Skip to main content
SpringerLink
Log in

Geometric Mechanics, Lagrangian Reduction, and Nonholonomic Systems

  • Chapter
Mathematics Unlimited — 2001 and Beyond

Abstract

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

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Abraham, R. and J. E. Marsden [1978]: Foundations of Mechanics. Addison-Wesley, Second edition

    Google Scholar 

  • Abraham, R., J. E. Marsden and T.S. Ratiu [1988]: Manifolds, Tensor Analysis and Applications. Applied Mathematical Sciences, vol. 75. Springer, New York, Second edition

    Book  MATH  Google Scholar 

  • Alber, M. S., G. G. Luther, J. E. Marsden and J. M. Robbins [1998]: Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction. Physica D123, 271–290

    Article  MATH  MathSciNet  Google Scholar 

  • Alber, M. S., G. G. Luther, J. E. Marsden and J. W. Robbins [1998]: Geometry and control of three-wave interactions. Fields Inst. Commun. 24, 55–80

    MATH  MathSciNet  Google Scholar 

  • Arms, J. M., R. H. Cushman and M. Gotay [1991]: A universal reduction procedure for Hamiltonian group actions. In: The Geometry of Hamiltonian systems, T. Ratiu, ed., MSRI Series 22, 33–52. Springer, New York

    Chapter  Google Scholar 

  • Arms, J. M., J. E. Marsden and V. Moncrief [1981]: Symmetry and bifurcations of momentum mappings. Comm. Math. Phys. 78, 455–478

    Article  MATH  MathSciNet  Google Scholar 

  • Arms, J. M., J. E. Marsden and V. Moncrief [1982]: The structure of the space solutions of Einstein’s equations: II Several Killings fields and the Einstein-Yang-Mills equations. Ann. Phys. 144, 81–106

    Article  MATH  MathSciNet  Google Scholar 

  • Arnold, V.I. [1966]: Sur la géométrie differentielle des groupes de Lie de dimenson infinie et ses applications à l’hydrodynamique des fluidsparfaits. Ann. Inst. Fourier, Grenoble16, 319–361

    Article  MathSciNet  Google Scholar 

  • Arnold, V. I., V. V. Kozlov and A. I. Neishtadt [1988]: Mathematical aspects of classical and celestial mechanics. In: Dynamical Systems III. V. I. Arnold, ed.. Springer

    Chapter  Google Scholar 

  • Arnold, V. I. [1989]: Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, vol. 60. Springer, Second edition

    Book  Google Scholar 

  • Bates, L. [1998]: Examples of singular nonholonomic reduction. Rep. Math. Phys. 42, 231–247

    Article  MATH  MathSciNet  Google Scholar 

  • Bates, L. and E. Lerman [1997]: Proper group actions and symplectic stratified spaces. Pacific J. Math. 181, 201–229

    Article  MATH  MathSciNet  Google Scholar 

  • Bates, L. and J. Sniatycki [1993]: Nonholonomic reduction. Rep. Math. Phys. 32, 99–115

    Article  MATH  MathSciNet  Google Scholar 

  • Bloch, A.M. and P. Crouch [1992]: On the dynamics and control of nonholonomic systems on Riemannian Manifolds. In: Proceedings of NOLCOS ’92, pp. 368–372

    Google Scholar 

  • Bloch, A. M. and P. Crouch [1994a]: Nonholonomic and vakonomic control systems on Riemannian manifolds. Fields Inst. Commun. 1, 25–52

    MATH  MathSciNet  Google Scholar 

  • Bloch, A. M. and P. E. Crouch [1994]: Reduction of Euler-Lagrange problems for constrained variational problems and relation with optimal control problems. Proc. CDC33, 2584–2590

    Google Scholar 

  • Bloch, A. M. and P. E. Crouch [1998]: Newton’s law and integrability of nonholonomic systems. SIAM J. Control Optim. 6, 2020–2039

    Article  MATH  MathSciNet  Google Scholar 

  • Bloch, A.M. and P.E. Crouch [1999]: Optimal control, optimization, and analytical mechanics. In: Mathematical Control Theory, J. Ballieul, ed. Springer, New York, pp. 268–321

    Chapter  Google Scholar 

  • Bloch, A. M., P. S. Krishnaprasad, J. E. Marsden and R. Murray [1996]: Nonholonomic mechanical systems with symmetry.Arch. Rational Mech. Anal. 136, 21–99

    Article  MATH  MathSciNet  Google Scholar 

  • Bloch, A.M., PS. Krishnaprasad, J.E. Marsden and T.S. Ratiu [1994]: Dissipation induced instabilities. Ann. Inst. H. Poincaré, Analyse Nonlinéaire 11, 37–90

