Lie Group and Lie Algebra Variational Integrators for Flexible Beam and Plate in R 3

Transcription

1 Lie Group nd Lie Algebr Vritionl Integrtors for Flexible Bem nd Plte in R 3 THÈSE N O 5556 PRÉSENTÉE le 6 novembre À LA FACULTÉ DE L'ENVIRONNEMENT NATUREL, ARCHITECTURAL ET CONSTRUIT LABORATOIRE DE CONSTRUCTION EN BOIS PROGRAMME DOCTORAL EN STRUCTURES ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES PAR Frnçois Mrie Alin Demoures cceptée sur proposition du ury: Prof. M. Bierlire, président du ury Prof. Y. Weinnd, Prof. T. Rtiu, directeurs de thèse Prof. M. Desbrun, rpporteur Dr F. Gy-Blmz, rpporteur Prof. J.-F. Molinri, rpporteur Suisse

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3 Contents Introduction xiii History nd bckground. History Discrete Lgrngin mechnics Energy computtion Discrete energy Discrete Hmiltonin Discrete energy on Q Q nd extended vritionl principle.4 Discrete forced Lgrngin systems Sphericl pendulum 3. Lie group vritionl integrtor Lie group vritionl integrtor Discrete momentum mp Simple sphericl pendulum Geometric mechnics First temporl discretiztion Alterntive temporl discretiztion Exmple Spring pendulum 9 3. Pendulum ttched to the origin by very stiff spring Geometric mechnics Lie group vritionl integrtor Exmple Lie group vritionl integrtor of geometriclly exct bem dynmics Lgrngin dynmics of bem in R Bsic kinemtics of bem Kinetic energy Potentil energy Eqution of motion Lie group vritionl integrtor for the bem iii

4 iv Contents 4.. The Lie group structure nd triviliztion Sptil discretiztion Lie group vritionl integrtors Lie group vritionl integrtor for the bem Including externl torques nd forces Exmples Bem with deformed initil configurtion Bem with concentrted msses Alterntive temporl discretiztion Lie group vritionl integrtor for the bem Discrete body moment nd Legendre trnsforms Asynchronous Lie group vritionl integrtor AVI Multisymplectic geometry Asynchronous Lie group vritionl integrtor for the bem Energy conservtion Exmple AVI Discrete ffine Euler-Poincré equtions Affine Euler-Poincré reduction Affine Euler-Poincré equtions Affine reduction for fixed prmeter Mteril nd convective Lgrngin dynmics of bem in R Deformtion expressed reltive to the inertil frme Description of the vribles nd functions involved Kinetic energy Potentil energy Advected vribles nd ffine ction for the bem Equtions of motion Discrete ffine Euler-Poincré reduction Review of the discrete Euler-Poincré equtions Discrete ffine Euler-Poincré reduction Discrete ffine Euler-Poincré reduction for fixed prmeter6 5.4 Hmiltonin pproch The ffine Lie-Poisson lgorithm Discrete Hmiltonin flow Poisson property of the discrete ffine Euler-Poincré flow t fixed prmeter The ssocited Lie-Poisson structure t fixed prmeter Discrete Hmiltonin flow t fixed prmeter Vritionl integrtor Alterntive temporl discretiztion Lie lgebr vritionl integrtor of geometriclly exct bem dynmics 9 6. Lgrngin dynmics of bem in R Lie Algebr vritionl integrtor for the bem Lie group structure

5 Contents v 6.. Sptil discretiztion Discrete Euler-Lgrnge equtions on Lie groups Lie lgebr vritionl integrtor for the bem Numericl simultions Exmple Lie lgebr vritionl integrtor of geometriclly exct plte dynmics Lgrngin dynmics of plte in R Bsic kinemtics of plte Deformtion expressed reltive to the inertil frme Kinetic energy Potentil energy Eqution of motions nd constrints for the plte Lie lgebr vritionl integrtor for the plte Lie group structure Constrined Lie lgebr vritionl integrtor Forced constrined discrete Euler-Lgrnge equtions Sptil nd temporl discretiztion Sptil discretiztion Temporl discretiztion of the Lgrngin Lie lgebr vritionl integrtor Appendix : intermedite clcultion Conclusions Dissiption nd discrete ffine Euler-Poincré Review of continuous Euler-Poincré systems with forces Forced Euler-Lgrnge equtions nd momentum mp conservtion Euler-Poincré reduction with forces Euler-Poincré reduction for semi-direct products with equivrint forces Affine Euler-Poincré reduction with forces Reduction of discrete forced Lgrngin systems with symmetries Discrete ffine Euler-Poincré reduction with forces Discrete mechnicl connection Discrete Euler-Lgrnge equtions Euler-Lgrnge vritionl opertor Discrete Euler-Lgrnge opertor Discrete Euler-Poincré equtions Euler-Poincré vritionl opertor Discrete Euler-Poincré opertor; Lie group version Discrete Euler-Poincré opertor; Lie lgebr version Discrete mechnicl connection Principl connections Mechnicl connection in geometric mechnics

6 vi Contents Discrete momentum mps Discrete mechnicl connection nd discrete vritionl mechnics Discrete horizontl spce Bibliogrphy 99

7 Résumé Le but de cette thèse est de développer des intégrteurs vritionnels synchrones ou bien synchrones, qui puissent être utilisés comme des outils pour étudier des structure complexes composées de plques et de poutres soumises à de grndes déformtions et sous contrintes. Les modèles de poutre et de plque sont les modèles géométriquement excts, dont l espce de configurtion sont des groupes de Lie. Ils sont dptés à l modélistion d obets soumis à de grndes déformtions, où l énergie de déformtion élstique choisie convient pour les types de mtériux correspondnt à notre domine d étude isotropes, ou composites. Les trvux de J. E. Mrsden, de ses doctornts et post-doctornts, ont servi de bse pour développer des intégrteurs vritionnels, qui sont symplectiques et conservent prfitement les symétries. En outre, pr une bonne discrétistion, l obectivité des modèles de poutre et de plque étudiés est conservée. L idée qui sous-tend ce trvil est de tirer profit des propriétés de ces intégrteurs pour définir l position d équilibre des structures, que l on ne connit générlement ps à priori, insi que pour déterminer les contrintes, tout en conservnt les invrints de l structure. Prllèlement à l résolution de cette problémtique, nous poursuivons l démrche de J.E. Mrsden qui consiste à poser les bses d une mécnique discrète, vec ses théorèmes, ses xiomes, ses définitions qui ont l même vleur que les lois de l mècnique des milieux continus mis pour un domine discret. C est à dire que les trectoires discrètes d un mouvement obtenues pr ces intégrteurs vritionnels vérifient ces lois discrètes. Mots clés : poutre, plque, intégrteurs mécniques, principe d Hmilton, conservtion des symétries, mécnique discrète, réduction, intégrteur synchrone vii

