x(s), u(s), v(s)) .

We shall often write ∂C l∂x ≡ ∂l

∂x , if no confusion is possible.

The covariant derivative on a given vector bundle, for instance g, induces a correspondingcovariant derivative on the dual bundle, in our case g∗. More precisely, let α(t) be a curvein g∗. We define the covariant derivative of α(t) in such a way that for any curve v(t) on g

such that both α(t) and v(t) project on the same curve x(t) on Q/G, we have

d

dt〈α(t), v(t)〉 =

⟨Dα(t)

Dt, v(t)

⟩+

⟨α(t),

Dv(t)Dt

⟩.


Likewise we can define the covariant derivative in the vector bundle T ∗(Q/G). Then weobtain a covariant derivative on the vector bundle T ∗(Q/G) ⊕ g∗.

It is in the sense of this definition that terms like DDt

∂l∂x in the second equation (which

defines the horizontal Lagrange-Poincare operator) and DDt

∂l∂v in the first equation (which

defines the vertical Lagrange-Poincare operator) of the following theorem should be inter-preted. In this case D/Dt means the covariant derivative in the bundle T ∗(Q/G). In thefirst equation D/Dt is the covariant derivative in the bundle g∗.

Theorem 3.10. Under the hypothesis of Theorem 3.7 we have the following:The vertical Lagrange-Poincare operator is given by

Ver(LP)(l) · η =(− D

Dt

∂l

∂v(x, x, v) + ad∗

v

∂l

∂v(x, x, v)

)· η

or simply,

Ver(LP)(l) = − D

Dt

∂l

∂v(x, x, v) + ad∗

v

∂l

∂v(x, x, v)

and the horizontal Lagrange-Poincare operator is given by

Hor(LP)(l) · δx =(

∂C l

∂x(x, x, v) − D

Dt

∂l

∂x(x, x, v)

)δx − ∂l

∂v(x, x, v)B(x)(x, δx)

or simply,

Hor(LP)(l) =∂C l

∂x(x, x, v) − D

Dt

∂l

∂x(x, x, v) − ∂l

∂v(x, x, v)B(x)(x, ·).

The next theorem summarizes Theorems 3.7 and 3.10.

Theorem 3.11. Let π : Q → Q/G be a principal bundle with structure group G actingon the left and let L : TQ → R be an invariant Lagrangian. The following conditions areequivalent:

(i) Hamilton’s principle holds: The curve q(t) is a critical point of the action functional∫ t1

t0

L(q, q)dt

on the space of all curves Ω(Q; q0, q1) in Q connecting q0 to q1, that is,

δ

∫ t1

t0

L(q, q)dt = 0

for arbitrary variations δq of the curve q such that δq(ti) = 0, for i = 0, 1.

(ii) The reduced variational principle holds: The curve x(t) ⊕ v(t) is a critical pointof the action functional

∫ t1

t0

l (x(t), x(t), v(t)) dt

on the reduced family of curves αA ([Ω(Q; q0, q1)]G), that is,

δ

∫ t1

t0

l (x(t), x(t), v(t)) dt = 0,


for variations δx ⊕ δAv of the curve x(t) ⊕ v(t), where δAv has the form

δAv =Dη

Dt+ [v, η] + B(δx, x),

with the boundary conditions δx(ti) = 0 and η(ti) = 0, for i = 0, 1.

(iii) The following vertical Lagrange-Poincare equations, corresponding to verticalvariations, hold:

D

Dt

∂l

∂v(x, x, v) = ad∗

v

∂l

∂v(x, x, v)

and the horizontal Lagrange-Poincare equations, corresponding to horizontalvariations, hold:

∂C l

∂x(x, x, v) − D

Dt

∂l

∂x(x, x, v) =

⟨∂l

∂v(x, x, v), ixB(x)

⟩.

In statement (ii), η can be always written η = [q, η]G, and the condition η(ti) = 0 fori = 0, 1, is equivalent to the condition η(ti) = 0 for i = 0, 1. Also, if x(t) = [q]G andv = [q, v]G where v = A (q, q), then variations δx ⊕ δAv such that

δAv =Dη

Dt+ [v, η] ≡ D[q, η]G

Dt+ [q, [v, η]]G

with η(ti) = 0 (or, equivalently, η(ti) = 0) for i = 0, 1, correspond exactly to verticalvariations δq of the curve q such that δq(ti) = 0 for i = 0, 1, while variations δx⊕ δAv suchthat δAv = B(δx, x) with δx(ti) = 0 for i = 0, 1, correspond exactly to horizontal variationsδq of the curve q such that δq(ti) = 0.

The Lagrange-d’Alembert-Poincare Operator. Next we shall see how the previousresults can be generalized for systems with nonholonomic constraints.

The next lemma can be easily proven using the results on higher order tangent bundlesand connection-like structures defined on them summarized before Theorem 3.3.

Lemma 3.12. Assume (A1) and (A2). Then

αA2

(T

(2)D Q

)= T (2)(Q/G) ×Q/G (2g)s,

where, by definition, (2g)s = s ⊕ Ds. In the above direct sum, the fiber of the vector bundleDs at the base point x = [q]G, is the vector space Dsx consisting of all vectors of the formDξ(t)

Dt

∣∣∣t=0

, where ξ(t) is a curve in s such that the base point of ξ(t) at t = 0 is x.