    Article  MATH  MathSciNet  Google Scholar 

  • Bloch, A.M., P.S. Krishnaprasad, J.E. Marsden and T. S. Ratiu [1996]: The Euler-Poincaré equations and double bracket dissipation. Comm. Math. Phys. 175, 1–42

    Article  MATH  MathSciNet  Google Scholar 

  • Bloch, A.M., P.S. Krishnaprasad, J.E. Marsden and G. Sánchez de Alvarez [1992]: Stabilization of rigid body dynamics by internal and external torques. Automatica 28, 745–756

    Article  MATH  MathSciNet  Google Scholar 

  • Bloch, A. M., N. Leonard and J. E. Marsden [1998]: Matching and stabilization by the method of controlled Lagrangians. Proc. CDC 37, 1446–1451

    Google Scholar 

  • Bloch, A.M., N. Leonard and J.E. Marsden [1999]: Controlled Lagrangians and the stabilization of mechanical systems I: The First Matching Theorem. IEEE Trans. Automat. Control (to appear)

    Google Scholar 

  • Bloch, A. M., M. Reyhanoglu and H. McClamroch [1992]: Control and stabilization of nonholonomic systems. IEEE Trans. Automat. Control37, 1746–1757

    Article  MATH  MathSciNet  Google Scholar 

  • Bobenko, A. I., B. Lorbeer and Yu. B. Suris [1998]: Integrable discretizations of the Euler top. J. Math. Phys. 39, 6668–6683

    Article  MATH  MathSciNet  Google Scholar 

  • Bobenko, A. I., A. G. Reyman and M. A. Semenov-Tian-Shansky [1989]: The Kowalewski Top 99 years later: A Lax pair, generalizations and explicit solutions. Comm. Math. Phys. 122, 321–354

    Article  MATH  MathSciNet  Google Scholar 

  • Bobenko, A.I. and Y.B. Suris [1999a]: Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top. Commun. Math. Phys. 204, 147–188

    Article  MATH  MathSciNet  Google Scholar 

  • Bobenko, A.I. and Y.B. Suris [1999b]: Discrete Lagrangian reduction, discrete Euler-Poincaré equations, and semidirect products. Lett. Math. Phys. 49, 79–93

    Article  MATH  MathSciNet  Google Scholar 

  • Bondi, H. [1986]: The rigid body dynamics of unidirectional spin. Proc. Roy. Soc. London 405, 265–274

    Article  MATH  MathSciNet  Google Scholar 

  • Bourbaki, N. [1983]: Variétés differentielles et analytiqes. Fascicule de résultats, Diffusion C. C. L. S., Paris

    Google Scholar 

  • Brockett, R. W. and L. Dai [1992]: Nonholonomic kinematics and the role of elliptic functions in constructive controllability. In: Nonholonomic Motion Planning, Z. Li and J. F. Canny, eds. Kluwer, pp. 1–22

    Google Scholar 

  • Bretherton, F. P. [1970]: A note on Hamilton’s principle for perfect fluids. J. Fluid Mech. 44, 19–31

    Article  MATH  Google Scholar 

  • Bryant, R. and P. Griffiths [1983]: Reduction for constrained variational problems and ∫ K2/2ds. Am. J. Math. 108, 525–570

    Article  Google Scholar 

  • Cannas da Silva, A. and Weinstein, A. [1999]: Geometric Models for Noncommutative Alebras. Berkeley Mathematics Lecture Notes, vol. 10. Amer. Math. Soc.

    Google Scholar 

  • Cantrijn, F., M. de León and D. Martin de Diego [1999]: On almost-Poisson structures in nonholonomic mechanics. Nonlinearity12, 721–737

    Article  MATH  MathSciNet  Google Scholar 

  • Cantrijn, F., Cortés, J., de Leon, M. and Martin de Diego, D. [2000]: On the geometry of generalized Chaplygin systems (preprint)

    Google Scholar 

  • Cantrijn, F., de León, M., Marrero, J.C. and Martin de Diego, D. [1998]: Reduction of nonholonoimc mechanical systems with symmetries. Rep. Math. Phys. 42, 25–45

    Article  MATH  MathSciNet  Google Scholar 

  • Cardin, F. and M. Favretti [1996]: On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints. J. Geom. Phys. 18, 295–325

    Article  MATH  MathSciNet  Google Scholar 

  • Cartan, E. [1928]: Sur la représentation géométrique des systèmes matèriels non holonomes. Atti. Cong. Int. Matem. 4, 253–261

    MathSciNet  Google Scholar 

  • Castrillón Lopez, M., Ratiu, T. S. and Shkoller, S. [2000]: Reduction in principal fiber bundles: CovariantEuler-Poincaré equations. Proc. Amer. Math. Soc. 128, 2155–2164; see http://www.ams.org/proc/