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9 Abstrct The purpose of this thesis is to develop vritionl integrtors synchronous or synchronous, which cn be used s tools to study complex structures composed of pltes nd bems subected to lrge deformtions nd stress. We consider the geometriclly exct models of bem nd plte, whose configurtion spces re Lie groups. These models re suitble for modeling obects subected to lrge deformtions, where the stored energy chosen is dpted for the types of mterils used in our field isotropic or composite. The work of J. E. Mrsden, nd of his doctorl nd post-doctorl students, were the bsis for the development of vritionl integrtors which re symplectic nd perfectly preserve symmetries. Furthermore, discrete mechnicl systems with symmetry cn be reduced. In ddition, by good discretiztion, the strin mesures re unchnged by superposed rigid motion. The ide behind this work is to tke dvntge of the properties of these integrtors to define the equilibrium position of structures, which re generlly unknown, s well s to determine the constrints, while preserving the invrints of the structure. Along with solving these problems, we continue to develop the ides of J.E. Mrsden who lid the foundtions of discrete mechnics, with its theorems, xioms, nd definitions, which prllel those in continuum mechnics but for discrete domin. Tht is, the discrete trectories of motion obtined by vritionl integrtors stisfy these discrete lws. Keywords : bem, plte, mechnicl integrtors, vritionl principles, conservtion properties, discrete mechnics, symmetry, reduction, synchronous ix

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11 Acknowledgments Mny people hve contributed to this work. In prticulr, I thnk Tudor S. Rtiu with whom it hs been true plesure nd honor to work, nd I cnnot forget the importnt contribution of Frnçois Gy-Blmz with whom I hd the chnce to work. I lso thnk Yves Weinnd for llowing me to get involved in this dventure. During the work on this thesis, I hd the gret fortune to meet Jerrold E. Mrsden. Ech time we tlked he gve me vluble dvice nd pointed me towrds the relevnt literture. Moreover I thnk my co-uthors Sigrid Leyendecker, Julien Nembrini, nd Sin Ober-Blöbum, ll of whom hve contributed mrkedly to my enoyment of reserch. In ddition I would like to thnk Mthieu Desbrun for his vluble suggestions nd help long the wy. Finlly, this work would not hve been possible without the understnding of my dughter Phénissi. xi

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13 Introduction This work cn be seen either in the context of solving complex problems of pplied mechnics, or, eqully well in the perspective of the development of the theory of discrete mechnics. The first point of view is ontologiclly linked to the origin of the thesis proect, tht is, the desire to explore complex forms composed of multiple singulr points nd free forms with multiple contct points. The im is to find the equilibrium position s well s to clculte the stress to which the mteril is subected. Indeed, these forms re so complex nd flexible tht we cnnot guess t first glnce their equilibrium positions. The second perspective is directly relted to the properties enoyed by the mthemticl obects we develop, nmely, the conservtion of symmetries nd the study of the sttics nd the dynmics of the mteril. The mthemticl obects with which we work re worth studying nd nturlly fit the obects under study. This thesis further develops the subect of Lie group nd Lie lgebr vritionl integrtors s it pplies to exct models of bems nd pltes subect to lrge deformtions. Moreover, dditionl topics connected with discrete mechnics will be developed since they re needed in our development. The point of view tken in this thesis is tht we do not discretize the equtions but the problem itself. This is chieved by discretizing spce nd time in the setup of the of the studied obect. We will use vritionl principles for ll the problems tht we considered nd hence the gol is to discretize the vritionl principle in order to get discrete equtions of motion. Then, we shll formulte theorems in this discrete setting tht re nlogues of the clssicl continuous time nd spce sttements. A very successful nd well developed technique in numericl nlysis is the finite element method. It uses simplicil decomposition of the given domin nd discretizes the locl lw of the continuous problem. Thus, for mny importnt problems, especilly long time simultions for conservtive systems, the development of stble finite element methods remins extremely chllenging or even out of rech, the underlying geometric or vritionl structures of the simulted continuous systems being often destroyed. We believe tht this problem cn be circumvented by the use of vritionl integrtors. The geometric formultion of the continuous theory is used to guide the development of discrete nlogues of the geometric structure, such s discrete conservtion lws, xiii

14 xiv Introduction discrete multisymplectic forms, nd discrete vritionl principles. The pst yers hve seen mor developments in discrete vritionl mechnics nd corresponding numericl integrtors. The theory of discrete vritionl mechnics hs its roots in the optiml control literture of the 96 s. The vritionl view of discrete mechnics nd its numericl implementtion hs been developed in the pst ten yers minly by Jerrold Mrsden of Cltech, his students, postdocs, nd collbortors. Discrete mechnics ws born s result of the interply of clssicl theoreticl mechnics, numericl nlysis, nd computer science. It hs become incresingly importnt in concrete pplictions s different s the modelistion of specific physicl systems, nimtion, computer vision nd grphics, imge processing, shocks between elstic solids, tmospheric nd ocenogrphic simultions of Lgrngin coherent sttes, spcecrft mission design, nd mny others. Remrkbly, to our knowledge, there is no mor ppliction of these discrete mechnics techniques to civil engineering. In prticulr, we re not wre of ny ppliction of discrete mechnics to the study of surfces formed by pltes tied by multi-edges nd exhibiting shrp corners. Understnding nd controlling mny physicl systems typiclly requires numericl simultions of dynmics tht occurs over wide rnge of time nd spce scles. Recent yers hve seen n explosive growth of discrete mechnics, discrete exterior clculus, nd corresponding integrtors preserving vrious geometric structures. There hs been growing reliztion tht stbility of numericl methods cn be obtined by methods which re comptible with these structures in the sense tht mny discrete vritionl integrtors re symplecticmomentum methods, tht is, they preserve the symplectic structure on phse spce nd momentum mps rising from the symmetries of the system see e.g. Mrsden, nd West [9]. A lrge number of mechnicl systems in nture re governed by Hmilton s vritionl principle. The bsic ide of underlying discrete vritionl integrtors is to discretize the vritionl principle rther thn discretizing the equtions of motion themselves which is the stndrd pproch tken by the finite element method. Furthermore, well-known result Ge, nd Mrsden [35] sttes tht integrtors with fixed time step typiclly cnnot simultneously preserve energy, the symplectic structure, nd ll conserved quntities. But one cn still chieve this if one uses time step dptive schemes s in Kne, Mrsden, nd Ortiz [56] nd Lew, Mrsden, Ortiz, nd West [69] who developed the theory of AVIs bsed on the introduction of spcetime discretiztion llowing different time steps for different elements in given finite element. The first mthemticl model expressed in terms of discrete mechnics uses discrete vritionl integrtors; see Mrsden, nd West [9]. However, in order to hve the flexibility to focus on prts of the phse portrit where dynmics is more complicted, synchronous vritionl integrtors AVIs hve proved to be more effective. These integrtors re bsed, s mentioned erlier, on the introduction of spcetime discretiztion llowing different time steps for different

