Nonholonomic Integrators

Transcription

1 Nonolonomic Integrators J. Cortés an S. Martínez Laboratory of Dynamical Systems, Mecanics an Control, Instituto e Matemáticas y Física Funamental, CSIC, Serrano 13, 8006 Mari, SPAIN j.cortes@imaff.cfmac.csic.es, s.martinez@imaff.cfmac.csic.es Abstract. We introuce a iscretization of te Lagrange- Alembert principle for Lagrangian systems wit nonolonomic constraints, wic allows us to construct numerical integrators tat approximate te continuous flow. We stuy te geometric invariance properties of te iscrete flow wic provie an explanation for te goo performance of te propose meto. Tis is teste on two examples: a nonolonomic particle wit a quaratic potential an a mobile robot wit fixe orientation. Submitte to: Nonlinearity AMS classification sceme numbers: 37J60, 37M15

2 Nonolonomic Integrators 1. Introuction Discrete mecanics as become a fiel of intensive researc activity in te last years. Te increasing interest in te subject is mainly ue to its ual caracter. On te one an, iscrete mecanics allows for te construction of integration scemes, te so-calle mecanical integrators, tat turn out to be numerically competitive in many situations. On te oter an, many of te geometric properties of mecanical systems in te continuous case amit an appropriate counterpart in te iscrete setting, wic makes it a ric area to be explore. Bot aspects of iscrete mecanics mutually interact, since te geometric properties of te iscrete moel play a key role in te explanation of te goo beaviour of te integrators erive from it in a number of situations. Mecanical integrators preserve some of te invariants of te mecanical system, suc as energy, momentum or te symplectic form. It is well known tat if te energy an te momentum map inclue all integrals belonging to a certain class see [11], ten one cannot create constant time step integrators tat are simultaneously symplectic, energy preserving an momentum preserving, unless tey integrate te equations exactly up to a time reparametrization. Recently, it as been sown tat te construction of energy-symplectic-momentum integrators is inee possible if one allows varying time steps [15]. Tis justifies te focus on mecanical integrators tat are eiter symplectic-momentum or energy-momentum preserving altoug oter types may also be consiere, suc as metos preserving reversing symmetries. Base on certain applications, suc as molecular ynamics simulation, te necessity of treating olonomic constraints in iscrete mecanics as also been iscusse in te literature. For example, te popular Verlet algoritm for unconstraine mecanical systems was aapte to anle olonomic constraints, resulting in te Sake algoritm [30] an te Rattle algoritm [1] see [1] for a iscussion of te symplectic caracter of tese metos. Te case of general Hamiltonian systems i.e. not necessarily mecanical subject to olonomic constraints as also been stuie [14, 31, 3]. A ifferent approac, base on te Dirac teory of constraints to fin unconstraine formulations in wic te constraints appear as invariants, may be foun in [0]. Energy-momentum integrators erive from iscrete irectional erivatives an iscrete versions of Hamiltonian mecanics ave also been recently aapte to eal wit olonomic constraints [1, 13]. Variational integrators are symplectic-momentum mecanical integrators erive from a iscretization of Hamilton s principle [, 8, 34, 35]. Tis iscrete variational principle leas to te obtention of te iscrete Euler-Lagrange DEL equations. Different iscrete Lagrangians result in ifferent variational integrators, incluing te Verlet algoritm an te wole family of Newmark algoritms [16] use in structural mecanics. Variational integrators anle constraints in a simple an efficient manner by using Lagrange multipliers [36]. It is wort mentioning tat, wen treate variationally, olonomic constraints o not affect te symplectic or conservative nature of te algoritms, wile oter tecniques can run into trouble in tis regar [0].

3 Nonolonomic Integrators 3 In tis paper, we aress te problem of constructing integrators for mecanical systems wit nonolonomic constraints. Tis problem as been state in a number of recent papers [8, 36], incluing te presentation of open problems in symplectic integration given in [7]. In nonolonomic Lagrangian mecanics, te symplectic form constructe from te Lagrangian is no longer preserve as in te unconstraine case. Moreover, in te case of a nonolonomic system wit symmetry, te momentum map is not conserve in general, ue to te presence of te constraint force. However, one can consier a nonolonomic momentum map along te symmetry irections compatible wit te constraints, an verify tat its evolution along te integral curves of te constraine system is given by te nonolonomic momentum equation [4, 7, 9]. On te oter an, at least in te case of linear or, more generally, omogeneus constraints, te energy is still a conservation law for te system. Consequently, two of te tree cornerstones on wic te construction of mecanical integrators for unconstraine systems relies i.e. preservation of symplectic structure an momentum are lacking in te nonolonomic case. Our starting point to evelop integrators in te presence of nonolonomic constraints is te introuction of a iscrete version of te Lagrange- Alembert principle. Tis follows te iea tat, by respecting te geometric structure of nonolonomic systems, one can create integrators capturing te essential features of tis kin of systems. Inee, we sow tat te nonolonomic integrators erive from tis iscrete principle preserve te structure of te evolution of te symplectic form along te trajectories of te system. We also prove tat, for nonolonomic systems wit symmetry, te nonolonomic integrators give rise to a iscrete version of te nonolonomic momentum equation. Moreover, in te presence of orizontal symmetries, te iscrete flow exactly preserves te associate momenta. We also treat te case were no nonolonomic momentum map exists, ue to te absence of symmetry irections fulfilling te constraints. Tis situation, known in te literature as te vertical or purely kinematic case [4, 7, 10, 18], allows one to reuce te continuous flow to tat of an unconstraine system wit a nonconservative force. We sow tat te nonolonomic integrator also passes to te iscrete reuce space, yieling a generalize variational integrator in te sense of [16]. In case te nonconservative force vanises, we prove tat te reuce nonolonomic integator is inee a variational integrator. Te paper is organise as follows. In Section, we give a brief review of variational integrators. Tis serves as a motivation to introuce te iscrete Lagrange- Alembert principle in Section 3, from wic we erive te iscrete Lagrange- Alembert DLA equations. In case te constraints are olonomic, we sow tat te DLA algoritm is just a variational integrator. Section 4 eals wit te construction of integrators, by way of appropriate iscretizations of te Lagrangian an te constraints. Generalizing a teorem presente in [36], we prove tat if te configuration manifol Q can be embee into a linear space V, ten te DLA algoritm on Q is equivalent to te DLA algoritm on V subject to te nonolonomic constraints plus te olonomic ones efining Q as a subspace of V. As expecte, te equations resulting from te nonolonomic