    Article  MATH  MathSciNet  Google Scholar 

  • Cendra, H., D. D. Holm, M. J. W. Hoyle and J. E. Marsden [1998]: The Maxwell-Vlasov equations in Euler-Poincaré form. J. Math. Phys. 39, 3138–3157

    Article  MATH  MathSciNet  Google Scholar 

  • Cendra, H., D.D. Holm, J.E. Marsden and T.S. Ratiu [1998]: Lagrangian Reduction, the Euler-Poincaré equations and semidirect products. Amer. Math. Soc. Transl. 186, 1–25

    MATH  Google Scholar 

  • Cendra, H., A. Ibort and J.E. Marsden [1987]: Variational principal fiber bundles: a geometric theory of Clebsch potentials and Lin constraints. J. Geom. Phys. 4, 183–206

    Article  MATH  MathSciNet  Google Scholar 

  • Cendra, H. and J. E. Marsden [1987]: Lin constraints, Clebsch potentials and variational principles. Physica D 27, 63–89

    Article  MATH  MathSciNet  Google Scholar 

  • Cendra, H., J. E. Marsden and T. S. Ratiu [2000]: Lagrangian reduction by stages. Mem. Amer. Math. Soc. (to appear). See http://www.cds.caltech.edu/~marsden/

    Google Scholar 

  • Chaplygin, S. A. [1897a]: On the motion of a heavy body of revolution on a horizontal plane. In: Physics Section of the Imperial Society of Friends of Physics, Anthropology and Ethnographics, Moscow, 9, 10–16 [Reproduced in Chaplygin (1954), pp. 413–425]

    Google Scholar 

  • Chaplygin, S. A. [1897b]: On some feasible generalization of the theorem of area, with an application to the problem of rolling spheres. Mat. SbornikXX, 1–32 [Reproduced in Chaplygin (1954), pp. 434–454]

    Google Scholar 

  • Chaplygin, S. A. [1903]: On a rolling sphere on a horizontal plane. Mat. SbornikXXIV, 139–168 [Reproduced in Chaplygin (1949), pp. 72–99, and Chaplygin (1954), pp. 455–471]

    Google Scholar 

  • Chaplygin, S. A. [1911]: On the theory of the motion of nonholonomic systems. Theorem on the reducing factor. Mat. SbornikXXVIII, 303–314 [Reproduced in Chaplygin (1949), pp. 28–38 and Chaplygin (1954), pp. 426–433]

    Google Scholar 

  • Chaplygin, S. A. [1949]: Analysis of the Dynamics of Nonholonomic Systems. Classical Natural Sciences, Moscow

    Google Scholar 

  • Chaplygin, S.A. [1954]: Selected Works on Mechanics and Mathematics. State Publ. House, Technical-Theoretical Literature, Moscow

    MATH  Google Scholar 

  • Chetayev, N. G. [1941]: On the equations of Poincaré. J. Appl. Math. Mech. 5, 253–262

    MathSciNet  Google Scholar 

  • Chetayev, N. G. [1961]: The Stability of Motion. Pergamon

    Google Scholar 

  • Chetayev, N. G. [1989]:Theoretical Mechanics. Springer

    Google Scholar 

  • Courant, T. [1990]: Dirac manifolds. Trans. Amer. Math. Soc. 319, 631–661

    Article  MATH  MathSciNet  Google Scholar 

  • Crabtree, H. [1909]: Spinning Tops and Gyroscopic Motion. Chelsea

    MATH  Google Scholar 

  • Cushman, R. and L. Bates [1997]: Global Aspects of Classical Integrable Systems. Birkhäuser, Boston

    Book  MATH  Google Scholar 

  • Cushman, R., J. Hermans and D. Kemppainen [1995]: The rolling disc. In: Nonlinear dynamical systems and chaos (Groningen, 1995), Progr. Nonlinear Differential Equations Appl., vol. 19. Birkhäuser, Basel, pp. 21–60

    Google Scholar 

  • Cushman, R., Kemppainen, D., Śniatycki, J. and Bates, L.M. [1995]: Geometry of nonholonomic constraints. Rep. Math. Phys. 36, 275–286

    Article  MATH  MathSciNet  Google Scholar 

  • Ebin, D. G. and J. E. Marsden [1970]: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92, 102–163

    Article  MATH  MathSciNet  Google Scholar 

  • Ge, Z. and J. E. Marsden [1988]: Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory. Phys. Lett. A 133, 134–139

    Article  MATH  MathSciNet  Google Scholar 

  • Getz, N. H. and J. E. Marsden [1994]: Symmetry and dynamics of the rolling disk. CPAM Preprint, 630

    Google Scholar 

  • Getz, N. H. and J. E. Marsden [1995]: Control for an autonomous bicycle. In: International Conference on Robotics and Automation, IEEE