15 xv elements in finite element mesh long with the derivtion of time integrtion lgorithms in the context of discrete mechnics, i.e., the lgorithm is given by spcetime version of the discrete Euler-Lgrnge DEL equtions of discrete version of Hmilton s principle. The dvntge of these discrete vritionl integrtors is tht they preserve the symplectic structure clssicl property of mechnicl systems, nd preserve moment for systems with symmetry, hve excellent energy behvior even with some dissiption dded, nd llow the usge of different time steps t different points. These properties significntly enhnce the efficiency of these lgorithms. We shll use discrete vritionl integrtors in the study of bems nd shells. The second mthemticl model used to study thin-shells is bsed on the mthemticl formultion of three dimensionl elsticity s developed, for exmple, in Mrsden, nd Hughes [83]. This pproch tightly links elsticity theory with geometric mechnics nd symplectic geometry see, e.g., Abrhm, nd Mrsden []. Unfortuntely, our knowledge of nonlinerly elstic, lminted, or composite mterils, their dynmics, nd their behvior ner corners is very limited. Stndrd demonstrtions of the utility of given rod or shell theory for effectively pproximting limited number of problems should not led to the impression tht ll of these problems hve good numericl simultions. Much recent work on rod nd shell theory hs been motivted by developments in numericl nlysis nd computtionl techniques. Advntges of the discrete mechnics point of view. The finite element method is n importnt computtionl tool to study the dynmics nd the sttics of bems nd pltes. However, even with significnt dvnces in error control, convergence nd stbility of these finite pproximtions, the invrint geometric structures cn be lost. For exmple, in finite element pproximtion of the motion of the free rigid body, one cn gin or lose momentum nd thereby fil to preserve fundmentl geometric nd topologicl structures underlying the continuous model. The min problem with this method is tht it discretizes the differentil equtions of continuum mechnics in order to obtin position, discrete trectory, moment, or other relevnt quntities relevnt to the motion of the system. It is not t ll sure tht the solutions thus obtined stisfy some of the fundmentl properties of the continuum mechnicl model. A key point of this thesis is to work both on discrete theory of mechnics nd to use these results to study bems nd pltes. Tht is, s soon s one uses vritionl integrtor, the theoreticl results re checked, something tht is fr from being trivil since this work involves, for exmple, reduction theory, n indispensble tool in the study of stbility of reltive equilibri, nd multisymplectic theory, where one replces the discrete time point with mesh in spcetime thus llowing different time steps for different elements of the mesh when synchronous vritionl integrtors AVI re used. Thus, in this thesis we continue modestly the work begun by J. Mrsden nd his PhD students, tht is, to develop the generl theory of discrete mechnics.

16 xvi Introduction From theory to lgorithm. One chooses configurtion spce Q with coordintes {q } tht describes the configurtion of the system under study. The discrete version of the tngent bundle T Q of the configurtion spce Q is Q Q. Given n priori choice of time intervl t, point q, q Q Q corresponds to tngent vector t q. Given smooth Lgrngin L : T Q R, usully the kinetic minus the potentil energy, one ssocites to it discrete Lgrngin L d : Q Q R nd discrete ction functionl S d = N = L d q, q +, t. The discrete vritionl principle sttes tht δs d =, which mens tht one seeks sequences {q } k N for which the functionl S d is sttionry under vritions of q with fixed endpoints q nd q N. As in the smooth cse, the discrete vritionl principle leds to the discrete Euler-Lgrnge equtions D L d q, q, t + D L d q, q +, t =, where D k denotes the kth prtil derivtive, k =,. In this wy n updte rule q, q q, q + is obtined; this is the vritionl integrtor. If one uses time step dptive schemes s in Kne, Mrsden, nd Ortiz [56] we obtin vritionl integrtor for conservtive mechnicl systems tht re symplectic, energy, nd momentum conserving. Indeed, whtever the choice of the discrete Lgrngin, for the non-dissiptive nd non-forced systems, vritionl integrtors re symplectic nd conserve the symmetries. The symplectic nture of the integrtor is given by the conservtion of the discrete two-form Ω d = D D L d dq dq + on Q Q, which ppers s integrnd in the boundry terms of the discrete vritionl principle when endpoints re llowed to vry. Moreover, the energy behvior is remrkbly stble in the conservtive cse, s proved by Hirer, Lubich nd Wnner [4]. In describing the dynmic response of elstic bodies under loding, one begins by selecting reference configurtion B R 3 of the body t initil time t. The motion of the body is described by the deformtion mpping ϕ : B R 3. Let T be tringultion of B. A key observtion underlying the formultion of vritionl integrtors is tht, owing to the extensive chrcter of the Lgrngin, the following element-by-element dditive decomposition holds : L = L K, K T where L K is the contribution of the element K T to the totl Lgrngin L. Another key feture is the existence of synchronous vritionl integrtors where the elements K nd nodes defining the tringultion of the body re updted synchronously in time; ech element K crries its own set of time steps Θ K, which induces set of time steps Θ for ech node. The discrete Euler-Lgrnge equtions D L d x, x + D L d x, x + = re pplied to ech node, where x is the position of the node t time t Θ. One obtins n updte rule ssocited to this node nd thus the discrete trectories of the system.