Definition 3.13. Leti∗s : T ∗(Q/G) ⊕ g

∗ → T ∗(Q/G) ⊕ s∗

be the natural projection, namely, the dual of the natural inclusion

is : T (Q/G) ⊕ s → T (Q/G) ⊕ g,

and leti(2)s

: T (2)(Q/G) ×Q/G (2g)s → T (2)(Q/G) ×Q/G 2g,


be the natural inclusion. The Lagrange-d’Alembert-Poincare operator is the operator

LDP(l) : T(2)D (Q/G) ×Q/G (2g)s → T ∗(Q/G) ⊕ s

∗,

defined by

LDP(l) = i∗s LP(l) i(2)s

The Lagrange-d’Alembert-Poincare equation is the equation LDP(L) = 0.

Now we shall state one of our main results.

Theorem 3.14. Assume (A1) and (A2) and let L : TQ → R be a G-invariant Lagrangian.Choose a principal connection A on Q and identify the bundles D/G and T (Q/G)⊕ s usingthe isomorphism αA and also the bundles T

(2)D Q/G and T (2)(Q/G)×Q/G (2g)s using the iso-

morphism αA2 . Thus, in particular, an element [q, q]G of D/G can be written, equivalently,as an element (x, x, v) of T (Q/G) ⊕ s. Let l : T (Q/G) ⊕ g → R be the reduced Lagrangian.Then the Lagrange-d’Alembert-Poincare operator


∗,

satisfies the condition that, for any curve q satisfying the constraints, which means that(q(t), q(t)) ∈ Dq(t), for all t, and any variation δq of q vanishing at the endpoints, and alsosatisfying the constraints, that is, δq(t) ∈ Dq(t), for all t, the corresponding reduced curve[q, q]G = (x, x, v), where v = [q, A(q, q)]G, and covariant variation δx ⊕ δAv, where

δAv(t) =Dη

Dt(t) + [v(t), η(t)] + B(δx(t), x(t)),

with η(t) = [q(t), η(t)]G ∈ Sq(t), for all t, and δx(t) = Tπ(δq(t)), we have

EL(L)(q(t), q(t), q(t)) · δq(t) = LDP(l)(x(t), x(t), v(t)) · (δx(t) ⊕ η(t))

for all t.

The proof of this theorem is entirely similar to the proof of Theorem 3.7, keeping trackof the conditions imposed on the curves and on the variations by the constraint.

Definition 3.15. The Lagrange-d’Alembert-Poincare operator


∗,

given the decomposition of its range space as a direct sum, naturally decomposes as

LDP(l) = Hor(LDP)(l) ⊕ Ver(LDP)(l),

which defines the horizontal Lagrange-d’Alembert-Poincare operator and the verti-cal Lagrange-d’Alembert-Poincare operator. The horizontal Lagrange-d’Alembert-Poincare equation and vertical Lagrange-d’Alembert-Poincare equation are theequations Hor(LDP)(l) = 0 and Ver(LDP)(l) = 0.

We will omit the proof of the following theorem, which is entirely similar to the proof ofTheorem 3.10, keeping track of the conditions imposed by the constraints.


Theorem 3.16. Under the hypothesis of Theorem 3.14 we have the following:The vertical Lagrange-d’Alembert-Poincare operator is given by

Ver(LDP)(l) · η =(− D

Dt

∂l

∂v(x, x, v) + ad∗

v

∂l

∂v(x, x, v)

)· η,

where v ∈ s and η ∈ s, or, simply,

Ver(LDP)(l) =(− D

Dt

∂l

∂v(x, x, v) + ad∗

v

∂l

∂v(x, x, v)

)∣∣∣∣s

.

The horizontal Lagrange-d’Alembert-Poincare operator is given by

Hor(LDP)(l) · δx =(

∂C l

∂x(x, x, v) − D

Dt

∂l

∂x(x, x, v)

)δx − ∂l

∂v(x, x, v)B(x)(x, δx),

where v ∈ s, or simply,

Hor(LDP)(l) =∂C l

∂x(x, x, v) − D

Dt

∂l

∂x(x, x, v) − ∂l

∂v(x, x, v)B(x)(x, .)

Summary. The next theorem is the main result of this section and it summarizes theprevious results. It contains Theorem 3.11 as the particular case in which D = TQ.

Theorem 3.17. Let q(t) be a curve in Q such that (q(t), q(t)) ∈ Dq(t) for all t and let(x(t), x(t), v(t)) = αA ([q(t), q(t)]G) be the corresponding curve in T (Q/G)⊕s. The followingconditions are equivalent.

(i) The Lagrange-d’Alembert principle holds:

δ

∫ t1

t0

L(q, q)dt = 0

for variations δq of q(t) such that δq(ti) = 0, for i = 0, 1, and δq(t) ∈ Dq(t) for all t.

(ii) The reduced Lagrange-d’Alembert principle holds: The curve x(t)⊕ v(t) satisfies

δ

∫ t1

t0

l (x(t), x(t), v(t)) dt = 0,

for variations δx ⊕ δAv of the curve x(t) ⊕ v(t), where δAv has the form

δAv =Dη

Dt+ [v, η] + B(δx, x),

with boundary conditions δx(ti) = 0 and η(ti) = 0, for i = 0, 1, and where η(t) ∈ sx(t).

(iii) The following vertical Lagrange-d’Alembert-Poincare equations, correspondingto vertical variations, hold:

D

Dt

∂l

∂v(x, x, v) = ad∗

v

∂l

∂v(x, x, v)

and the horizontal Lagrange-d’Alembert-Poincare equations, corresponding tohorizontal variations, hold:

∂C l

∂x(x, x, v) − D

Dt

∂l

∂x(x, x, v) =

⟨∂l

∂v(x, x, v), ixB(x)

⟩.