4 Nonolonomic Integrators 4 integrator inerit some of te geometric caracteristics of te continuous system. Tis is investigate in Section 5, were we stuy te invariance properties relate to te nonolonomic momentum equation an to te kinematic case, proviing complementary insigts into te geometric structure of iscrete nonolonomic mecanics. Finally, in Section 6, we present some numerical tests in te examples of a nonolonomic particle wit a quaratic potential an a mobile robot wit fixe orientation, illustrating te goo performance of te meto wen compare to te stanar 4 t orer Runge-Kutta.. Variational Integrators Mecanical integrators base on te Veselov iscretization tecnique [8, 34, 35] ave been stuie intensively in te last years an are by now well known [5, 15, 16, 6, 36]. We briefly review ere te main ieas of tis approac. Let Q be a n-imensional configuration manifol an L : Q Q R a smoot map playing te role of a iscrete Lagrangian. Te action sum is te map S : Q N+1 R efine by S = N 1 k=0 L q k, q k+1, 1 were q k Q for k {0, 1,..., N} an k is te iscrete time. Te iscrete variational principle states tat te evolution equations extremize te action sum, given fixe en points q 0, q N. Tis leas to te iscrete Euler-Lagrange DEL equations: D 1 L q k, q k+1 + D L q k 1, q k = 0. Uner appropriate regularity assumptions on te iscrete Lagrangian L, te DEL equations efine a map Φ : Q Q Q Q, Φq k 1, q k = q k, q k+1 wic escribes te iscrete time evolution of te system. Now, efine te fiber erivative or iscrete Legenre transform corresponing to L by FL : Q Q T Q q, q q, D L q, q, an te -form Ω L on Q Q by pulling back te canonical -form Ω Q = Θ Q from T Q, Ω L = FL Ω Q. Te alternative iscrete fiber erivative FL q, q = q, D 1 L q, q may also be use an te results obtaine will be essentially uncange. A funamental fact is tat te algoritm Φ exactly preserves te symplectic form Ω L, tat is, Φ Ω L = Ω L see [36]. If we furter assume tat te iscrete Lagrangian is invariant uner te action of a Lie group G on Q, one can prove tat te associate iscrete momentum map, J : Q Q g were g enotes te ual of te Lie algebra g of G efine by J q, q, ξ = D L q, q, ξ Q q,

5 Nonolonomic Integrators 5 is exactly preserve by te algoritm Φ [36]. Here, ξ Q enotes te funamental vector fiel corresponing to te element ξ g. Moreover, wen regaring te iscrete mecanical moel as an approximation to a continuous system, one can verify tat te constant value of te iscrete momentum map approaces te value of its continuous counterpart, as te time step ecreases. Consequently, variational integrators are symplectic-momentum integrators. 3. Discrete Lagrange- Alembert Principle In tis section, we propose a iscrete version of te Lagrange- Alembert principle for nonolonomic systems. Before oing so, we first recall te general picture in te continuous case. Consier a istribution D on te configuration space Q, escribing some kinematic constraints impose on a Lagrangian system. We say tat a curve qt in Q satisfies te constraints if qt D qt for all t. Te ynamics of te nonolonomic system is etermine by a Lagrangian L : T Q R troug te application of te Lagrange Alembert principle, wic states tat a curve qt is an amissible motion of te system if δj = δ b a Lqt, qtt = 0, for all variations suc tat δqt D qt, a t b, an if it satisfies te constraints. It is wort noting tat te Lagrange- Alembert principle is not variational, since we impose te constraints on te curve after extremizing te functional J. Te inverse proceure, tat is, imposing te constraints before extremizing J, results in a ifferent set of equations tis time truly variational terme vakonomic. Some straigtforwar manipulations sow tat te principle ols precisely wen L δl = t q L δq i = 0, i q i for all te variations δq D qt. If {ω a = ω a i q i } m a=1 is a set of m inepenent 1-forms efining te anniilator D o of D, we arrive at te equations escribing te nonolonomic ynamics L t q L i q = λ aω a i i, 3 ωi a q i = 0, 4 were λ a, a {1,..., m}, is a set of Lagrange multipliers. Te rigt-an sie of equation 3 represents te force of constraint. If we introuce coorinates q i = r α, s a on Q, were α {1,..., n m}, in terms of wic ω a takes te form ω a q = s a + A a αr, sr α,

6 Nonolonomic Integrators 6 ten te Lagrange multipliers are exactly given by λ a = L t ṡ L a s, a an te constraint force reas L F = t ṡ L ω a. a s a Now, we turn to te iscrete version of nonolonomic mecanics. Consier as before a iscrete Lagrangian L : Q Q R an te associate action sum S = N 1 k=0 L q k, q k+1, 5 were q k Q an k {0, 1,..., N} is te iscrete time. In te unconstraine iscrete mecanics case cf. Section, we ave seen tat one extremizes te action sum wit respect to all possible sequences of N 1 points, given fixe en points q 0, q N. Tis means tat at eac point q Q, te allowe variations are given by te wole tangent space T q Q. However, in te nonolonomic case, we must restrict te allowe variations. Tese are exactly given by te istribution D. In aition, we will consier a iscrete constraint space D Q Q wit te same imension as D an suc tat q, q D for all q Q. Tis iscrete constraint space imposes constraints on te solution sequence {q k }, namely, q k, q k+1 D. Later, wen regaring te iscrete principle as an approximation of te continuous one, we sall impose more conitions on te selection of D in orer to obtain a consistent iscretization of te continuous equations of motion. Consequently, to evelop te iscrete nonolonomic mecanics, one nees tree ingreients: a iscrete Lagrangian L, a constraint istribution D on Q an a iscrete constraint space D. Notice tat te iscrete mecanics can also be seen witin tis framework, were D = T Q an D = Q Q. Ten, we efine te iscrete Lagrange- Alembert principle to be te extremization of 5 among te sequence of points q k wit given fixe en points q 0 an q N, were te variations must satisfy δq k D qk an q k, q k+1 D, for all k {0,..., N 1}. Tis leas to te set of equations D 1 L q k, q k+1 + D L q k 1, q k i δq i k = 0, 1 k N 1, were δq k D qk, along wit q k, q k+1 D. If ω a : Q Q R, a {1,..., m}, are functions wose anniilation efines D, wat we ave got is te following iscrete Lagrange- Alembert DLA algoritm { D 1 L q k, q k+1 + D L q k 1, q k = λ a ω a q k ω aq 6 k, q k+1 = 0. Notice tat te iscrete Lagrange- Alembert principle is not truly variational, as te continuous principle. Alternatively, we will refer to te DLA algoritm 6 as a nonolonomic integrator, by analogy wit te unconstraine case. Note also tat, uner appropriate regularity assumptions, te implicit function teorem ensures us tat