    Google Scholar 

  • Golubitsky, M., J. E. Marsden, I. Stewart and M. Dellnitz [1995]: The constrained Liapunov Schmidt procedure and periodic orbits. Fields Inst. Commun. 4, 81–127

    MATH  MathSciNet  Google Scholar 

  • Golubitsky, M. and D. Schaeffer [1985]: Singularities and Groups in Bifurcation Theory, vol. 1. Applied Mathematical Sciences, vol. 69. Springer

    Google Scholar 

  • Golubitsky, M. and I. Stewart [1987]: Generic bifurcation of Hamiltonian systems with symmetry. Physica D24, 391–405

    Article  MATH  MathSciNet  Google Scholar 

  • Golubitsky, M., I. Stewart and D. Schaeffer [1988]: Singularities and Groups in Bifurcation Theory, vol. 2. Applied Mathematical Sciences, vol. 69. Springer

    Google Scholar 

  • Guichardet, A. [1984]: On rotation and vibration motions of molecules. Ann. Inst. H. Poincaré40, 329–342

    MATH  MathSciNet  Google Scholar 

  • Guillemin, V., E. Lerman and S. Sternberg [1996]: Symplectic Fibrations and Multiplicity Diagrams. Cambridge University Press

    Book  MATH  Google Scholar 

  • Guillemin, V. and E. Prato [1990]: Heckman, Kostant and Steinberg formulas for symplectic manifolds. Adv. Math. 82, 160–179

    Article  MATH  MathSciNet  Google Scholar 

  • Guillemin, V. and S. Sternberg [1978]: On the equations of motions of a classic particle in a Yang-Mills field and the principle of general covariance. Hadronic J.1, 1–32

    MATH  MathSciNet  Google Scholar 

  • Guillemin, V. and S. Sternberg [1980]: The moment map and collective motion. Ann. of Phys. 1278, 220–253

    Article  MATH  MathSciNet  Google Scholar 

  • Guillemin, V. and S. Sternberg [1982]: Convexity properties of the moment map. Invent.Math. 67, 491–513; also 77, 533–546

    Article  MATH  MathSciNet  Google Scholar 

  • Guillemin, V. and S. Sternberg [1984]: Symplectic Techniques in Physics. Cambridge University Press

    MATH  Google Scholar 

  • Hamel, G. [1904]: Die Lagrange-Eulerschen Gleichungen der Mechanik. Z. für Mathematik u. Physik50, 1–57

    MATH  Google Scholar 

  • Hermans, J. [1995]: A symmetric sphere rolling on a surface. Nonlinearity 8, 1–23

    Article  MATH  MathSciNet  Google Scholar 

  • Hernandez, A. [2000]: Singular Reduction and Blowing Up. PhD Thesis, Caltech

    Google Scholar 

  • Holm, D. D. and B.A. Kupershmidt [1983]: Poisson brackets and Clebsch representations for magnetohydrodynamics, multifluid plasmas and elasticity. Physica D6,347–363

    Article  MATH  MathSciNet  Google Scholar 

  • Holm, D.D. and B.A. Kupershmidt [1983]: Poisson structures and superconductors.Lett. A93, 177–181

    Article  MathSciNet  Google Scholar 

  • Holmes, P. J. and J. E. Marsden [1983]: Horseshoes and Arnold diffusion for Hamiltonian systems on Lie groups. Indiana Univ. Math. J. 32, 273–310

    Article  MATH  MathSciNet  Google Scholar 

  • Holm, D. D., J. E. Marsden and T. S. Ratiu [1986a]: The Hamiltonian structure of continuum mechanics in material, spatial and convective representations. In: Séminaire de mathématiques supérieures 100, 11–122. Les Presses de L’Univ. de Montréal

    MATH  Google Scholar 

  • Holm, D. D., J. E. Marsden and T. S. Ratiu [1986b]: Nonlinear stability of the Kelvin-Stuart cat’s eyes flow. SIAM, Lects. in Appl. Math. 23, 171–186

    MATH  MathSciNet  Google Scholar 

  • Holm, D. D., J.E. Marsden and T. S. Ratiu [1998a]: Euler-Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 349, 4173–4177

    Article  Google Scholar 

  • Holm, D. D., J. E. Marsden and T. S. Ratiu [1998b]: The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137, 1–8

    Article  MATH  MathSciNet  Google Scholar 

  • Holm, D.D., J.E. Marsden and T. Ratiu [1999]: The Euler-Poincaré equations in geophysical fluid dynamics. In: Proceedings of the Isaac Newton Institute Programme on the Mathematics of Atmospheric and Ocean Dynamics, Cambridge University Press (to appear)

    Google Scholar 

  • Holm, D. D., J. E. Marsden, T. S. Ratiu and A. Weinstein [1985]: Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1–6