In part (ii), if v = [q, v]G with v = A(q, q) then η can be always written η = [q, η]G, andthe condition η(ti) = 0 for i = 0, 1, is equivalent to the condition η(ti) = 0 for i = 0, 1.Also, if x(t) = [q]G and v = [q, v]G where v = A (q, q), then variations δx ⊕ δAv such that

δAv =Dη

Dt+ [v, η] ≡ D[q, η]G

Dt+ [q, [v, η]]G

with η(ti) = 0 (or, equivalently, η(ti) = 0) for i = 0, 1, and η(t) ∈ sx(t) correspond exactlyto vertical variations δq of the curve q such that δq(ti) = 0 for i = 0, 1, and δq(t) ∈ Sq(t),while variations δx ⊕ δAv such that δAv = B(δx, x) with δx(ti) = 0 for i = 0, 1, correspondexactly to horizontal variations δq of the curve q such that δq(ti) = 0.

4 The Local Momentum and Horizontal Equation

The momentum equation, found in BKMM, is an important equation for understandinglocomotion in nonholonomic systems (see also Marsden and Ostrowski [1998] for furtherinformation and references).

The momentum equation in body representation on the principal bundle Q → Q/Gis the equation (4.5) below. Moreover, the momentum equation in this representation isindependent of, that is, decouples from, the group variables g. In this section we shall showthat this equation can be directly obtained from the formalism we have introduced as thevertical Lagrange-d’Alembert-Poincare equation.

We start by reviewing the local form of both the vertical and the horizontal Lagran-ge-Poincare operator, following CMR. After this, we explain how the local expressions forthe vertical and horizontal Lagrange-d’Alembert-Poincare operators can be easily derivedfrom the local form of the vertical and horizontal Lagrange-Poincare operators by restrictingthem to satisfy the conditions imposed by the constraints.

The Local Vertical Lagrange-Poincare Equation. Following CMR we shall now de-rive coordinate expressions for the vertical Lagrange–Poincare equations. The expressionsthat we obtain coincide with or can be easily derived from the ones obtained in BKMM,with some changes in the notation.

Let π : X×G → X be a trivial principal left G-bundle, where X ⊂ Rn is open. Let A be

a given principal connection. Denote by xα the coordinates on X and choose the standardflat connection ∇ on X. Then, at any tangent vector (x, g, x, g) ∈ T(x,g) (X × G), we have

A(x, g, x, g) = Adg (Ae(x) · x + v) ,

where Ae is a g-valued 1-form on X defined by Ae(x) · x = A(x, e, x, 0) and v = g−1g.Observe that, in this case, the adjoint bundle g is the trivial bundle X × g. The vectorbundle isomorphism αA in this case becomes

αA ([x, g, x, g]G) = (x, x) ⊕ v,

where v = (x, Ae(x) · x+v). We will often write (x, x, v) instead of (x, x)⊕ v, and sometimes,simply v = Ae(x) · x + v. Let us choose maps eb : X → g, where b = 1, ...,dim(G),such that, for each x ∈ X, the set eb(x) | b = 1, ...,dim(G) is a basis of g. For eachb = 1, ..,dim(G), let eb(x) be the section of g given by eb(x) = [x, e, eb(x)]G ≡ (x, eb(x)).Let us call p = p(x, x, v) the vertical momentum of the reduced system, that is, by definition,

p(x, x, v) =∂l

∂v(x, x, v).


Let us call pb = p(eb) ≡ 〈p, eb〉. We want to find an equation for the evolution of pb. Wehave

d

dtpb =

d

dt〈p, eb〉 =

⟨D

Dtp, eb

⟩+

⟨p,

D

Dteb

⟩. (4.1)

Using the vertical Lagrange-Poincare equation we immediately obtain⟨D

Dtp, eb

⟩= 〈p, [v, eb]〉 = 〈p, (x, e, [Ae(x) · x + v, eb])〉 ≡ 〈p, [Ae(x) · x + v, eb]〉 . (4.2)

Lemma 2.3.4 of CMR gives the general formula for calculating the covariant derivative of agiven curve [q(t), ξ(t)]G in g, namely

D[q(t), ξ(t)]GDt

=[q(t),−[A (q(t), q(t)) , ξ(t)] + ξ(t)

]G

. (4.3)

We apply this formula to the curve eb (x(t)) = [x(t), e, eb (x(t))]G ≡ (x(t), eb (x(t))) in g.Note that the tangent vector to the curve q(t) ≡ (x(t), e) is (q(t), q(t)) ≡ (x(t), e, x(t), 0)and hence A (q(t), q(t)) ≡ Ae (x(t)) · x. Using equation (4.3) we obtain

D

Dteb = [x, e,−[Ae(x) · x, eb] + eb]G ≡ (x,−[Ae(x) · x, eb] + eb) (4.4)

From equations (4.1), (4.2), and (4.4) we obtain the momentum equation

dpb

dt= 〈p, [v, eb] + eb〉 . (4.5)

This equation coincides with the momentum equation (4.4.2) of BKMM.Using this equation we can easily find an expression in coordinates for the momentum

equation, or, which is equivalent, as we have just shown, to the vertical Lagrange-Poincareequation. Let us choose the functions eb(x) to be constant functions, therefore, we haveeb = 0 and the momentum equation becomes

dpb

dt= 〈p, [v, eb]〉 . (4.6)

Recall that v − Ae(x) · x = v; thus equation (4.6) becomes

dpb

dt= 〈p, [v − Ae(x) · x, eb]〉 . (4.7)

Let Cabd be the structure constants of the Lie algebra g. For the given local coordinates xα on

X, let Aaα(x) be the coefficients of Ae(x), that is, by definition, (Ae(x) · x)aea = Aa