7 Nonolonomic Integrators 7 we ave obtaine a well-efine algoritm Φ : Q Q Q Q, Φq k 1, q k = q k, q k+1. In fact, tis is guarantee if te matrix D 1 D L q k, q k+1 ω a q k D ω aq 7 k, q k+1 0 is invertible for eac q k, q k+1 in a neigbouroo of te iagonal of Q Q. Remark 3.1 Assume we are given a continuous nonolonomic problem wit ata L : T Q R an D T Q. In te following section, we sall iscuss some types of iscretizations of tis problem. To guarantee tat te DLA algoritm approximates te continuous flow witin a esire orer of accuracy, one soul select te iscrete Lagrangian L : Q Q R an te iscrete constraint space D in a consistent way. Tis essentially means tat if ω 1,..., ω m are 1-forms on Q wose anniilation locally efine te constraint istribution D, one performs te same type of iscretization of bot te Lagrangian L : T Q R an te 1-forms interprete as functions linear in te velocities, ω a : T Q R. For instance, if L is constructe by means of a iscretization mapping Ψ : Q Q T Q efine on a neigbouroo of te iagonal of Q Q, tat is, L = L Ψ, ten D must locally be efine by te anniilation of te functions ω a = ωa Ψ. State oterwise, D soul be suc tat ΨD = D. Remark 3. Consier te continuous nonolonomic problem given by L an D, an let L an D be appropriate iscrete versions of tem. Ten, if te matrix D 1 D L q k, q k ω a q k D ω aq k, q k 0 is invertible for eac q k Q, a sufficiently small stepsize guarantees tat te matrix 7 is also nonsingular an ence te DLA algoritm is solvable for q k+1. Remark 3.3 Te olonomic case Let us examine te nonolonomic integrator wen te constraints are olonomic, tat is, te case wen te istribution D is integrable. Assume tat tere exists a function g : Q R l wose level surfaces are precisely te integral manifols of D, i.e. for eac r R l, N r = g 1 r is a submanifol of Q suc tat T q N r = D q for all q N r. Ten, we can consier as constraint space te following subspace of Q Q, D = r R ln r N r. Observe tat if we take q 0 N 0, ten q 0, q 1 D is equivalent to q 1 N 0. We ten fin tat te nonolonomic integrator for an initial pair q 0, q 1 N 0 becomes { D 1 L q k, q k+1 + D L q k 1, q k = λ a Dg a q k 8 gq k+1 = 0, were g a : Q R enotes te a component of g. Notice tat 8 is just a variational integrator [36]. It is known tat for an appropriate iscrete Lagrangian, one recovers te Sake algoritm [1, 30], written in terms of position variables. Te Sake algoritm is very useful in molecular ynamics simulation.

8 Nonolonomic Integrators 8 4. Construction of integrators In te unconstraine case [36], tere are mainly two ways of constructing mecanical integrators, epening on weter Q is seen as a manifol in its own rigt te intrinsic point of view or as being embee in a larger space te extrinsic point of view. Assume tat we ave a continuous nonolonomic problem given by L : T Q R an D T Q. Wen aopting te intrinsic point of view, one makes use of coorinate carts on Q to construct te iscrete Lagrangian. Let ϕ : U Q R n be a local cart wose coorinate omain U contains q k an asume tat q k+1 U a conition guarantee by a sufficiently small timestep. A coice of iscrete Lagrangian is te following L α q k, q k+1 = L ϕ 1 1 αϕq k + αϕq k+1, ϕ 1 ϕqk+1 ϕq k, 9 were 0 α 1 is an interpolation parameter an te ifferential ϕ 1 is taken at te point x = 1 αϕq k + αϕq k+1. Of course tere are oter possible coices of iscretizations, as for instance, L sym,α q k, q k+1 = 1 L ϕ 1 1 αϕq k + αϕq k+1, ϕ 1 ϕqk+1 ϕq k L ϕ 1 αϕq k + 1 αϕq k+1, ϕ 1 ϕqk+1 ϕq k. In te unconstraine case, te coice 10 always yiels secon orer accurate numerical metos, wereas in general tis is only guarantee for te iscretization 9 if α = 1 altoug for natural Lagrangians of te form L = 1 qm q V q, 9 also gives secon orer numerical metos [16]. Tis approac is calle te Generalize Coorinate Formulation. Remark 4.1 In general, tis viewpoint is necessarily local, since te iscretizations are only vali in te coorinate omain U of ϕ. If we coose an atlas of carts covering te wole manifol Q, we cannot guarantee tat te construction of te iscrete Lagrangian L will coincie on te cart overlaps. Tere are certain cases, owever, in wic tis is inee possible. For example, if we can fin an atlas {U s, ϕ s } suc tat for any two overlapping carts, ϕ s1 an ϕ s, te local iffeomorpism ϕ s1 s = ϕ s1 ϕ 1 s verifies ϕ s1 s 1 αx + αy = 1 αϕ s1 s x + αϕ s1 s y, for any x, y ϕ s U s an ϕ s1 s = i, ten it is easy to see tat one can paste te local constructions 9 respectively 10 to ave a well-efine iscrete Lagrangian on a neigbouroo of te iagonal of Q Q. [A simple example of tis situation is given by te manifol S 1, wit te local carts ϕ 1 z 1, z = arcsinz /z 1 0, π an ϕ z 1, z = arcsinz /z 1 π, π]. Remark 4. Anoter way of constructing a well-efine iscrete Lagrangian on a neigbouroo of te iagonal of Q Q is te following. Assume tat tere exist a q 0 Q an a ifferentiable mapping Υ : Q DiffQ suc tat Υqq = q 0, for

9 Nonolonomic Integrators 9 all q Q. In tis case, we can efine L for eac q, q accoring to eiter 9 or 10 by means of ϕ = ϕ 0 Υq, were ϕ 0 is a local cart wose coorinate omain contains q 0. It is important to note tat in tis construction te mapping ϕ 0 Υq varies wit te pair q, q. Tis is te case for instance of finite imensional Lie groups Q = G, were one can take q 0 = e, te ientity element, Φg = L g 1 for eac g G an ϕ 0 = exp 1 e see [6]. We sall make use of tis construction in Section Te extrinsic point of view assumes tat Q is embee in some linear space V an tat we ave a Lagrangian L : T V R suc tat L T Q = L. In aition, it is assume tat tere exists a vector value constraint function g : V R l, suc tat g 1 0 = Q V, wit 0 a regular value of g. Accoring to 9 an 10, we can efine te following iscrete Lagrangians on V V L α v k, v k+1 = L 1 αv k + αv k+1, v k+1 v k, 11 an L sym,α v k, v k+1 = 1 L 1 αv k + αv k+1, v k+1 v k L αv k + 1 αv k+1, v k+1 v k. In tis way, we can sum an subtract points in Q because we are regaring tem as vectors in V by means of te natural inclusion j : Q V. Of course, we must ensure tat te points obtaine by te algoritm all belong to Q. Ten, te solution sequence v k will extremize te action sum S = N 1 k=0 L v k, v k+1 subject to te olonomic constraints impose by g. Tis leas to te iscrete equations { D 1 L v k, v k+1 + D L v k 1, v k = λ l Dg l v k gv k+1 = 0. Tis approac is calle te Constraine Coorinate Formulation. Bot formulations are sown to be equivalent in te omain of efinition of te local cart ϕ selecte in te Generalize Coorinate Formulation see [36], wereby te following ientification is unerstoo: L q k, q k+1 = L jq k, jq k+1, wic is vali for coices of te cart U, ϕ in te efinition of L suc tat te map J = j ϕ 1 : ϕu R n V is linear. Notice tat tis assumption is not at all restrictive, since j is an injective immersion an suc a cart U, ϕ can always be cosen. In te nonolonomic case, we can construct an appropriate aaptation of bot formulations. In te Generalize Coorinate Formulation, we introuce D as follows. Take a local basis of 1-forms of te anniilator of te constraint istribution D, {ω 1,..., ω m } D o. Tese 1-forms can be intreprete as functions linear in te velocities, locally efine on T Q. Ten, we iscretize tem accoring to te previous iscretizations of te Lagrangian, tat is, we take eiter ωq a k, q k+1 = ω a ϕ 1 1 αϕq k + αϕq k+1, ϕ 1 ϕqk+1 ϕq k, 13