17 xvii Method used to study complex structures. The principle is to consider the studied obect s system oscillting bout the equilibrium position under the influence of its lod. After pplying certin kind of dissiption which conserve the symmetries, we get equilibrium position. Then we obtin the strin nd the stress for the obtined deformtion. The obects studied re the exct nonliner models of bem nd plte of Simo, where the spce configurtion is Lie group, tht is SE3 or SO3 R 3. For the sptil discretiztion of this model we tke into ccount the developments in [4] to obtin perfect obectivity. We will develop Lie group nd Lie lgebr vritionl integrtors for given clssicl discrete Lgrngin, i.e., it equls kinetic minus potentil energy. These lgorithms re obtined by forming discrete version of Hmilton s vritionl principle. For dissiptive or forced systems, one uses the Lgrnge-d Alembert principle. Vritionl integrtors exhibit remrkble properties. For non-dissiptive nd non-forced problems, no mtter the choice of the discrete Lgrngin, they re symplectic nd momentum conserving. Moreover, with good dissiption, the momentum mps re conserved. In ddition, vritionl integrtors hve remrkbly good energy behvior see Hirer, Lubich, nd Wnner [4]. Orgniztion of the thesis. The thesis consists of nine chpters. In the first chpter, the theory of discrete mechnics is reviewed nd the necessry bckground is developed. In the second nd third chpters, we present two simple exmples, the sphericl pendulum nd the spring pendulum, in order to fmilirize the reder with vritionl integrtors. We lso compre two different time discretiztions. Chpter four is devoted to the numericl study of the Simó bem model. We develop severl Lie group vritionl integrtors, with two different time discretistions, both for synchronous nd synchronous integrtors. In chpter five, we develop discrete version of ffine Euler-Poincré equtions, extending discrete Euler-Poincré equtions for semi-direct products to the cse of n ffine representtion of the Lie group configurtion spce on the vector spce. This yields vritionl integrtor for bems. Associted to this theory, discrete Lie-Poisson reduction for semi-direct products is lso developed. In chpter six, we develop discrete Lie lgebr vritionl integrtor motivted by the fct tht, if pplicble, these integrtors re esier to implement thn the Lie group vritionl integrtors. We pply it to the Simó bem model in this chpter nd to the Simó plte model in chpter seven. In this second exmple, we need to hndle lso nturl holonomic constrint inherent to the model. In chpter eight, we ddress the problem of dissiption. We construct specific discrete model of dissiption such tht energy is dissipted but ngulr momentum is conserved. We lso estblish discrete ffine Euler-Poincré reduction with forces. This theory is pplied to bem nd plte models. Up to this point, ll mechnicl systems considered hd s configurtion spce Lie group, possibly infinite dimensionl. Chpter nine ddresses the generl problem nd is devoted to the discretiztion of the reduction process

18 xviii Introduction for mechnicl systems whose configurtion spce is generl mnifold. In this context, the stndrd continuous theory uses in n essentil mnner connection on principl bundle. Thus, we introduce the discrete mechnicl connection which enbles us to split the discrete trectory into its horizontl nd verticl prts, thereby obtining pir of discrete Lgrnge-Poincré equtions. This lso llows us to study the stbility of the motion nd, in prticulr, to dissocite mechnicl instbilities from instbilities due to the implementtion. Exmples of splitting of discrete trectories re given.

19 Chpter History nd bckground. History During the lst decde, mor developments hve been done in the re of discrete vritionl mechnics nd their corresponding numericl integrtors. The theory of discrete vritionl mechnics hs its roots in the optiml control literture of the 96 s. For instnce systems described by non-liner difference equtions, by Jordn nd Polk [54], mximum principle by Hwng nd Fn [47], discrete clculus of vritions by Cdzow []. In ddition, studies relevnt to the discrete mechnics begn in the 97 s : discrete time systems by Cdzow [], invrince properties of the discrete Lgrngin by Logn [77], discrete Lgrngin systems with symmetries by Med [79; 8; 8], time discretiztion by Lee [63]. This theory ws then developed in systemtic wy. A formultion of the discrete Hmilton s principle, discrete symplectic form, discrete momentum mp nd Noether theorem were given by Wendlndt nd Mrsden [5; 6], nd the time step dpttion in order to get symplectic-energy-momentum preserving vritionl integrtors by Kne, Mrsden nd Ortiz [56]. Discrete nlogues of Euler-Poincré nd Lie-Poisson reduction theory with discrete Lgrngin were developed by Mrsden, Pekrsky nd Shkoller [86], discretiztion of the Lgrnge d Alembert principle s well s vritionl formultion of dissiption by Kne, Mrsden, Ortiz nd West [57], long time behviour of symplectic methods by Hirer nd Lubitch [4] bckwrd error nlysis by Benettin nd Giorgilli [5], Hirer [38], Hirer nd Lubitch [39], Reich [98]. And to conclude this period, Mrsden nd West [9], gve n importnt review of integrtion lgorithms for finite dimensionl mechnicl systems, tht re bsed on the discrete vritionl principle. From this time, bsed on different vritionl formultions e.g. Lgrnge, Hmilton, Lgrnge-d Alembert, Hmilton-Pontrygin, etc., vritionl integrtors hve been developed in vrious fields : The integrtors hve been extended to non smooth frmework by Kne, Repetto, Ortiz nd Mrsden [58], by Fetecu, Mrsden, Ortiz nd West [3],