    Article  MATH  MathSciNet  Google Scholar 

  • Ibort, A., M. De Leon, J. C. Marrero and D. Martin De Diego [1999]: Dirac brackets in constrained dynamics. Fortschr. Phys. 30, n 8, 459–492

    Article  MATH  MathSciNet  Google Scholar 

  • Iwai, T. [1982]: The symmetry group of the harmonic oscillator and its reduction. J.Math. Phys. 23, 1088–1092

    Article  MATH  MathSciNet  Google Scholar 

  • Iwai, T. [1985]: On reduction of two degrees of freedom Hamiltonian systems by an S1 action and SO (1, 2) as a dynamical group. J. Math. Phys. 26, 885–893

    Article  MATH  MathSciNet  Google Scholar 

  • Iwai, T. [1987]: A geometric setting for classical molecular dynamics. Ann. Inst. Henri Poincaré, Phys. Th. 47, 199–219

    MATH  MathSciNet  Google Scholar 

  • Iwai, T. [1990]: On the Guichardet/Berry connection. Phys. Lett. A 149, 341–344

    Article  MathSciNet  Google Scholar 

  • Jalnapurkar, S. M., M. Leok, J. E. Marsden and M. West [2000]: Discrete Routh reduction (preprint)

    MATH  Google Scholar 

  • Jalnapurkar, S. M. [1994]: Modeling of Constrained Systems.http://www.cds.caltech.edu/~smj/

    Google Scholar 

  • Jalnapurkar, S. M. and J. E. Marsden [1999]: Stabilization of Relative Equilibria II. Regul.Chaotic Dyn. 3, 161–179

    Article  MATH  MathSciNet  Google Scholar 

  • Jalnapurkar, S. M. and J. E. Marsden [2000]: Reduction of Hamilton’s variational principle.Dynam. Stability Systems(to appear)

    Google Scholar 

  • Jurdjevic, V. [1993]: The geometry of the plate-ball problem. Arch. Rational Mech. Anal.124, 305–328

    Article  MATH  MathSciNet  Google Scholar 

  • Kazhdan, D., B. Kostant and S. Sternberg [1978]: Hamiltonian group actions and dynamical systems of Calogero type. Comm. Pure Appl. Math. 31, 481–508

    Article  MATH  MathSciNet  Google Scholar 

  • Kane, C, J.E. Marsden, M. Ortiz and M. West [2000]: Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems.Int. J.Num. Math. Eng. (to appear)

    Google Scholar 

  • Karapetyan, A. V. [1994]: On the specific character of the application of Routh’s theory to systems with differential constraints. J. Appl. Math. Mech. 58, 387–392 and J. Appl Math. Mech. 51 (1987), 431–436

    Article  MATH  MathSciNet  Google Scholar 

  • Karapetyan, A. V. and V. V. Rumyantsev [1990]: Stability of conservative and dissipative systems. In: Applied Mechanics: Soviet Reviews, G. K. Mikhailov and V. Z. Parton,eds., 1. Hemisphere, New York

    Google Scholar 

  • Kelly, S. D. and R. M. Murray [1995]: Geometric phases and robotic locomotion. Journal of Robotic Systems(to appear)

    Google Scholar 

  • Kirillov, A. A. [1962]: Unitary representations of nilpotent Lie groups. Russian Math.Surveys 17, 53–104

    Article  MATH  MathSciNet  Google Scholar 

  • Kirillov, A. A. [1976]: Elements of the Theory of Representations. Grundlehren der mathematischen Wissenschaften, vol. 220. Springer

    Google Scholar 

  • Kirillov, A. A. [1976]: Local Lie Algebras. Russian Math. Surveys31, 55–75

    Article  MATH  MathSciNet  Google Scholar 

  • Kirk, V., J. E. Marsden and M. Silber [1996]: Branches of stable three-tori using Hamiltonian methods in Hopf bifurcation on a rhombic lattice. Dyn. and Stab, of Systems11, 267–302

    Article  MATH  MathSciNet  Google Scholar 

  • Koiller, J. [1992]: Reduction of some classical nonholonomic systems with symmetry.Arch. Rational Mech. Anal. 118, 113–148

    Article  MATH  MathSciNet  Google Scholar 

  • Kozlov, V. V. and N. N. Kolesnikov [1978]: On theorems of dynamics. PMM 42, 28–33

    MathSciNet  Google Scholar 

  • Koon, W. S. and J. E. Marsden [1997a]: Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction. SIAM J. Control and Optim. 35, 901–929

    Article  MATH  MathSciNet  Google Scholar 

  • Koon, W. S. and J. E. Marsden [1997b]: The geometric structure of nonholonomic mechanics.Proc. CDC 36, 4856–4862

    Google Scholar 

  • Koon, W. S. and J. E. Marsden [1997c]: The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems. Rep. Math. Phys. 40, 21–62