10 Nonolonomic Integrators 10 or ωq a k, q k+1 = 1 ωa ϕ 1 1 αϕq k + αϕq k+1, ϕ 1 ϕqk+1 ϕq k + 1 ωa ϕ 1 1 αϕq k + αϕq k+1, ϕ 1 ϕqk+1 ϕq k In tis way we obtain te functions ω a : Q Q R wose anniilation efines D Q Q. As in te unconstraine case, it is not ar to prove tat te iscretization 10 togeter wit14 yiels secon orer accurate approximations to te continuous flow, wereas tis is only guarantee for te iscretization 9, 13 if α = 1. In te Constraine Coorinate Formulation, we assume tat tere exist local 1- forms on V efining D o, { ω 1,..., ω m } suc tat ω a q TqQ = ω a q for q Q. Ten, we iscretize tem accoring to ω v a k, v k+1 = ω a 1 αv k + αv k+1, v k+1 v k, 15 an ω v a k, v k+1 = 1 ωa 1 αv k + αv k+1, v k+1 v k + 1 ωa αv k + 1 αv k+1, v k+1 v k Observe tat we can ientify ω a q k, q k+1 = ω a jq k, jq k+1 in te same way as we ave one for te iscrete Lagrangians. Ten, te iscrete Lagrange- Alembert principle wit te olonomic constraints g an te nonolonomic constraints ω 1,..., ω m leas us to te equations D 1 L v k, v k+1 + D L v k 1, v k = λ l Dg l v k + µ a ω a v k gv k+1 = 0 ω av k, v k+1 = 0. Te following teorem, analogous to te one presente in [36], ensures tat bot formulations 6 an 17 are inee equivalent in te same sense as before, as one migt expect. We prove it for te iscretizations 9, 13. Te proof for te symmetric iscretizations 10, 14 is analogous. Teorem 4.3 Let ϕ : U Q R n be a local cart of Q suc tat J = j ϕ 1 is linear. Ientify U wit ϕu an j U wit J U troug ϕ. Let q k 1, q k be two initial points in te coorinate cart an let v k 1 = Jq k 1, v k = Jq k. Ten, te Generalize Coorinate Formulation 6 as a solution q k+1, µ k a Coorinate Formulation 17 as a solution v k+1, λ k l = µ k an µ k a a.. 17 if an only if te Constraine, µ k a. Inee, v k+1 = Jq k+1 Proof: To establis te equivalence, we first expan equations 6 an 17 in terms of L an te 1-forms { ω a } m a=1. Let v, v enote te canonical coorinates of T V.

11 Nonolonomic Integrators 11 Equations 6 become, wen written in matrix form, { [ 1 L D T Jq k v a k, b k L ] v a k+1, b k α L v a k+1, b k+1 + α L } v a k, b k + µ k a ω a Jq k = 0 ω ajq k, Jq k+1 = ω a a k+1, b k+1 = 0, 18 were a k = αjq k + 1 αjq k 1 an b k = 1 Jq k Jq k 1. Note tat we are using te ientifications ω a = ωa Q Q, L α Q Q = L α an ωa T Q = ωa. If ω a = ω a l vl an ω a = ω a i q i, we ave tat ω a i q k q i = J ω a l v l q k = J l q i q k ω a l Jq k q i, wic can be written in a more compact way as ω a q k = D T Jq k ω a Jq k. Here an in te following te superscript T refers to te transpose of a matrix. On te oter an, equation 17 can be written as 1 [ L v v k 1+α, v k v k 1 L v + α L v gv k+1 = 0 v k+α, v ] k+1 v k + 1 α L v v k 1+α, v k v k 1 v k+α, v k+1 v k + µ k a ω a v k = λ k l Dg l v k 19 l ω av k, v k+1 = ω l av vk+1 v k k+α = 0, were te sortan notation v k+α = 1 αv k + αv k+1 is use. Now, assume tat v k+1, λ k l, µ k a is a solution of 19 wit v k = Jq k an v k 1 = Jq k 1. Te fact tat gv k+1 = 0 implies tat v k+1 belongs to te image of J. Let q k+1 = J 1 v k+1. Multiplying te first equation of 19 by D T Jq k an making te corresponing substitutions, one obtains for te pair q k+1, µ k a just te first equation of 18, since te term D T Jq k D T gv k cancels ue to g J = 0. Conversely, if q k+1, µ k a is a solution of 18, ten one can fin Lagrange multipliers λ k l, suc tat v k+1 = Jq k+1, λ k l, µ k a is a solution of 19 as follows. Te secon an te tir equation of 19 are automatically satisfie because of v k+1 Q an taking into account te secon equation of 18. Moreover, as DJq k an Dgq k are assume to ave full rank, we ave tat T vk V = RDJq k N D T Jq k, were RDJq k an N D T Jq k refer to te range an te kernel, respectively, of te operator uner consieration. Since RD T gq k N D T Jq k an im RD T gq k = im N D T Jq k, we can write T vk V = RDJq k RD T gq k. Now, te left-an sie of te first equation of 19 can be ecompose into a part belonging to RDJq k an a part belonging to RD T gq k. But te part in RDJq k is zero, because of te first equation of 18. Consequently, te entire expression belongs to RD T gq k, an tus tere exist some λ k l suc tat [ 1 L v k 1+α, v k v k 1 L v v v k+α, v k+1 v k ] + 1 α L v v k+α, v k+1 v k