20 Chpter. History nd bckground nd by Pndolfi, Kne, Mrsden, nd Ortiz [96]. The theory of Lgrngin mechnics on Lie groups, with discrete Lgrngin reduction, discrete Euler-Poincré equtions, nd semi-direct product ws developed by Bobenko nd Suris [; ], nd by Mrsden, Pekrsky nd Shkoller [87]. Thereby Lee, Leok nd McClmroch studied vritionl pproch on the Lie group of rigid bodies configurtions, for exmple, under their mutul grvity in [64]. In multisymplectic geometry, Mrsden, Ptrick nd Shkoller [85] hve investigted spcetime multisymplectic formultion. And new clss of synchronous vritionl integrtors AVI for non-liner elstodynmics hs been introduced by Lew, Mrsden, Ortiz nd West [7; 7]. A Lie-Poisson integrtor for Lie-Poisson Hmiltonin system ws developed by M nd Rowley [78]. In stochstic mechnics discrete Lgrngin theory for stochstic Hmiltonin system hs been exhibited by Bou-Rbee nd Owhdi [4]. In order to solve optiml control problems for mechnicl systems, Ober- Blöbum, Junge nd Mrsden [94] proposed optimiztion lgorithm, which lets the discrete solution directly inherit chrcteristic structurl properties from the continuous one. Furthermore Kobilrov nd Mrsden [6] constructed necessry conditions for optiml trectories, with mechnicl systems on Lie groups. To study mechnicl systems with holonomic nd non holonomic constrints, where there re bundnce of importnt models, Kobilrov, Mrsden, nd Sukhtme [59] proposed verticl nd horizontl splitting of the vritionl principle with non-holonomic constrints. And, with holonomic constrints, using the discrete null spce method, Leyendecker, Mrsden, nd Ortiz [74], s well s Leyendecker, Ober-Blöbum, Mrsden, nd Ortiz [76] hve eliminted the constrint forces nd reduced the system to its miniml dimension. Multiscle systems with fst vribles which hve computtionl cost determined by slow vribles were exmined by To, Owhdi, nd Mrsden [4]. As consequence of these developments, vritionl integrtors hve become incresingly importnt in concrete pplictions such s nimtion, computer vision nd grphics, imge processing, shocks between elstic solids, tmospheric nd ocenogrphic simultions of Lgrngin coherent sttes, spcecrft mission design. In prticulr, we mention the works of Gwlik, Mullen, Pvlov, Mrsden, nd Desbrun [3], nd those of Pvlov, Mullen, Tong, Knso, Mrsden nd Desbrun [97] in fluid mechnics; tht of Ryckmn nd Lew [3] in contct problems; nd one of Bergou, Wrdezky, Robinson, Audoly nd Grinspun [6] in computer science. However, these new tools hve not yet been fully explored in the context engineering sciences nd this work ims to contribute in this direction.

21 .. Discrete Lgrngin mechnics 3. Discrete Lgrngin mechnics In this section we briefly review some bsic fcts bout discrete Lgrngin mechnics, following Mrsden, nd West [9]. Let Q be the configurtion mnifold of mechnicl system. Suppose tht the dynmics of this system is described by the Euler-Lgrnge equtions ssocited to Lgrngin L : T Q R defined on the tngent bundle of the configurtion mnifold Q. Recll tht these equtions chrcterize the criticl curves of the ction functionl ssocited to L, nmely d L dt q L T q = δ Lqt, qtdt =, for vritions of the curve vnishing t the endpoints. Recll tht the Legendre trnsform ssocited to L is the mpping FL : T Q T Q tht ssocites to velocity its corresponding conugte momentum, where T Q denotes the cotngent bundle of Q. It is loclly given by q, q q, L q. Symmetries of the systems re given by Lie group ctions Φ : G Q Q, g, q Φ g q under which the Lgrngin is invrint. In this cse, the Noether theorem gurntees tht the ssocited momentum mp J : T Q g, given by Jα q, ξ = α q, ξ Q q α q T Q, ξ g.. is conserved quntity, where g denotes the Lie lgebr of the Lie group G, g its dul, nd the vector field ξ Q on Q is the infinitesiml genertor of the ction ssocited to ξ g, tht is, ξ Q q := d dε Φ expεξ q, ε= where exp : g G is the exponentil mp of the Lie group G. Discrete Euler-Lgrnge equtions. We shll now recll the discrete version of this pproch see e.g. [9]. Suppose tht time step t hs been fixed, denote by {t = t =,..., N} the sequence of time, nd by q d : {t } N = Q, q dt = q discrete curve. Let L d : Q Q R, L d = L d q, q + be discrete Lgrngin which we think of s pproximting the ction integrl of L long the curve segment between q nd q +, tht is, we hve L d q, q + t + t Lqt, qtdt, where qt = q nd qt + = q +. The discrete Euler-Lgrnge equtions re obtined by pplying the discrete Hmilton s principle to the discrete ction S d q d = N = L d q, q +.

22 4 Chpter. History nd bckground The resulting equtions D L d q, q + D L d q, q + =, for =,..., N... re clled the discrete Euler-Lgrnge equtions. The discrete Legendre trnsforms F + L d, F L d : Q Q T Q ssocited to L d re defined by F + L d q, q + := D L d q, q + T q +Q F L d q, q + := D L d q, q + Tq Q,..3 so tht the discrete Euler-Lgrnge eqution cn be equivlently written s F + L d q, q = F L d q, q +, for =,..., N...4 If both discrete Legendre trnsforms re loclly isomorphisms for nerby q nd q +, then we sy tht L d is regulr. When the discrete Lgrngin L d is regulr, the discrete Euler-Lgrnge equtions define well-defined discrete Lgrngin evolution opertor X Ld : Q Q Q Q Q Q, X Ld q, q = q, q, q, q +, nd well-defined discrete Lgrngin flow F Ld : Q Q Q Q, F Ld q, q = q, q +. Similrly s in the continuous cse, the discrete Lgrngin one forms Θ + L d nd Θ L d on Q Q re obtined by pulling-bck the cnonicl one-form Θ on T Q vi the Legendre trnsform, tht is Θ ± L d = F ± L d Θ,..5 where we recll tht Θ is defined by Θα q, w αq = T πq w αq, α q, with π Q : T Q Q the cotngent bundle proection. We thus hve the locl formuls Θ + L d q, q + = D L d q, q + dq +, Θ L d q, q + = D L d q, q + dq,..6 where Θ + L d q, q + Tq Q nd Θ + L d q, q + Tq Q. Note tht dl d = Θ + L d Θ L d so tht dθ + L d = dθ L d. Thus there only one single discrete Lgrngin symplectic two form Ω Ld := dθ + L d = dθ L d nd we hve Ω Ld = F ± L d Ω,..7 where Ω = dθ is the cnonicl symplectic form on T Q, nd where both F + L d nd F L d cn be used to define Ω Ld. A mp f : Q Q Q Q is sid to be specil discrete symplectic mp if f Θ L d = Θ L d nd f Θ + L d = Θ + L d. It is clled discrete symplectic mp