α(x)xαea.Then equation (4.7) becomes

dpb

dt= pa

(Ca

dbvd − Ca

dbAdαxα

), (4.8)

or, more explicitly,

dpb(x, x, v)dt

= pa(x, x, v)(Ca

dbvd − Ca

dbAdα(x)xα

). (4.9)

This equation coincides with equation (5.3.3) of BKMM as well as with equation (3.2) inKoon and Marsden [1997a]. Since v = Ae(x) · x + v, we also obtain

dpb(x, x, Ae(x) · x + v)dt

= pa(x, x, Ae(x) · x + v)Cadbv

d. (4.10)


Observe that one can calculate the momentum pb by taking the derivative of l with respectto either vb or vb. In Bloch, Krishnaprasad, Marsden and Murray [1996], the variable vb iscalled Ωb and is interpreted as the locked body angular velocity. This variable is intrinsic,given the choice of a connection, whereas vb depends on the local trivialization. In thespecial case in which the connection is trivial, i.e., Ae = 0, v = v, so we get the Poincareequation, which is one of the equations in Hamel’s equations.

In the case of nonholonomic systems, v must belong to S(x, e), which is equivalent to v ∈s. Recall that we are assuming (A1) and, therefore, dim(S(x, e)) =: s is constant. Choosethe gx-basis eb(x) | b = 1, ...,dim(G), for each x, in such a way that, eb(x) | b = 1, ..., sgenerates sx. Equation (4.5) becomes thus

dpb(x, x, v)dt

= pa(x, x, v)(

Cacbv

c − CadbA

dα(x)xα +

(∂eb

∂xα

)a

xα

), (4.11)

where (∂eb/∂xα)a is the a-component of ∂eb/∂xα. Note that a, c = 1, ...,dim(G), whereasb, d = 1, ..., s. The nonholonomic momentum equation (4.4.2) of BKMM, coincides withequation (4.5), or, equivalently, with equation (4.11), for the indices b = 1, ..., s. Observethat this is also the local expression of the vertical Lagrange-d’Alembert-Poincare equation.If the second term of the right hand side of equation (4.11) is calculated in coordinatesexplicitly, one obtains equation (7.2.2) of BKMM.

The Local Horizontal Lagrange-Poincare Equation. To calculate the horizontalLagrange-Poincare equation we shall first determine ∂C l

∂x (x, x, v) · δx. By definition, we have

∂C l

∂x(x, x, v) · δx =

d

dλl (x + λδx, x, w(λ))

∣∣∣∣λ=0

,

where w(λ) is a curve such that w(λ) ∈ gx+λδx for each λ, w(0) = v, and Dw(λ)Dλ = 0. If

w(λ) = (x + λδx, e, w(λ)), we can deduce from equation (4.3)

Dw(λ)Dλ

=(

x + λδx, e,− [Ae (x + λδx) · δx, w(λ)] +dw(λ)

dλ

). (4.12)

Therefore we must have dw(λ)dλ = [Ae (x + λδx) · δx, w(λ)] . We then obtain

∂C l

∂x(x, x, v) · δx =

∂l

∂x(x, x, v) · δx +

∂l

∂v(x, x, v) · [Ae(x) · δx, v] .

On the other hand, it is easy to see that B(x)(x, δx) = (x, e, B(x, e)(x, δx)). Then thehorizontal Lagrange-Poincare equation is

(∂l

∂x(x, x, v) − d

dt

∂l

∂x(x, x, v)

)· δx =

⟨∂l

∂v(x, x, v), B(x, e)(x, δx) + [v, Ae(x) · δx]

⟩.

(4.13)

As we did with the vertical Lagrange-Poincare operator, it is convenient to rewrite thisequation explicitly in coordinates and we easily obtain

∂l

∂xα(x, x, v) − d

dt

∂l

∂xα(x, x, v) =

∂l

∂va(x, x, v)

(Ba

βα(x, e)xβ + Cadbv

dAbα(x)

)(4.14)

where a fixed basis ea of g has been chosen and, in this basis, v = vaea. This equationcoincides with equation (5.3.2) of BKMM and equation (3.1) of Koon and Marsden [1997a].

5 The Snakeboard 31

We remark that in these papers the convention for the sign of the curvature Baαβ is the

opposite to the one used in this paper.For nonholonomic systems, v must belong to S(x, e), which is equivalent to v ∈ s. Recall

that we are assuming (A1) and, therefore, dim(S(x, e)) =: s is constant. Choose the gx-basiseb(x) | b = 1, ...,dim(G), for each x, in such a way that, eb(x) | b = 1, ..., s generates sx.The corresponding horizontal Lagrange-d’Alembert-Poincare equation is simply equation(4.14) with restriction on the indices a = 1, ...,dim(G), b, d = 1, ..., s.

Summary. The Lagrange-d’Alembert-Poincare equations in a frame ea | a = 1, ...,dim(G)adapted to the constraints, that is, ea, a = 1, ..., s, generate s at every point, are

dpb

dt= pa

(Ca

cbvc − Ca

dbAdαxα +

(∂eb

∂xα

)a

xα

)

∂l

∂xα− d

dt

∂l

∂xα=

∂l

∂va

(Ba

βαxβ + Cadbv

dAbα

),

where, as usual, a summation is implied over repeated indices. In these equations the indicesare a, d = 1, ...,dim(G), b, c = 1, ...., s, α = 1, ...,dim(Q/G).

We refer to BKMM for other coordinate representations of these equations.

5 The Snakeboard

We now describe the snakeboard, following BKMM in which further references can be found.See also Koon and Marsden [1997c] for further information. Our purpose is to use it toillustrate the formalism we have developed so far.