12 Nonolonomic Integrators 1 + α L v k 1+α, v k v k 1 + µ k a ω a v k = λ k l Dg l v k, v wic is precisely te first equation of 19. QED Te relevance of Teorem 4.3 becomes apparent wen anling concrete examples. Generally, it is easier to treat te nonolonomic integrator following te Constraine Coorinate Formulation, since points in Q can be treate as points in some R s, an tis is a efinite avantage for te numerical implementation. On te oter an, te geometric stuy of te properties of iscrete nonolonomic mecanics is carrie out from te intrinsic point of view, so tings are prove for te Generalize Coorinate Formulation. 5. Geometric invariance properties In te unconstraine case, one can stuy iscrete mecanics by itself, starting from a given iscrete Lagrangian L an investigating te geometric properties tat te iscrete flow enjoys, suc as te preservation of te symplectic form or of te momentum in te presence of symmetry. Furtermore, wen one regars a iscrete mecanical system as an approximation of a continuous one, it turns out tat te symplectic-momentum nature of te variational integrators makes te ifference in capturing te essential features of Lagrangian systems. In te following, we provie some geometric arguments for te goo performance of te DLA algoritm wen compare to oter stanar iger orer numerical metos, suc as te 4 t orer Runge-Kutta, as will be sown in Section 6. Of course, a more toroug error analysis woul be of interest, but ere we focus our attention on te invariance properties tat te iscrete nonolonomic mecanics possesses, as a sign of its appropriateness for approximating te continuous counterpart. As we ave mentione above, in nonolonomic mecanics te symplectic form is not preserve by te flow of te system, so one can not expect te iscrete version to preserve it. However, we will sow in Section 5.1 tat te iscrete flow preserves te structure of te evolution of te symplectic form along te trajectories of te system. Tis property generalizes te symplectic caracter of variational integrators systems an, in fact, one precisely recovers te preservation of te symplectic form in te absence of constraints. Moreover, uner te action of a Lie group G on te configuration manifol Q, leaving invariant te Lagrangian L : T Q R an te constraints D T Q, te associate momentum J : T Q g in general will not be conserve eiter. Te evelopment of te reuction teory of nonolonomic Lagrangian systems wit symmetry as rawn on a careful examination of te compatibility of te symmetry irections an te constraints, wic is encoe in te intersection V D. Koiller [18] starte wit te so-calle vertical or purely kinematic case, V D = 0, an subsequent works [4, 7, 10, 9] ave treate te orizontal, V D, an te general cases 0 V D V oter relevant contributions make use, among oters, of te Hamiltonian formalism [3] or of Poisson metos [19, 5], among oters. An important

13 Nonolonomic Integrators 13 geometric object in tis reuction teory is te so-calle nonolonomic momentum map, wic correspons to te usual momentum map restricte to te symmetry irections compatible wit te constraints. Tis momentum map can be use to augment te constraints an provie a principal connection on Q Q/G, te nonolonomic principal connection, a fact wit important applications, for instance, to te control of nonolonomic systems [9]. In aition, one can measure te evolution of tis mapping along te integral curves of te Lagrange- Alembert equations. Tis measurement constitutes te nonolonomic momentum equation [4]. We sall sow in Section 5. tat te DLA algoritm satisfies a iscrete version of te nonolonomic momentum equation. In aition, in te presence of orizontal symmetries, we sall sow tat te associate momenta are actually conservation laws. In te vertical or purely kinematic case, tere are no symmetry irections lying in te constraint istribution an one oes not ave any nonolonomic momentum. Nonolonomic systems of Caplygin type form te most representative class of systems falling into tis category. In Section 5.3, we will iscuss ow, for Caplygin systems, te DLA algoritm passes to te reuce space Q/G an yiels a variational integrator in te sense of [16]. In some cases in agreement wit te continuous counterpart, tis reuce formulation exactly yiels a stanar variational integrator Te symplectic form In tis section, we investigate te beaviour of te DLA algoritm wit respect to te iscrete symplectic form Ω L efine in Section. In oing so, we first recall te properties of te continuous flow in tis regar, an ten sow tat te iscrete algoritm follows te same pattern. Te nonolonomic equations of motion 3 can be written in a coorinate free form [] in a symplectic context. To o so, we nee to introuce some geometric objects. In terms of te tangent bunle coorinates q A, q A, let us enote by = q A te q A ilation or Liouville vector fiel on T Q see [4] an by S = q A te canonical q A vertical enomorpism see [3]. Te action of S on a 1-form will be enote by S. Ten we can efine te Poincaré-Cartan 1-form an -form, corresponing to a given Lagrangian L, by Θ L = S L an Ω L = Θ L, respectively. We furter ave tat E L = L L represents te energy function of te system. If te Lagrangian L is regular, wic will always be tacitly assume in te sequel, Ω L is symplectic. Te equations of motion for te nonolonomic system are ten given by { i X Ω L E L D S T D o, 0 X D T D. Te integral curves of te ynamical vector fiel X satisfy precisely te nonolonomic equations 3. From 0, we can write i X Ω L = E L + β, wit β S T D o. Tis implies tat te evolution of te symplectic form along te trajectories of te system is given by L X Ω L = i X Ω L + i X Ω L = β, 1

14 Nonolonomic Integrators 14 were L enotes te Lie erivative. Te DLA algoritm also preserves tis structure for te evolution of te iscrete symplectic form Ω L. Inee we ave tat Φ Ω L = Φ FL Θ Q = FL Φ Θ Q = FL Φ Θ Q = FL Θ Q β, were β D o an in te last equality we ave use te efinition of te iscrete principle 6. Finally, we get Φ Ω L = Ω L + β, wic is te iscrete version of Eq. 1. Note tat in te absence of constraints, we precisely recover te conservation of te iscrete symplectic form. 5.. Te momentum In nonolonomic mecanics, te momentum associate to a symmetry group G of te system in general is not a conserve quantity. Instea, one consiers a nonolonomic momentum map J n, wic is te usual one restricte to te symmetry irections compatible wit te constraints, an erives a momentum equation escribing te evolution of J n. Wat we evelop in te following is a iscrete version of te nonolonomic momentum map an we sow tat te nonolonomic integrator 6 fulfills a iscrete version of te momentum equation. Let us briefly recall ow te teory is evelope in te continuous picture [4, 7]. Consier a Lie group G acting on te configuration manifol Q, suc tat te Lagrangian L : T Q R an te constraints D T Q are G-invariant. For eac q Q, te following subspace of te Lie algebra of G, g q = {ξ g / ξ Q q D q }, is introuce, were ξ Q enotes te funamental vector fiel associate to te element ξ g. Denote by g D te isjoint union of all suc subspaces, g D = q Q g q. Ten, we ave a generalize bunle g D Q wic captures at eac point q Q te symmetry irections wic lie in te constraint istribution. Define ten J n : T Q g D v q J n v q : g q R ξ J n v q, ξ = FLv q, ξ Q q. Note in passing tat te nonolonomic momentum map coincies wit te usual momentum map along te bunle g D. Assume we ave a smoot section ξ of te bunle g D Q an consier te function J ξ n : T Q R given by J ξ n = J n, ξ. By means of te Lagrange- Alembert principle one can now prove te following