23 .. Discrete Lgrngin mechnics 5 if f Ω Ld = Ω Ld. For exmple, the discrete Lgrngin flow F Ld is discrete symplectic mp: F Ld Ω Ld = Ω Ld. The discrete Hmiltonin mp F Ld : T Q T Q is defined by F Ld := F ± L d F Ld F ± L d, where F Ld is the discrete Lgrngin flow. The fct tht the discrete Hmiltonin mp cn be equivlently defined with either discrete Legendre trnsform is consequence of the fct tht the following digrm commute. F L d T Q F Ld T Q F + L d Q Q Q Q F Ld F L d q, p F Ld q, p F Ld q +, p + F + L d F F L d + L d F L d F L d q, q q, q + q +, q + F Ld F Ld Figure..: Properties of the discrete Legendre trnsforms nd discrete flows Discrete Lgrngin systems with symmetries. Let Φ be group ction of Lie group G on Q with the infinitesiml genertor ξ Q q ssocited to the Lie lgebr element ξ g. There is nturlly induced ction on Q Q given by Φ Q Q g q, q + := Φ g q, Φ g q +, with the infinitesiml genertor ξ Q Q q, q + = ξ Q q, ξ Q q +. Given discrete Lgrngin L d : Q Q R not necessrily G-invrint, the discrete Lgrngin momentum mps J + L d, J L d : Q Q g re defined by J + L d q, q +, ξ = Θ + L d q, q +, ξ Q Q q, q + = F + L d q, q +, ξ Q q + J L d q, q +, ξ = Θ L d q, q +, ξ Q Q q, q +..8 Note tht we hve = F L d q, q +, ξ Q q. J ± L d = F ± L d J, where J : T Q g is the cotngent lift momentum mp given by Jα q, ξ = α q, ξ Q q. It is importnt to note tht if the discrete curve {q } N = verifies the discrete Euler-Lgrnge then we hve the equlity J + L d q, q = J L d q, q +, for ll =,..., N...9

24 6 Chpter. History nd bckground T Q J g J ± L d Q Q F ± L d g g J + L d J L d Q Q Q Q J L d F Ld µ µ J + J L d L J d L d q, q q, q + F Ld Figure..: On the left: the definition of the discrete momentum mps. Two digrms on the right: illustrtion of the equlity..9 When G cts on Q Q by specil discrete symplectic mps, tht is, if Φ Q Q g Θ ± L d = Θ ± L d, then the discrete Lgrngin momentum mps re G- equivrint, tht is, J + L d Φ Q Q g = Ad g J+ L d, J L d Φ Q Q g = Ad g J L d. This hppens for exmple if the discrete Lgrngin L d is G-invrint, since in this cse Φ Q Q g is specil discrete symplectic mp. Moreover, in this cse the two momentum mps coincide: J + L d = J L d, nd therefore, from..9 we obtin the discrete Noether s theorem... Theorem Discrete Noether s theorem Consider given discrete Lgrngin system L d : Q Q R which is invrint under the lift of the left ction Φ : G Q Q. Then the corresponding discrete Lgrngin momentum mp J Ld : Q Q g is conserved quntity of the discrete Lgrngin mp F Ld : Q Q Q Q, tht is, J Ld F Ld = J Ld..3 Energy computtion Review of geometric mechnics. Given the Lgrngin L : T Q R we define the ction A : T Q R, v q FLv q, v q with v q = q, q, nd the energy by E = A L. The Lgrngin L is sid hyperregulr if the Legendre trnsform FL is diffeomorphism. Then we hve the following theorem.3. Theorem The hyperregulr Lgrngins L on T Q nd hyperregulr Hmiltonin H on T Q correspond in biective mnner : H is constructed from L by mens of H = E FL, nd L from H by mens of L = A E = A H FH, where FH : T Q T Q T Q is the fiber derivtive of H : T Q R. Thus, in this cse, we cn clculte the energy of the system, t time t, both using the energy E s well s the Hmiltonin H with the sme result. But with the discrete configurtion Q Q we cnnot define discrete ction. On the other hnd we cn dd the discrete kinetic energy with the discrete potentil energy nd obtin in somewy discrete energy. However if the discrete

25 .3. Energy computtion 7 Lgrngin is regulr, we cn go from the discrete structure in time on Q Q to the continuous structure in time on T Q by the discrete Legendre trnsform nd we obtin the discrete energy E d or the discrete Hmiltonin H d. In this cse if we wnt to do this process properly, it seems necessry to define the generl frmework in terms of chin complexes. Chin complexes. Let set S = {,..., p } of p + independent points in R n. The geometric p-simplex in R n is the set of ll points of the p-dimensionl hyperplne H p, contining S, for which the brycentric coordintes with respect to S re ll non-negtive. The p-simplex, with n ordering of its vertices, is denoted p =,..., p. We obtin geometric -complex X by quotienting collection of disoint simplices identified by some fces vi homeomorphisms preserving the ordering of vertices. The -complex re denoted simplicil complex when simplices re uniquely determined by their vertices. Then we consider prticulr set of morphisms σ i : p X, for ll p {,..., n}, which is the orienttion of the fces of the simplexes with respect to ech other in X, such tht σ i p = p i. A p-dimensionl chin on the -complex X with coeficients in group G is function c p on the oriented p-simplexes of X with vlues in the group G such tht if c p p i = g i, then c p p i = g i. The collection of ll such p- dimensionl chins on X is denoted p-chin complex C p X, G. This p-chin complex C p X, G provided with the boundry opertor p : p X p X, which verify p p =, forms n obect of the ctegory of the chin complexes Ch. Where the morphisms ϕ p : C p X, G C p X, G of this ctegory re the morphisms of belin groups such tht the commuttive reltion p ϕ p c p = ϕ p p c p holds for ech chin c p C p X, G. We hve the commuttive digrm in Figure p+ C p X, R ϕ p p C p X, R p... ϕ p... p+ C p X, R p C p X, R p... Figure.3.: Chin complex Ch.3. Discrete energy Let configurtion Q : C D, G, where D is compct with piecewise boundry, nd G given Lie group. Let n hyper-regulr Lgrngin L : T Q R. We ssume tht there is G-invrint Riemnnin metric γ on the configurtion