The snakeboard is a modified version of a skateboard in which the front and back pairsof wheels are independently actuated; see Figure 5.1. The degree of freedom ψ, whilesimultaneously moving the wheels with the proper phase relationship enables the rider togenerate forward motion.

φ

φ

ψ

Figure 5.1: The variables in the snakeboard

One of the interesting features of the snakeboard is that it leads to a nontrivial mo-mentum equation, which has terms that are linear in p and also quadratic in x. Assumingφ1 = −φ2 = φ for simplicity, the momentum equation is

p = 2J0(cos2 φ)φψ − (tanφ)pφ,

where φ and ψ represent the internal variables (the x in the theory) of the system and J0

is the rotor inertia. We shall say more about the modeling of this example shortly. Animportant point to recognize is that this equation does not depend on the rotational andtranslational position of the system, i.e., there is no explicit g dependence which means,

5 The Snakeboard 32

in this case, no x, y or θ dependence, which parameterize overall translations and rotationsof the system. Thus, if one has a given internal motion, this equation can be solved for pand from it, the attitude and position of the snakeboard calculated by means of anotherintegration using the reconstruction equation for g−1g.

The momentum for the snakeboard is closely related to the angular momentum of thesystem about the point P shown in Figure 5.2. See Marsden and Ostrowski [1998] for furtherdiscussion and references.

Figure 5.2: The angular momentum about the point P plays an important role in the analysis of thesnakeboard.

Other examples with a similar structure for the momentum equation are the roller racer(see Tsikiris [1995], Zenkov, Bloch and Marsden [1998]) and the bicycle (see the papers ofGetz and Marsden [1995] and Koon and Marsden [1997c]).

We model the snakeboard as a rigid body (the board) with two sets of independentlyactuated wheels, one on each end of the board. The human rider is modeled as a momentumwheel which sits in the middle of the board and is allowed to spin about the vertical axis.Spinning the momentum wheel causes a counter-torque to be exerted on the board. Theconfiguration of the board is given by the position and orientation of the board in the plane,the angle of the momentum wheel, and the angles of the back and front wheels. Thus theconfiguration-space is Q = S1×S1×S1×SE(2), and we shall consider it as a principal bundlewith structure group SE(2) acting on the left. We let (θ, x, y) represent the orientation andthe position of the center of the board, ψ the angle of the momentum wheel relative to theboard, and φ1 and φ2 the angles of the back and front wheels, also relative to the board.We take the distance between the center of the board and the wheels as r.

The Lagrangian for the snakeboard consists only of kinetic energy terms. We take thesimplest possible model for the various mass distributions and write the Lagrangian as

L(q, q) = 12m(x2 + y2) + 1

2Jθ2 + 12J0(θ + ψ)2 + 1

2J1(θ + φ1)2 + 12J2(θ + φ2)2,

where m is the total mass of the board, J is the inertia of the board, J0 is the inertia of therotor, and Ji, i = 1, 2, is the inertia corresponding to φi. The Lagrangian is independent ofthe configuration of the board and hence it is invariant to all possible group actions.

The rolling of the front and rear wheels of the snakeboard is modeled by using non-holonomic constraints which allow the wheels to spin about the vertical axis and roll inthe direction that they are pointing. The wheels are not allowed to slide in the sidewaysdirection. This gives constraint one-forms

ω1(q) = − sin(θ + φ1)dx + cos(θ + φ1)dy − r cos φ1dθ,

ω2(q) = − sin(θ + φ2)dx + cos(θ + φ2)dy + r cos φ2dθ.(5.1)

5 The Snakeboard 33

These constraints are invariant under the SE(2) action given by

(α, a, b) · (ψ, φ1, φ2, θ, x, y)= (ψ, φ1, φ2, θ + α, x cos α − y sinα + a, x sinα + y cos α + b),

where (α, a, b) ∈ SE(2), and also under the S1 action defined by

δ · (ψ, φ1, φ2, θ, x, y) = (ψ + δ, φ1, φ2, θ, x, y).

We consider the SE(2) symmetry corresponding to the situation in which the S1 symmetryis destroyed by the controls. The constraints determine the kinematic distribution Dq:

Dq = span

∂

∂ψ,

∂

∂φ1,

∂

∂φ2, a

∂

∂x+ b

∂

∂y+ c

∂

∂θ

, (5.2)

where a, b, and c, are given by

a = −r(cos φ1 cos(θ + φ2) + cos φ2 cos(θ + φ1)),b = −r(cos φ1 sin(θ + φ2) + cos φ2 sin(θ + φ1)), (5.3)c = sin(φ1 − φ2).

The tangent space to the orbit of the SE(2) action is given by

Tq(Orb(q)) = span

∂

∂θ,

∂

∂x,

∂

∂y

(5.4)

(note that this is not a left-invariant basis). The intersection between the tangent space tothe group orbit and the fiber of the constraint distribution is thus given by

Dq ∩ Tq(Orb(q)) = span(

c∂

∂θ+ a

∂

∂x+ b

∂

∂y

). (5.5)

We construct the momentum by choosing a section of D ∩ T Orb(q) regarded as a bundleover Q. Since Dq ∩ Tq Orb(q) is one-dimensional, we choose the section to be

a∂

∂x+ b

∂

∂y+ c

∂

∂θ, (5.6)

which is invariant under the lifted action of SE(2) on TQ.It is convenient to rewrite some of the previous formulas in a more compact form, by

introducing complex variables. First, we identify S1 with the unit circle in the complexplane C. Thus a typical element of the base space X of the bundle Q will be written as(eiψ, eiφ1 , eiφ2) and a typical tangent vector (x, x) ∈ TxX will be written as

(x, x) = (eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, eiφ2iφ2),

or, sometimes, simply as (eiψiψ, eiφ1iφ1, eiφ2iφ2).