15 Nonolonomic Integrators 15 Teorem 5.1 [4] Any solution qt of te Lagrange- Alembert equations for te nonolonomic system must satisfy te momentum equation J n = FL qt, t ξ t ξqt = L [ ] i Q q i t ξqt. 3 Q In some cases, it can appen tat an element ξ of te Lie algebra belongs to g q, for all q Q. Tis ten efines a constant section ξ of te bunle g D an ξ is calle a orizontal symmetry. As a consequence of te momentum equation 3, we get Corollary 5. If ξ is a orizontal symmetry, ten J n ξ is a conservation law. Next, we investigate tese issues for te iscrete Lagrange- Alembert principle evelope in Section 3. First, given te iscrete Lagrangian L : Q Q R, efine te iscrete momentum map by J n : Q Q g D q k 1, q k J nq k 1, q k : g q R ξ D L q k 1, q k, ξ Q q k. Take, as in te continuous case, a smoot section ξ of te bunle g D an consier te function J n ξ on Q Q. Ten, one fins tat te nonolonomic integrator fullfils te following iscrete version of te momentum equation. Teorem 5.3 Te flow q k 1, q k q k, q k+1 of te iscrete Lagrange- Alembert equations verifies J n ξq k, q k+1 J n ξq k 1, q k = D L q k, q k+1 ξqk+1 ξq k q k+1. 4 Proof: Te invariance of te iscrete Lagrangian L implies tat Lexps ξq k q k, exps ξq k q k+1 = Lq k, q k+1. Differentiating wit respect to s an setting s = 0 yiels D 1 L q k, q k+1 ξqk q k + D L q k, q k+1 ξqk Q Q Q q k+1 = 0. 5 On te oter an, te iscretization of te Lagrange- Alembert principle 6 implies tat D 1 L q k, q k+1 ξqk q k + D L q k 1, q k ξqk q k = 0. 6 Q Q Subtracting equation 5 from equation 6, we fin tat D L q k, q k+1 ξqk q k+1 = D L q k 1, q k ξqk Finally, te result follows from 7 since J n ξq k, q k+1 J n ξq k 1, q k = = D L q k, q k+1 ξqk+1 = D L q k, q k+1 ξqk+1 Q Q Q q k+1 D L q k 1, q k ξqk q k+1 D L q k, q k+1 ξqk Q q k. 7 Q Q q k q k+1.

16 Nonolonomic Integrators 16 QED In te presence of orizontal symmetries we fin tat te algoritm 6 exactly preserves te associate components of te momentum. Corollary 5.4 If ξ is a orizontal symmetry, nonolonomic integrator. ten J n ξ is conserve by te 5.3. Caplygin systems Consier te situation in wic te action of te Lie group G as no symmetry irection lying in D, i.e. V D = 0, were V enotes te vertical bunle of te projection π : Q Q/G. Uner te common assumption tat D q + V q = T q Q for all q Q imension assumption, one as inee a splitting of te tangent bunle at eac point q Q, T q Q = D q V q. Te istribution D being G-invariant, tis situation correspons precisely to te notion of a principal connection on te principal fiber bunle π : Q Q/G. Suc systems are known in te literature as generalize Caplygin systems [6, 18]. It is known tat one of te peculiarities of nonolonomic Caplygin systems is tat, after reuction by te Lie group G, tey take on te form of an unconstraine system, subject to an external force of a special type. In te following, we briefly review tese facts for te sake of clarity of te exposition Reuction in te continuous case In te Caplygin case, reuction can be acieve as follows. Consier te lifte action of te Lie group G on T Q an enote by ρ : T Q T Q te associate projection. Define by F te subbunle of T T Q along D wose anniilator is given by S T D o. Ten, V D = 0 implies tat F V ρ = 0 an terefore T D = F T D V ρ. Tis means tat on te principal bunle ρ D : D D T Q/G we ave anoter principal connection. Denote by : T D F T D an v : T D V ρ te orizontal an vertical projectors, respectively. Ten, we consier te 1-form α = i X i Θ L i Θ L, were i : D T Q is te canonical inclusion. On te oter an, te Lagrangian L inuces a Lagrangian L : T Q/G R by L q, v q = Lq, v q, were πq = q an v q enotes te unique vector in D q suc tat π v q = v q. Tis function is well-efine because of te G-invariance of L. Ten, one can prove tat te solution X of 0 is ρ-projectable an tat te projecte ynamics X = ρ X satisfies i X ω L = E L + α, 8

17 Nonolonomic Integrators 17 were α is te projection of te 1-form α. Moreover, one can sow tat te contraction of α wit X vanises, tat is, i X α = 0. Hence, te nonconservative force represente by α is of gyroscopic type. Tis implies in particular tat te energy E L is a conserve quantity of te reuce ynamics. Alternatively, a principal connection can be caracterize by a g-value 1-form A on Q te connection 1-form satisfying Aξ Q q = ξ for all ξ g an Aφ g X = A g AX for all X T Q. Te constraint istribution is precisely given by te orizontal space D q = {v q T q Q : Av q = 0}. Te principal bunle structure implies tat te configuration manifol Q can be locally seen as Q/G G. In te sequel, we will not make a notational istinction between r, g consiere as a point on te prouct manifol an consiere as te corresponing aapte coorinates. In eac case, te precise meaning soul be clear from te context. In terms of aapte bunle coorinates, te G-action on Q reas φ r, g = r, g an te projection π : Q Q/G is given by πr, g = r. Te coorinate expression for te connection 1-form ten reas Ar, gṙ, ġ = A g g 1 ġ + Ar, eṙ, so tat te constraint 1-forms become ω = g 1 g + Ar, er. If we fix a basis {e 1,..., e m } of te Lie algebra g, ten we can write ω = g 1 g + A b βrr β e b. 9 Te G-invariance of te Lagrangian yiels Lr, g, ṙ, ġ = Lr, e, ṙ, g 1 ġ. Denote ten by lr, ṙ, ξ te projection of L onto T Q/G. It follows tat te reuce Lagrangian L on T Q/G is given by L r, ṙ = lr, ṙ, A b β rṙβ e b. Te gyroscopic 1-form can be locally written as l A a β α = ξ a r Aa γ γ r + β ca bca b βa c γ ṙ γ r β, were te * on te rigt-an sie inicates tat, after computing te erivative of l wit respect to ξ b, one replaces te ξ a everywere by A a β rṙβ. Te constants c b ac appearing in te last term on te rigt-an sie are te structure constants of g wit respect to te cosen basis, i.e. [e b, e c ] = c a bc e a. Note in passing tat te expressions A b β r Ab γ γ r + β cb aca a βa c γ are te coefficients of te curvature of te principal connection A in local form. Consequently, te integral curves of te projecte solution X verify te equations L L t ṙ β r = α β β, were β {1,..., n m} Reuction of te iscrete principle Next, we examine te possibility of passing te iscrete nonolonomic principle to te reuce space Q/G. Consier a iscrete Lagrangian L : Q Q R an a iscrete space D, escribe by te anniilation of some constraint functions ω a : Q Q R, a {1,..., m}. Assume tat bot