    Article  MATH  MathSciNet  Google Scholar 

  • Koon, W. S. and J. E. Marsden [1998]: The Poisson reduction of nonholonomic mechanical systems. Reports on Math. Phys. 42, 101–134

    Article  MATH  MathSciNet  Google Scholar 

  • Kobayashi, S. and K. Nomizu [1963]: Foundations of Differential Geometry. Wiley

    MATH  Google Scholar 

  • Korteweg, D. [1899]: Ueber eine ziemlich verbreitete unrichtige Behandlungsweise eines Problemes der rollenden Bewegung und insbesondere über kleine rollende Schwingungen um eine Gleichgewichtslage. Nieuw Archief voor Wiskunde 4, 130–155

    MATH  Google Scholar 

  • Kouranbaeva, S. [1999]: The Camassa-Holm equation as a geodesic flow on the diffeomorphism group. J. Math. Phys. 40, 857–868

    Article  MATH  MathSciNet  Google Scholar 

  • Krishnaprasad, P.S. [1989]: Eulerian many-body problems. Contemp. Math. 97, 187–208

    Article  MATH  MathSciNet  Google Scholar 

  • Krishnaprasad, P. S. [1990]: Geometric phases and optimal reconfiguration for multibody systems. Proc. Am. Control Conf., pp. 2440–2444

    Google Scholar 

  • Kummer, M. [1981]: On the construction of the reduced phase space of a Hamiltonian system with symmetry. Indiana Univ. Math. J. 30, 281–291

    Article  MATH  MathSciNet  Google Scholar 

  • Kummer, M. [1990]: On resonant classical Hamiltonians with n frequencies. J. Diff.Eqns. 83, 220–243

    Article  MATH  MathSciNet  Google Scholar 

  • Kupershmidt, B. A. and T. Ratiu [1983]: Canonical maps between semidirect products with applications to elasticity and superfluids. Comm. Math. Phys. 90, 235–250

    Article  MATH  MathSciNet  Google Scholar 

  • Lagrange, J. L. [1788]: Mécanique Analytique. Chez la Veuve Desaint

    Google Scholar 

  • Le, H. and D. G. Kendall [1993]: The Riemannian structure of Euclidean shape spaces:a novel envoronment for statistics. Ann. Statistics21, 1225–1271

    Article  MATH  MathSciNet  Google Scholar 

  • Leonard, N. E. and J. E. Marsden [1997]: Stability and drift of underwater vehicle dynamics:mechanical systems with rigid motion symmetry. Physica D105, 130–162

    Article  MATH  MathSciNet  Google Scholar 

  • Lerman, E., R. Montgomery and R. Sjamaar [1993]: Examples of singular reduction.In: Symplectic Geometry, London Math. Soc. Lecture Note Ser., vol. 192. Cambridge Univ. Press, Cambridge, pp. 127–155

    Google Scholar 

  • Leonard, N. E. [1997]: Stability of a bottom-heavy underwater vehicle. Automatica J.IFAC33, 331–346

    Article  MATH  MathSciNet  Google Scholar 

  • Lewis, A. D. [1996]: The Geometry of the Gibbs-Appell equations and Gauss’ principle of least constraint. Rep. Math. Phys. 38, 11–28

    Article  MATH  MathSciNet  Google Scholar 

  • Lewis, A. D. [2000]: Towards F = ma in a general setting for Lagrangian mechanics (to appear in Annales Henri Poincaré)

    MATH  Google Scholar 

  • Lewis, A.D. and R.M. Murray [1995]: Variational principles for constrained systems: theory and experiment. Int. J. Non-Linear Mech. 30, 793–815

    Article  MATH  MathSciNet  Google Scholar 

  • Lewis, D. [1992a]: Lagrangian block diagonalization. Dyn. Diff. Eqns. 4, 1–42

    Article  MATH  MathSciNet  Google Scholar 

  • Lewis, D. [1992b]: Bifurcation of liquid drops. Nonlinearity 6, 491–522

    Article  MATH  MathSciNet  Google Scholar 

  • Lie, S. [1890]: Theorie der Transformationsgruppen, Zweiter Abschnitt. Teubner, Leipzig

    MATH  Google Scholar 

  • Libermann, P. and C. M. Marie [1987]: Symplectic Geometry and Analytical Mechanics. Kluwer Academic Publishers

    Book  Google Scholar 

  • Littlejohn, R. and M. Reinch [1997]: Gauge fields in the separation of rotations and internal motions in the n-body problem. Rev. Mod. Phys. 69, 213–275

    Article  MathSciNet  Google Scholar 

  • Marie, C.-M. [1995]: Reduction of constrained mechanical systems and stability of relative equilibria. Comm. Math. Phys. 174, 295–318