Second, the Euclidean group SE(2) is the semidirect product G = S1 C where thesemidirect product group operation is given by

[(eiθ1 , z1), (eiθ2 , z2)] = (ei(θ1+θ2), eiθ1z2 + z1).

A typical tangent vector to S1C at the point (eiθ, z) will be written (eiθ, z, eiθiθ, z) or,simply, (eiθiθ, z). The Lie algebra bracket is given by

[(iθ1, z1), (iθ2, z2)] = (0, iθ1z2 − iθ2z1).

5 The Snakeboard 34

Third, a typical element of Q = X × (S1C) will be written (eiψ, eiφ1 , eiφ2 , eiθ, z) and atypical element of TQ as (eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, e

iφ2iφ2, eiθ, z, eiθiθ, z), or, sometimes,

for simplicity, (eiψiψ, eiφ1iφ1, eiφ2iφ2, e

iθiθ, z). Using the de Moivre formulas, the numbersa, b, and c defined above can be conveniently written as

a + bi = −reiθ

(ei(φ1+φ2) +

12

(ei(φ1−φ2) + e−i(φ1−φ2)

))(5.7)

and

c =12i

(ei(φ1−φ2) − e−i(φ1−φ2)

). (5.8)

Note that c only depends on φ1 − φ2 and a + ib only depends on φ1, φ2 and θ.Thus, the bundle S is the line bundle generated by the section of TQ given by

(eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, eiφ2iφ2, e

iθ, z, eiθic, a + ib).

Next, we shall choose the connection A and then find the Lagrange-Poincare bundle,the bundle s, and the Lagrange-d’Alembert-Poincare vertical and horizontal operators. It isvery easy to see that the trivial connection A on the trivial bundle Q satisfies the dimensionassumption. In fact, we have that D = H⊕ S where

H = span

∂

∂ψ,

∂

∂φ1,

∂

∂φ2

is the horizontal space of the connection A. The connection A is given by right translationto the neutral element on TS1C, that is,

A(eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, eiφ2iφ2, e

iθ, z, eiθiθ, z) = (iθ,−iθz + z).

The Lagrange-Poincare bundle is TX ⊕ g where the bundle g can be naturally identifiedwith the trivial bundle X × g. With this identification, a typical element of g is written as(x, v) = (eiψ, eiφ1 , eiφ2 , iθ, z).

Recall that, for a trivial bundle Q = X × G, where the group acts on the left, the mapαA for the trivial connection A is always given by left translation on the group factor, thatis, αA(x, x, g, g) = (x, x, v), where v = g−1g. In our case we have

(x, x) = (eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, eiφ2iφ2)

and v = (iθ, e−iθ z). The vector bundle isomorphism αA is given by

αA([eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, eiφ2iφ2, e

iθ, z, eiθiθ, z]G)

= (eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, eiφ2iφ2, iθ, e

−iθ z).

Now we shall identify the bundle s. It is the line subbundle of g generated by thesection of g, call it (eiψ, eiφ1 , eiφ2 , f), which is the image under αA of the section thatgenerates the bundle S/SE(2) defined above. Since the left translation of the tangent vector(eiθ, z, eiθic, a+ib) to the neutral element is

(ic, e−iθ(a + ib)

), we obtain the following section

that generates s,

(eiψ, eiφ1 , eiφ2 , f) =(eiψ, eiφ1 , eiφ2 , ic, e−iθ(a + ib)

).

A typical element of TX ⊕ s is therefore written as

(eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, eiφ2iφ2, λf),

5 The Snakeboard 35

where λ is a real number.The reduced Lagrangian at any element (eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, e

iφ2iφ2, v), wherev = (iθ, e−iθ z), is easily seen to be

l(eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, eiφ2iφ2, v)

=12J(θ)2 +

12m|z|2 +

12J0(θ + ψ)2 +

12J1(θ + φ1)2 +

12J2(θ + φ2)2.

The reduced Lagrangian at a point (eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, eiφ2iφ2, λf) of the bundle

s is

l(eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, eiφ2iφ2, λf)

=12J(λc)2 +

12mλ2(a2 + b2) +

12J0(λc + ψ)2 +

12J1(λc + φ1)2 +

12J2(λc + φ2)2.

To calculate the horizontal Lagrange-d’Alembert-Poincare equation, first note that

∂l

∂ψ=

∂l

∂φ1=

∂l

∂φ2= 0.

On the other hand, since the connection A is trivial, its curvature is 0, therefore, thehorizontal Lagrange-d’Alembert-Poincare equation gives the conserved quantities

∂l

∂ψ= J0(θ + ψ);

∂l

∂φ1

= J1(θ + φ1);∂l

∂φ2

= J2(θ + φ2). (5.9)

Since the Lagrangian is the kinetic energy in this example and there is no potential energy,the Lagrangian itself is conserved, then the previous equations immediately imply that

12J(θ)2 +

12m|z|2 (5.10)

is also a preserved quantity.Now we shall calculate the vertical Lagrange-d’Alembert-Poincare equation. An impor-

tant observation in this example is the obvious fact that the coadjoint operation in the fiberof the bundle g, restricted to the bundle s, is zero, because the fibers of s are 1-dimensional.On the other hand, since the connection A is trivial, the covariant derivative of a given curvein the trivial bundle g is the usual derivative. Therefore, the vertical Lagrange-d’Alembert-Poincare equation, according to Theorem 3.16, becomes,

d

dt

∂l

∂v(x, x, v) · δη = 0

for all δη ∈ s. Since s is generated by f , the vertical Lagrange-d’Alembert-Poincare equationin this example is equivalent to

d

dt

∂l

∂v(eiψ, eiφ1 , eiφ2 , eiψiψ, eiφ1iφ1, e

iφ2iφ2, v) · f = 0.