18 Nonolonomic Integrators 18 te Lagrangian an te constraints are G-invariant uner te iagonal action of te Lie group on te manifol Q Q. Te DLA algoritm ten becomes { D 1 L r k, g k, r k+1, g k+1 + D L r k 1, g k 1, r k, g k = λ a ω a r k, g k ω ar 30 k, g k, r k+1, g k+1 = 0, were it soul be recalle tat D 1 enotes te erivative wit respect to q k = r k, g k. Tese equations can be rewritten, using te expression 9 for te constraint 1-forms, in te following form L r r β k, g k, r k+1, g k+1 + L r k r β k 1, g k 1, r k, g k = k L = r k, g k, r k+1, g k+1 + L r k 1, g k 1, r k, g k A b 31 g k g βr k L gk e b k ω ar k, g k, r k+1, g k+1 = 0, were L g enotes te left multiplication by g in G. It must be note tat in te rigtan sie of te first equation, a sortan notation is use to enote te natural pairing between tangent vectors an covectors on G. Observe tat D can be locally ientifie wit Q/G Q/G G via te assignement r k, g k, r k+1, g k+1 D r k, g k, r k+1, since g k+1 is uniquely etermine by te equations ω ar k, g k, r k+1, g k+1 = 0, a {1,..., m}. In aition, te G-invariance of te constraint functions implies tat g k+1 r k, g k, r k+1 = g k g k+1 r k, e, r k+1. Let us consier te restriction of L : Q Q R to D, L c : D R. Te G-invariance of L an D implies te G-invariance of L c. Define a iscrete Lagrangian on te reuce manifol as L L : Q/G Q/G R r k, r k+1 L c r k, e, r k+1. Now, we sall write te DLA algoritm 31 in terms of te constraine iscrete Lagrangian L c an ten examine te possibility of passing te equations to Q/G, in terms of te reuce iscrete Lagrangian L. First, we ave tat L c r β k L c r β k+1 Seconly, we also ave 0 = Lc = L r β k + L +1, +1 r β k = L + L +1. r β g k+1 k+1 r β k+1 = L + L = L + R L g +1, k +1 were R g enotes te rigt multiplication in te Lie group by te element g G. In view of tis, we see tat te nonolonomic integrator can be expresse in te following way D 1 L c r k, g k, r k+1 + D L c r k 1, g k 1, r k = F q k, q k+1 + F + q k 1, q k,

19 Nonolonomic Integrators 19 were F q k, q k+1 = L +1 + L r +1 r β k, g k, r k+1, g k+1 A b g βr k L gk e b k k L +1 = r +1 r β k, r k+1 + L A b g βr k L gk e b, k k F + q k 1, q k = L r β k = L r β k + L r k 1, g k 1, r k, g k A b βr k L gk e b r k 1, r k L gk 1 e b L 1 A b βr k L gk 1 A gk r k 1,r k e b. Note tat bot iscrete forces, F an F +, are G-invariant. Tis can be seen as follows. As L is G-invariant, we ave tat L r k, g k, r k+1, g k+1 = L r k, e, r k+1, g 1 k g k+1 = l r k, r k+1, f k,k+1, were we use te sortan notation f k,k+1 = g 1 k g k+1. From ere, one can erive tat L r k, g k, r k+1, g k+1 = L R l g g 1 f k k k,k+1 f k,k+1 L r k, g k, r k+1, g k+1 = L l. g g 1 k+1 k f k,k+1 Moreover, if r k, g k, r k+1, g k+1 D, ten f k,k+1 = g 1 k g k+1 = g k+1 r k, r k+1. Terefore, substituting in te expressions for te iscrete forces, one verifies tat F q k, q k+1 = l r k, r k+1, f k,k+1 +1 r k, r k+1 f k,k+1 r β k Rf l k,k+1 r k, r k+1, f k,k+1 A b f βr k e b, k,k+1 F + q k 1, q k = l r f k 1,k r β k 1, r k k l + L gk r k 1,r k r k 1, r k, f k 1,k A b f βr k e b. k 1,k Terefore, we can write a well-efine algoritm on Q/G of te form D 1 L r k, r k+1 + D L r k 1, r k = F r k, r k+1 + F + r k 1, r k. 3 Equation 3 belongs to te type of iscretization generalizing variational integrators for systems wit external forces evelope in [16]: δ L q k, q k+1 + F q k, q k+1 δq k + F + q k, q k+1 δq k+1 = 0, 33 were F, F + are te left an rigt iscrete friction forces. Equation 33 efines an integrator q k 1, q k q k, q k+1 given implicitly by te force iscrete Euler-Lagrange equations D 1 L q k, q k+1 + D L q k 1, q k + F q k, q k+1 + F + q k 1, q k = We summarize te above iscussion in te following result.

20 Nonolonomic Integrators 0 Teorem 5.5 Consier a iscrete nonolonomic problem wit ata L : Q Q R, a istribution D on Q an a iscrete constraint space D. Let G be a Lie group acting freely an properly on Q, leaving D invariant an suc tat T q Q = D q V q, for all q Q, were V enotes te vertical bunle of te G-action. Assume furter tat bot L an D are invariant uner te iagonal action of a Lie group G on te manifol Q Q. Ten, te DLA algoritm 31 passes to te reuce space Q/G, yieling a generalize variational integrator in te sense of [16]. We call te algoritm on Q/G te reuce iscrete Lagrange- Alembert algoritm RDLA. So far, we ave obtaine tat te DLA algoritm respects te structure of te evolution of te symplectic form along te flow of te system cf. Eq an tat, in te presence of symmetries, it satisfies a iscrete version of te nonolonomic momentum equation. In aition, we ave been able to establis in te two extremal cases orizontal an vertical Corollary 5.4 an Teorem 5.5, respectively. Tese results are important, bot from a geometrical an from a numerical perspective. On te one an, tey sow interesting interactions between te iscrete unconstraine an nonolonomic mecanics, similar to tose occuring in te continuous case. On te oter an, wen regaring te iscrete version of mecanics as an approximation of te continuous one, tey provie goo arguments to consier te propose DLA algoritm 6 as an appropriate in a symplectic-momentum sense iscretization of te continuous flow. It is wort noting, toug, tat wen regaring te iscrete nonolonomic mecanics as an approximation of te continuous one, one cannot expect te iagram LA RLA DLA RDLA to be commutative in general, because te two orizontal arrows symbolize processes tat are of a ifferent matematical nature iscrete an continuous, respectively. For instance, tere exist some special situations in wic te reuce Caplygin system amits a Hamiltonian escription, tat is, te gyroscopic force F vanises [6]. But in general, te RDLA will not be a stanar variational integrator. In te following section we sow tat, uner strong assumptions on te linearity of te geometric operations involve, te mipoint RDLA algoritm [wic correspons to take α = 1/ in Eq. 9 or in Eq. 10, since in tis case bot iscretizations coincie] yiels inee a variational integrator, i.e. te iagram is commutative. Te ypotesis on te linearity are justifie by te fact tat te iagram involves bot iscrete an continuous systems. Tis result provies an aitional reason since we alreay know tat tis type of iscretization always guarantees a secon orer accurate numerical approximation to te continuous flow to consier te mipoint rule as a reliable integrator.