26 8 Chpter. History nd bckground spce Q, nd tht the Lgrngin is of the form Lq, q = γ q, q V q, for potentil V : Q R. The ssocited Legendre trnsform FL : T Q T Q becomes in this cse FLv q, w q = γv q, w q. After sptil discretiztion of D, we obtin set of oriented simplices n i with nodes, tht we will denote for simplifiction by K. Given the configurtion q Q t nodes K, we get by interpoltion n hyper-regulr Lgrngin L K : T Q R on K s L K q K, q K := K γ q, q V K q K, where q K = {q } K. Such tht we hve the pproximtion γ q, q V q dv D γ q, q V K q K. K T K The set of simplices K correctly ssembled, together with the Lgrngin L K tking vlues in R, forms n-chin complex T C n D, R, where the morphisms re the mps which tke n oriented simplicil complex t time t nd brings it t time t +, nd which my be continuous or not. Then we pply temporl discretiztion by constructing n incresing sequence of times {t = t =,, N} R from the time-step t, nd obtin the discrete regulr Lgrngin L d q K, q+ K : Q Q R which is time discretiztion of t + L t K q K, q K dt, with q K = q Kt, nd q + K = q Kt +, L d q K, q+ K t + t L K q K, q K dt. In such wy tht L d s well s L K re defined on T. Furthermore, becuse of the discrete regulrity of L d, there exists locl isomorphism F L d : {q } Q T q Q, nd nother one F + L d : Q {q + } T q +Q, where : T Q T Q is the inverse of the index lowering opertor : T q Q v v, T q Q. See digrm.3.. R L d Q Q F ± L d T Q T Q L K R.3. Thus we cn define n energy E d = A L K on T Q t time t for simplex

27 .3. Energy computtion 9 K, to be E d q K F, L d K = FL K F L d K = γ F L d K, F L d, F L d with { q K F, L d = q, F L d K L K q K, F L d K + V K q K,.3. with F ± L d := F± L d q, q +. Moreover n energy, t time t +, cn be equivlently defined with F + L d..3. Discrete Hmiltonin We know tht vritionl integrtor on Q Q preserves the discrete symplectic form Ω d = F ± L d Ω, where Ω is the cnonicl two-form on T Q. Then the discrete Hmiltonin flow F Ld = F + L d F L d will preserve the pushforwrds of these structures see Mrsden nd West [9]. Therefore the discrete Hmiltonin flow } K q, F L d q, q + q +, F + L d q, q + is symplectic with respect to the Poisson brcket {, } on T Q. And it is possible to define n Hmiltonin function H d = E d FL K on T Q such tht H d q K F, L d K = F L d, FL K F L d L K q K, F L d K K = F L d, FL K F L d + V K q K..3.3 K An Hmiltonin function, t time t +, cn be equivlently defined with F + L d. Moreover we know by.3. tht the discrete energy nd the discrete Hmiltonin hve the sme vlue for given discrete Legendre trnsform F ± L d., Almost-conservtion of energy. The min feture of the numericl scheme q, q q, q + given by solving the discrete Euler-Lgrnge equtions is tht the ssocited scheme q, p q +, p + induced on the phse spce T Q through the discrete Legendre trnsform defines symplectic integrtor. Here we supposed tht the discrete Lgrngin L d is regulr, tht is, both discrete Legendre trnsforms F + L d, F L d : Q Q T Q re loclly isomorphisms

28 Chpter. History nd bckground for nerby q nd q +. The symplectic chrcter of the integrtor is obtined by showing tht the scheme q, q q, q + preserves the discrete symplectic two-forms Ω ± L d := F ± L d Ω cn, where Ω cn is the cnonicl symplectic form on T Q, so tht q, p q +, p + preserves Ω cn nd is therefore symplectic, see Mrsden, nd West [9], Lew, Mrsden, Ortiz, nd West [7]. It is known, see Hirer, Lubich, nd Wnner [4], tht given Hmiltonin H, symplectic integrtor for H is exctly solving modified Hmiltonin system for Hmiltonin H which is close to H. So the discrete trectory hs ll the properties of conservtive Hmiltonin system, such s conservtion of the energy H. The sme conclusion holds on the Lgrngin side for vritionl integrtors see e.g. Lew, Mrsden, Ortiz, nd West [7]. This explins why energy is pproximtely conserved for vritionl integrtors, nd typiclly oscilltes bout the true energy. We refer to Hirer, Lubich, nd Wnner [4] for detiled ccount nd full tretment of bckwrd error nlysis for symplectic integrtors..3.3 Discrete energy on Q Q nd extended vritionl principle We cn lso clculte the energy on Q Q, by extending the configurtion in time, in the frmework of the multysymplectic geometry, see Mrsden, Ptrick, nd Shkoller [85], Lew, Mrsden, Ortiz nd West [69; 7] mong other ppers. The discrete Lgrngin is now defined on Q Q R L d q, q +, t + t : Q Q R R. Then we extend the discrete vritionl principle using positions nd time, nd we get system of two equtions D L d q, q, t t + D L d q, q +, t + t =, D 3 L d q, q, t t D 3 L d q, q +, t + t =. The second eqution mens tht following energy E d defined t time t is conserved E d = D 3L d q, q +, t + t..3.4 This discrete energy generlly represents the sum of discrete kinetic nd potentil energy. And we note tht this definition is more generl thn previous ones s it does not require the discrete regulrity of L d..4 Discrete forced Lgrngin systems To integrte discrete Lgrngin with discrete externl forcing it is possible to extend the discrete vritionl frmework to include forcing, s ws done in Mrsden, nd West [9]. In presence of n externl force field, given by

29 .4. Discrete forced Lgrngin systems fiber preserving mp F : T Q T Q, Hmilton s principle is replced by the Lgrnge-d Alembert principle T T δ Lqt, qtdt + F qt, qt δq dt =, where F q, q δq is the virtul work done by the force field F with virtul displcement δq. This principle yields the Lgrnge-d Alembert equtions d L dt q L = F q, q q Review of discrete forced Lgrngin systems. structures, there re two discrete Lgrngin forces As with other discrete F ± d : Q Q T Q, which re fiber preserving in the sense tht π q F ± d = π± Q, where π Q : T Q Q is the cotngent bundle proection nd π ± Q : Q Q Q re defined by π Q q, q + = q nd π + Q q, q + = q +. Thus, in locl coordintes, we hve F d q, q + = q, F d q, q + nd F + d q, q + = q +, F + d q, q + We now recll from Mrsden nd West [9] the discrete Lgrnge-d Alembert principle..4. Theorem Discrete Lgrnge-d Alembert principle Let L d : Q Q R be discrete Lgrngin, nd consider the discrete Lgrngin forces F ± d : Q Q T Q. Then the following re equivlent: i The discrete curve {q } stisfies the discrete Euler-Lgrnge equtions for L d with forcing: D L d q, q + D L d q, q + + F + d q, q + F d q, q + =, for ll =,..., N. ii The discrete Lgrnge d Alembert principle L d q, q + + N δ = N = [ F d q, q + δq + F + d q, q + δq +] =, holds for vritions δq with fixed endpoints δq = δq N =..4.