More explicitly, we have

∂l

∂θ= Jθ;

∂l

∂x= mx;

∂l

∂y= my,

from which we obtain

d

dt

∂l

∂θ= Jθ;

d

dt

∂l

∂x= mx;

d

dt

∂l

∂y= my.


Therefore the vertical Lagrange-d’Alembert-Poincare equation becomes

Jθc + mxa + myb = 0. (5.11)

Since (θ, x, y) satisfies the constraints, we have, for some real valued function λ(t),

θ = λc; x = λa; y = λb.

Then we can write

θ = λc + λc; x = λa + λa; y = λb + λb.

Replacing these in the vertical Lagrange-d’Alembert-Poincare equation (5.11) and rearrang-ing one obtains the equation

λ(Jc2 + ma2 + mb2) +12λ

d

dt(Jc2 + ma2 + mb2) = 0.

Solving this equation for λ one gets

λ = K(Jc2 + m(a2 + b2)

)− 12 , (5.12)

where K is a constant. Using equations (5.7) and (5.8) one can easily see that λ does notdepend on θ. We remark that equation (5.12) does not give more information than the onecontained in the conservation of energy equation, namely, equation (5.10). In fact, the sameexpression for λ can be easily obtained by replacing θ = λc, x = λa, and y = λb in equation(5.10) and solving for λ.

Now we can combine equation (5.12), equation θ = λc, and equations (5.9) to obtain

ψ = −cK(Jc2 + ma2 + mb2)−12 + C (5.13)

φ1 = −cK(Jc2 + ma2 + mb2)−12 + C1 (5.14)

φ2 = −cK(Jc2 + ma2 + mb2)−12 + C2, (5.15)

where C, C1, and C2 are constants. Due to the very special structure of this system ofthree differential equations it is possible to reduce it to a single equation. In fact, it canbe easily deduced from these equations that φ1 − ψ = ω1t + E1 and φ2 − ψ = ω2t + E2,where ωi, Ei are constants for i = 1, 2. Then we can replace these expressions for φ1 andφ2 in the equation (5.13) to obtain the equation ψ = F (ψ, t), where the function F can becalculated; it is a simple elementary function. A more detailed study of this equation andits application to concrete questions about the motion of the snakeboard, as well as a studyof the Hamiltonian counterpart, will be the purpose of future work.

6 Miscellany and Future Directions

Systems with Affine Constraints In the previous sections we studied systems withnonholonomic constraints D, where D is a vector subbundle of TQ. Now assume that P isa given section of TQ, that is, a vector field. Then we have an affine subbundle P +D, thatis, at each point q ∈ Q, P (q) + Dq is an affine subspace of TqQ.

The previous geometric theory of systems with nonholonomic constraints given by avector subbundle D can be generalized for the more general case of systems with affineconstraints. The basic idea is that one should replace the condition q ∈ D by the conditionq ∈ P + D, while the condition δq ∈ D stays.


More precisely, we have the following Lagrange-d’Alembert principle:

A curve q(t) ∈ Q, t ∈ [t0, t1], is an actual motion of the system if and only if q(t) ∈P (q(t)) + D(q(t)) for all t and, besides, for any deformation q(t, λ) of q(t) such that thecorresponding variation

δq(t) =∂q(t, λ)

∂λ

∣∣∣λ=0

satisfies δq(t) ∈ Dq(t) for all t, the following condition holds

δ

∫ t1

t0

L(q, q)dt = 0.

By definition, the dimension assumption for a given affine constraint P + D holds,if D satisfies (A1). Let P + D be a given affine constraint on a principal bundle Q withstructure group G, as before. By definition, P + D is invariant if it is preserved by theaction of G on TQ. Under both the dimension assumption and G-invariance for a givenaffine constraint, we can obtain a generalization of all the results in this paper for systemswith affine constraints.

Almost Poisson and Dirac Structures. As in Koon and Marsden [1998] (and otherreferences therein), nonholonomic systems have an interesting almost Poisson structure (“al-most” meaning that Jacobi’s identity can fail), as in Cannas da Silva and Weinstein [1999]and Cantrijn, Leon and Diego [1999]. It would be interesting to develop that geometry inthe context of the bundle picture presented in this paper. The relations between Poissongeometry that are developed for the bundle picture in CMR should be useful in advancingthis endeavor. The use of the Dirac theory of constraints to approach this problem is ofcourse also very interesting, and is explored in van der Schaft and Maschke [1994] and Ibort,Leon, Marrero and Diego [1999].

Lagrange-d’Alembert-Routh Reduction. One knows from the work on the nonholo-nomic energy-momentum method of Zenkov, Bloch and Marsden [1998] that the Routhianplays an important role in the stability theory of nonholonomic systems, despite the factthat the momentum obeys a momentum equation rather than a conservation law. In ad-dition, as the cited paper shows, the Routhian can be used to cast the reduced equationsinto an interesting form. Thus, it would be of interest to develop the intrinsic geometry forreduction of nonholonomic systems using the Routhian.

The Hamiltonian Bundle Picture. In connection with the preceding remarks aboutLagrange-d’Alembert-Routh reduction, one should note on the Lagrangian side, we choosea connection on the bundle πQ,G : Q → Q/G and realize TQ/G as the Whitney sum bundleT (Q/G)⊕ g over Q/G. Correspondingly, on the Hamiltonian side we realize T ∗Q/G as theWhitney sum bundle T ∗(Q/G)⊕ g∗ over Q/G. The reduced Poisson structure on this space,as we have mentioned already, has been investigated by Montgomery, Marsden and Ratiu[1984], Montgomery [1986], CMR, and Zaalani [1999].