    Article  MathSciNet  Google Scholar 

  • Marie, C.-M. [1998]: Various approaches to conservative and nonconservative nonholonomic systems. Rep. Math. Phys. 42, 211–229

    Article  MATH  MathSciNet  Google Scholar 

  • Marsden, J. E. [1992]: Lectures on Mechanics. London Math. Soc. Lecture Note Ser., vol. 174. Cambridge University Press

    Google Scholar 

  • Marsden, J., G. Misiolek, M. Perlmutter and T. S. Ratiu [1998]: Symplectic reduction for semidirect products and central extensions. Diff. Geom. Appl. 9, 173–212

    Google Scholar 

  • Marsden, J. E., G. Misiolek, M. Perlmutter and T. S. Ratiu [2000]: Reduction by stages and group extensions. Preprint

    Google Scholar 

  • Marsden, J. E., R. Montgomery, P. J. Morrison and W. B. Thompson [1986]: Covariant Poisson brackets for classical fields. Ann. Phys. 169, 29–48

    Article  MathSciNet  Google Scholar 

  • Marsden, J. E., R. Montgomery and T. S. Ratiu [1990]: Reduction, Symmetry and Phases in Mechanics. Memoirs Amer. Math. Soc. 436

    Google Scholar 

  • Marsden, J.E. and J. Ostrowski [1998]: Symmetries in motion: Geometric foundations of motion control. Nonlinear Sci. Today. http://link.springer-ny.com

    Google Scholar 

  • Marsden, J. E., G. W. Patrick and W. F. Shadwick (Eds.) [1996]: Integration Algorithms and Classical Mechanics. Fields Inst. Commun. 10, Am. Math. Soc.

    Google Scholar 

  • Marsden, J. E., G. W. Patrick and S. Shkoller [1998]: Mulltisymplectic Geometry, Variational Integrators and Nonlinear PDEs. Comm. Math. Phys199, 351–395

    Article  MATH  MathSciNet  Google Scholar 

  • Marsden, J. E. and M. Perlmutter [2000]: The Orbit Bundle Picture of Cotangent Bundle Reduction. C. R. Math. Rep. Acad. Sci. Canada22, 33–54

    MATH  MathSciNet  Google Scholar 

  • Marsden, J. E., S. Pekarsky and S. Shkoller [1999]: Discrete Euler-Poincaré and Lie-Poisson equations. Nonlinearity 12, 1647–1662

    Article  MATH  MathSciNet  Google Scholar 

  • Marsden, J.E. and T. S. Ratiu [1986]: Reduction of Poisson Manifolds. Lett. in Math. Phys. 11, 161–170

    Article  MATH  MathSciNet  Google Scholar 

  • Marsden, J. E. and T. S. Ratiu [1999]: Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, vol. 17. Springer, 1994; Second Edition, 1999

    Book  Google Scholar 

  • Marsden,J.E.,T.S.RatiuandJ.Scheurle[2000]:Reductiontheoryandthe Lagrange-Routh equations.J.Math. Phys.41,3379–3429

    Article  MATH  MathSciNet  Google Scholar 

  • Marsden,J. E., T.Ratiu and S.Shkoller [1999]:The geometry and analysis of the averaged Euler equations and a new diffeomorphism group.Geom. Funct.Anal. (to appear)

    Google Scholar 

  • Marsden, J. E., T. S. Ratiu and A. Weinstein [1984a]: Semi-direct products and reduction in mechanics. Trans. Amer. Math. Soc. 281, 147–177

    Article  MATH  MathSciNet  Google Scholar 

  • Marsden, J. E., T. S. Ratiu and A. Weinstein [1984b]: Reduction and Hamiltonian structures on duals of semidirect product Lie Algebras. Contemp. Math., Am. Math. Soc. 28, 55–100

    MATH  Google Scholar 

  • Marsden, J. E. and J. Scheurle [1993a]: Lagrangian reduction and the double spherical pendulum. ZAMP 44, 17–43

    Article  MATH  MathSciNet  Google Scholar 

  • Marsden, J. E. and J. Scheurle [1993b]: The reduced Euler-Lagrange equations. Fields Inst. Commun. 1, 139–164

    MATH  MathSciNet  Google Scholar 

  • Marsden, J. E. and S. Shkoller [1999]: Multisymplectic geometry, covariant Hamiltonians and water waves. Math. Proc. Camb. Phil. Soc. 125, 553–575

    Article  MATH  MathSciNet  Google Scholar 

  • Marsden, J.E. and A. Weinstein [1974]: Reduction of symplectic manifolds with symmetry.Rep. Math. Phys. 5, 121–130

    Article  MATH  MathSciNet  Google Scholar 

  • Marsden, J.E. and A. Weinstein [1982]: The Hamiltonian structure of the Maxwell-Vlasov equations. Physica D4, 394–406