21 Nonolonomic Integrators Te mipoint RDLA algoritm Consier a nonolonomic Caplygin system, wit te following ata: a principal G-bunle π : Q M = Q/G, associate to a free an proper action Ψ of G on Q, a Lagrangian L : T Q R wic is G-invariant wit respect to te lifte action on T Q, an linear nonolonomic constraints etermine by te orizontal istribution D of a principal connection γ on π. In tis section, we will focus our attention on te mipoint RDLA algoritm. For eac q = r k, g k Q, take a prouct cart ϕ = ϕ 1 ϕ, given by a cart ϕ 1 in Q/G an a cart ϕ in G. For te latter, we take see [6]: ϕ = exp 1 L g 1, wic k is efine in a neigbouro of g k, were exp : g G is te exponential mapping. Denote by η = 1 ϕ g k + 1 ϕ g k+1 = 1 logg 1 k g k+1, ζ = ϕ g k+1 ϕ g k = logg 1 k g k+1. We assume tat Q/G is itself a linear space, so tat we can always take te ientity cart ϕ 1 = i Q/G. Wit tis type of carts, we can construct te iscrete Lagrangian an te iscrete constraint istribution as explaine in Remark 4.. Te iscrete Lagrangian ten reas L 1 r k, g k, r k+1, g k+1 = L r k+ 1, ϕ 1 an te iscrete nonolonomic constraints β ζ + A b rk+1 r k βr k+ 1 e b = 0. η, r k+1 r k, ϕ 1 ζ, As before, te sortan notation r k+ 1 = 1 r k + 1 r k+1 is unerstoo. Te above iscretizations of te Lagrangian an of te constraints are G-invariant uner te iagonal action of te Lie group on te manifol Q Q. Here, we will make a ifferent ientification between D an Q/G Q/G G, taking into account te specific structure of te constraint functions. More precisely, we ientify D wit Q/G Q/G G via te assignment were r k, g k, r k+1, g k+1 D r k, r k+1, ĝ, ĝ = ϕ 1 η = L gk exp 1 ζ = g k exp 1 Ar k+ 1 r k+1 r k. Te inverse mapping r k, r k+1, ĝ r k, g k, r k+1, g k+1 D is given by g k+1 = ĝ exp 1 ζ, g k = ĝ exp 1 ζ. 35 Consier te restriction of L 1 : Q Q R to D, L c : D R. Define, as before, te iscrete Lagrangian L on te reuce manifol as L : Q/G Q/G R r k, r k+1 L c r k, r k+1, e.

22 Nonolonomic Integrators Ten, we ave tat L c r β k were, from 35, were r β k +1 r β k = L 1 r β k = 1 = 1 + L 1 r β k Analogously, we see tat L c r β k+1 r β k+1 +1 r β k+1 Seconly, we also ave 1 = L 1 r β k+1 = 1 = 1 A b βr k+ 1 1 A b βr k L 1 + L 1 +1, +1 r β k r β k+1 A b γ A b βr k+ 1 1 A b βr k = Lc ĝ = L ĝ + L ĝ 1 A b γ r r β k+ 1 r γ k+1 rγ k L gk e b, r r β k+ 1 r γ k+1 rγ k L gk+1 e b. + L 1 +1, +1 r β k+1 A b γ A b γ r r β k+ 1 r γ k+1 rγ k L gk e b, r r β k+ 1 r γ k+1 rγ k L gk+1 e b. = R L 1 exp 1 ζ 1 + R L. 36 exp 1 ζ +1 Now, we expan te term L 1 r k, g k, r k+1, g k+1 on te rigt-an sie of te first equation in te DLA algoritm 31 as L 1 r k, g k, r k+1, g k+1 = 1 L 1 r k, g k, r k+1, g k L 1 r k, g k, r k+1, g k+1, an ten make use of 36 to get te expression L 1 r k, g k, r k+1, g k+1 = 1 L 1 r k, g k, r k+1, g k+1 1 L 1 R expζ r k, g k, r k+1, g k Analogously, we fin for te oter term L 1 r k 1, g k 1, r k, g k = 1 L 1 r k 1, g k 1, r k, g k 1 L 1 R exp ζ r k 1, g k 1, r k, g k. 1 Ten, te iscrete forces in te RDLA algoritm take te form F q k, q k+1 = L 1 = 1 L 1 r β k + L r β k A b βr k L 1 A b γ r k, g k, r k+1, g k+1 A b g βr k L gk e b k r r β k+ 1 r γ k+1 rγ k + Ab βr k L gk e b

23 Nonolonomic Integrators L 1 A b βr k+ 1 1 A b γ r r β k+ 1 r γ k+1 rγ k L gk+1 e b +1 A b βr k L gk+1 A exp ζ e b, F + q k 1, q k = L r β k + L 1 r β k + L 1 r k 1, g k 1, r k, g k A b g βr k L gk e b k = 1 L 1 A b g βr k A b γ k 1 r r β k 1 r γ k rγ k 1 L gk 1 e b A b βr k L gk 1 A expζ e b + 1 L 1 A b βr k A b γ By te linear epenence of Ar on r, we ave tat r r β k 1 r γ k rγ k 1 + A b βr k L gk e b. A b βr k = A b βr k+ 1 1 A b β r r γ k+ 1 r γ k+1 rγ k, A b βr k = A b βr k A b β r r γ k 1 r γ k rγ k 1. Substituting into te expressions for te iscrete forces, we get F q k, q k+1 = 1 L 1 A b β r r γ k+ 1 r γ k+1 rγ k + 1 A b γ r r β k+ 1 r γ k+1 rγ k L gk e b + 1 L A b g βr k+ 1 1 A b γ k+1 r r β k+ 1 r γ k+1 rγ k L gk+1 e b A b βr k L gk+1 e b [ζ, e b ] = 1 L Ab β 4 r r γ k+ 1 r γ k+1 rγ k + Ab γ r r β k+ 1 r γ k+1 rγ k L gk e b L +1 A b β r r γ k+ 1 r γ k+1 rγ k Ab γ r r β k+ 1 r γ k+1 rγ k A a βr k A c γr k+ 1 c b car γ k+1 rγ k L gk+1 e b, F + q k 1, q k = 1 L A b g βr k A b γ k 1 r r β k 1 r γ k rγ k 1 L gk 1 e b A b βr k L gk 1 e b + [ζ, e b ] + 1 L 1 A b γ r r β k 1 r γ k rγ k A b β r r γ k 1 r γ k rγ k 1 L gk e b = 1 L Ab β 4 r r γ k 1 r γ k rγ k 1 + Ab γ r r β k 1 r γ k rγ k 1 1