The results of Marsden, Ratiu and Scheurle [2000] on Routh reduction show that on theLagrangian side, the reduced space J−1

L (µ)/Gµ is the fiber product T (Q/G) ×Q/G Q/Gµ.This is consistent (by taking the dual of our isomorphism of bundles) with the fact that thesymplectic leaves of (T ∗Q)/G can be identified with T ∗(Q/G)×Q/G Q/Gµ. The symplecticstructure on these leaves has been investigated by Zaalani [1999] and, from the point of viewof CMR, in Marsden and Perlmutter [2000].


Geometric Phases. The development of the theory of geometric phases in the Lagrangiancontext is natural to develop. As we have mentioned in the introduction, geometric phasesare central to many important questions in mechanics, such as locomotion generation. Thepaper of Marsden, Ratiu and Scheurle [2000] gives results for geometric phases in the con-text of Routh reduction for holonomic systems. As Koon and Marsden [1997b] indicates,geometric phases should be feasible and interesting in the nonholonomic context as well andthat the Lagrangian formalism is natural for doing this. In fact, the Lagrangian settingprovides natural connections and also a natural setting for averaging which is one of thebasic ingredients in geometric phases. The results of the present paper or future work onLagrange-d’Alembert-Routh reduction should be useful in this regard.

Variational Integrators and Discrete Reduction. As Weinstein [1996] points out,there is a more general context for Lagrangian mechanics that also includes discrete me-chanics in the sense of Veselov [1988, 1991]. It would certainly be interesting to developa discrete mechanics for the Lagrangian bundle picture both in the holonomic case (as inCMR) and in the nonholonomic case, as in the present paper.

One of the interesting developments in symplectic integration algorithms has been theprogress made in variational integrators. These integration algorithms are based on directdiscretizations of Hamilton’s principle following some of the ideas of Veselov [1988]. See,for example, Wendlandt and Marsden [1997], Marsden, Patrick and Shkoller [1998], and,for the inclusion of damping and forcing, Kane, Marsden, Ortiz and West [2000]. Thereis a discrete reduction theory for this point ov view of discrete mechanics as well that isstill under development. See Marsden, Pekarsky and Shkoller [1999], Bobenko, Lorbeer andSuris [1998], Bobenko and Suris [1999a, 1999b] and Jalnapurkar, Leok, Marsden and West[2000]. Of course it would be of interest to develop a discrete version of the nonholonomictheory and to implement this numerically.

Infinite Dimensional Examples. In this paper we dealt with Lagrange-d’Alembert re-duction theory in the context of finite dimensional manifolds. Of course, the theory formallyapplies to many interesting infinite dimensional examples, such as rolling elastic bodies. Inthe infinite dimensional context, many of the expressions that appear here as pure partialderivatives must be written in the notation of functional derivatives (see Marsden and Ratiu[1999] for some of the basic examples, an explanation of the functional derivative notationand additional references to the literature).

Multisymplectic Context. Another area of much current interest is that of multisym-plectic geometry. The history of this subject is very complex; modern accounts are, forexample, Marsden and Shkoller [1999] and Marsden, Patrick and Shkoller [1998]. Thistheory has both a Lagrangian and a Hamiltonian view and it has allowed, for example, adevelopment of the Moser-Veselov theory to the context of PDE’s. Reduction theory in thiscontext is in its infancy (see, for example, Marsden, Montgomery, Morrison and Thompson[1986] and Castrillon Lopez, Ratiu and Shkoller [2000]). Obviously, it would be of interestto develop such a theory for nonholonomic systems.

Singular Lagrange-d’Alembert Reduction. We mentioned the importance of singularreduction and some of the current literature in the introduction. Almost all of the theory ofsingular reduction is confined to the general symplectic category, with little attention paidto the tangent and cotangent bundle structure, as well as to the Lagrangian side, apartfrom that in Lewis [1992]. Explicit examples, even the spherical pendulum (see Lerman,Montgomery and Sjamaar [1993]) show that this cotangent bundle structure together with

REFERENCES 39

a “stitching construction” is important. It would be interesting to develop the generaltheory of singular Lagrangian reduction both for holonomic and nonholonomic systems using,amongst other tools, the techniques of blow up (see Marsden and Scheurle [1993a] andHernandez [2000]). In addition, this should be dual to a similar effort for the generaltheory of symplectic reduction of cotangent bundles. Surprisingly little is known aboutsingular nonholonomic reduction (see, e.g., Bates [1998]). We believe that the general bundlestructures in this paper will be useful for this endeavor.

Lagrange-d’Alembert-Poincare by Stages. In the introduction we discussed the cur-rent state of affairs in the theory of reduction by stages, both Lagrangian and Hamiltonian.The Lagrange-d’Alembert-Poincare counterpart of symplectic reduction is of course whatwe have developed here. Naturally, the development of this theory for reduction by stages(as in CMR) for group extensions would be very interesting.

Acknowledgements. We thank our many colleagues, collaborators and students for theirhelp, direct or indirect, with this paper. In particular, we would like to single out AnthonyBloch, Joel Burdick, Sameer Jalnapurkar, P.S. Krishnaprasad, Hans-Peter Kruse, MelvinLeok, Naomi Leonard, Richard Murray, Jim Ostrowski, Sergey Pekarsky, Matt Perlmutter,Jurgen Scheurle, Steve Shkoller, and Alan Weinstein. We also thank the referee for theirkind advice.

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