30 Chpter. History nd bckground Note tht in the discrete Lgrnge-d Alembert principle, the two discrete forces F + d n F d combine to give single one-form F d : Q Q T Q Q given by F d q, q + δq, δq + = F + d q, q + δq + + F d q, q + δq. The forced discrete Euler-Lgrnge equtions implicitly define the forced discrete Lgrngin mp F Ld : Q Q Q Q. Although in the continuous cse we used the stndrd Legendre trnsform for systems with forcing, in the discrete cse it is necessry to tke the forced discrete Legendre trnsforms to be F F L d q, q + = q, D L d q, q + + F d q, q + F F + L d q, q + = q +, D L d q, q + + F + d q, q As in..4, the discrete Euler-Lgrnge equtions with forces cn be equivlently written s F F + L d q, q = F F L d q, q +, for =,..., N. The forced discrete Hmiltonin mp is defined by := F F ± L d F Ld F F ± L d. F Ld We thus hve q +, p + = F Ld q, p where p = D L d q, q + F d q, q + nd p + = D L d q, q + +F + d q, q +. Discrete forced Noether s theorem. Consider n ction Φ : G Q Q, nd let L d : Q Q R be discrete Lgrngin. In the presence of forcing, the discrete momentum mps re defined by J F + L d q, q +, ξ = F F + L d q, q +, ξ Q q + J F L d q, q +, ξ = F F L d q, q +, ξ Q q. Note tht these expressions recover..8, when the forces re zero. If the discrete force F d is orthogonl to the group ction, so tht F q, ξ Q Q =, for ll ξ g, then we hve J F + L d = J F L d,.4.3 nd we denote by J F L d : Q Q g this unique mp. With this nottion, we hve the following result, which is the discrete nlog of Theorem Theorem Discrete forced Noether s theorem Let Φ : G Q Q be n ction nd let L d : Q Q R be G-invrint discrete Lgrngin system with discrete forces F + d, F d : Q Q T Q, such tht F d, ξ Q Q =, for ll ξ g. Then the discrete momentum mp J F L d : Q Q g will be preserved by the discrete Lgrngin mp, so tht J F L d F Ld = J F L d.

31 Chpter Sphericl pendulum Introduction The sphericl pendulum is composed of single mss which is fixed to pivot point. This is simple exmple where the potentil W ext is completely determined by the externl grvittionl field nd the symmetry is bout the verticl xis. During the motion, the mss spins round the verticl xis while oscillting between two prllel circles of the sphere. Thus we develop geometric vritionl discretiztion especilly well-suited for systems on Lie groups. Bsed on Moser, nd Veselov [93], discrete Euler- Lgrnge equtions for systems on Lie groups, nd the ssocited discrete Lgrngin reductions hve been crried out in Bobenko, nd Suris [; ], Mrsden, Pekrsky, nd Shkoller [86], nd further developed in Lee [66] nd pplied to mny exmples. These integrtors re referred to s Lie group vritionl integrtors. See lso Iserles, Munthe-Ks, Nørsett, nd Znn [5] for relted pproch for solving differentil equtions on Lie groups. With this simple exmple we highlight the properties of vritionl integrtors. Tht is, we observe the conservtion of symmetries i.e., of the momentum mps nd the lmost constnt behvior of the totl energy in time. Moreover, with respect to the work of Lee [66], we pid specil ttention to the discretiztion of the Lgrngin, in prticulr, to the speed..8 tht remins in the Lie lgebr fter discretiztion. One of the beutiful nd importnt things bout discrete mechnics is tht it permits to switch from the discrete to the continuous world by the Legendre trnsform. This is wht we did in writing the Hmiltonin equtions from the discrete vritionl point of view.. Lie group vritionl integrtor We recll briefly the Lie group vritionl integrtor presented in Bobenko nd Suris []. 3

32 4 Chpter. Sphericl pendulum.. Lie group vritionl integrtor Let configurtion spce Q = C D, G g which is Lie group, nd where D is compct domin. We suppose tht G cts on Q by left trnsltion, which is smooth mpping Φ : G Q Q h, g Φh, g =: hg,.. such tht i for ll g Q, Φe, g = g, where e is the identity element of G, nd ii for every h, h G, Φh, Φh, g = Φh h, g for ll g Q. Let L : T Q R be smooth regulr Lgrngin defined on the tngent bundle T Q to the Lie group Q. From now on, by convenience, we denote in this subsection the configurtion spce by G insted of Q. And, for convenience, we trnslte the vector ġ T g G to g, g ġ G T e G, by left triviliztion s described in Bobenko nd Suris []. We recll tht T e G =: g is the Lie lgebr of the Lie group G. By vector bundle isomorphism we trivilized the Lgrnge function L by pull-bck through G g g, g ġ ġ T G, which induces L : G g R defined by Lg, ξ := Lg, ġ, g ġ := ξ... Given n intervl of time [, T ], define the pth spce to be CG = C[, T ], G = {g : [, T ] G g is C curve}. Then curve g CG is sid to be solution of the Euler-Lgrnge equtions in terms of L, if g stisfies d L d L L ξ = g dt ξ ξ g. It is importnt to note tht by triviliztion we crry ġ T g G to g ġ g, which is expressed in body coordintes see Abrhm nd Mrsden []. By sptil nd temporl discretiztion we get, on the intervl of time [t, t + ], the discrete trivilized Lgrngin L d := L dg, f : G G R, with f = g g +. Let C d G = {g d : {t } N = G, t g := gt } be the discrete pth spce. The discrete ction mp S d L d : C d Q R is defined by S d L d := N = L d g, f. Let gε be deformtion of g in C d G, such tht gε = g nd gε N = g N for ny ε in n open intervl ] λ, λ[ nd g = g for ll =,,..., N.. Let δg := d dɛ gε T g G ɛ=