    Article  MATH  MathSciNet  Google Scholar 

  • Marsden, J. E. and A. Weinstein [1983]: Coadjoint orbits, vortices and Clebsch variables for incompressible fluids. Physica D 7, 305–323

    Article  MATH  MathSciNet  Google Scholar 

  • Marsden, J. E., A. Weinstein, T. S. Ratiu, R. Schmid and R. G. Spencer [1982]: Hamiltonian systems with symmetry, coadjoint orbits and plasma physics. In: Proc. IUTAMIS1 MM Symposium on Modern Developments in Analytical Mechanics (Torino, 1982), 117, 289–340, Atti della Acad. della Sc. di Torino

    Google Scholar 

  • Martin, J. L. [1959]: Generalized classical dynamics and the “classical analogue”of a Fermi oscillation. Proc. Roy. Soc. A 251, 536

    Article  MATH  Google Scholar 

  • Martinez, S., J. Cortés and M. De León [2000]: The geometrical theory of constraints applied to the dynamics of vakonomic mechanicalsystems: the vakonomic bracket. J. Math. Phys. 41, 2090–2120

    Article  MATH  MathSciNet  Google Scholar 

  • Meyer, K. R. [1973]: Symmetries and integrals in mechanics. In: Dynamical Systems, M. Peixoto, ed. Academic Press, pp. 259–273

    Chapter  Google Scholar 

  • Montgomery, R., J. E. Marsden and T. S. Ratiu [1984]: Gauged Lie-Poisson structures. Contemp. Math., Amer. Math. Soc. 28, 101–114

    Article  MATH  MathSciNet  Google Scholar 

  • Montgomery, R. [1984]: Canonical formulations of a particle in a Yang-Mills field. Lett. Math. Phys. 8, 59–67

    Article  MATH  MathSciNet  Google Scholar 

  • Montgomery, R. [1986]: The Bundle Picture in Mechanics. Ph.D. Thesis, University of California Berkeley

    Google Scholar 

  • Montgomery, R. [1988]: The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case. Comm. Math. Phys. 120, 269–294

    Article  MATH  MathSciNet  Google Scholar 

  • Montgomery, R. [1990]: Isoholonomic problems and some applications. Comm. Math Phys. 128, 565–592

    Article  MATH  MathSciNet  Google Scholar 

  • Montgomery, R. [1991]: Optimal control of deformable bodies and its relation to gauge theory. In: The Geometry of Hamiltonian Systems, T. Ratiu, ed. Springer

    Google Scholar 

  • Montgomery, R. [1993]: Gauge theory of the falling cat. Fields Inst. Commun. 1,193–218

    MATH  MathSciNet  Google Scholar 

  • Moser, J. and A. P. Veselov [1991]: Discrete versions of some classical integrable systems and factorization of matrix polynomials. Comm. Math. Phys. 139, 217–243

    Article  MATH  MathSciNet  Google Scholar 

  • Murray, R.M. and S. S. Sastry [1993]: Nonholonomic motion planning: steering using sinusoids. IEEE Trans, on Automatic Control 38, 700–716

    Article  MATH  MathSciNet  Google Scholar 

  • Nambu, Y. [1973]: Generalized Hamiltonian dynamics. Phys. Rev. D 7, 2405–2412

    Article  MATH  MathSciNet  Google Scholar 

  • Neimark, Ju. I. and N. A. Fufaev [1972]: Dynamics of Nonholonomic Systems, Translations of Mathematical Monographs, 33. Amer. Math. Soc., Providence, RI

    Google Scholar 

  • Newcomb, W. A. [1962]: Lagrangian and Hamiltonian methods in magnetohydrodynamics. Nuc. Fusion, Suppl. part 2, 451–463

    Google Scholar 

  • O’Reilly, O.M. [1996]: The dynamics of rolling disks and sliding disks. Nonlinear Dynamics 10, 287–305

    Article  Google Scholar 

  • Ortega, J.-P [1998]: Symmetry, Reduction, and Stability in Hamiltonian Systems. Thesis, UC Santa Cruz

    Google Scholar 

  • Ortega, J.-P. and T. S. Ratiu [1997]: Persistence and smoothness of critical relative elements in Hamiltonian systems with symmetry. C. R. Acad. Sci. Paris Sér. I Math. 325, 1107–1111

    Article  MATH  MathSciNet  Google Scholar 

  • Ortega, J.-P. and T. S. Ratiu [2001]: Hamiltonian Singular Reduction. Progress in Math. Birkhäuser(to appear)

    Google Scholar 

  • Ostrowski, J. [1998]: Reduced equations for nonholonomic mechanical systems with dissipative forces. Rep. Math. Phys. 42, 185–209

    Article  MATH  MathSciNet  Google Scholar 

  • Ostrowski, J., J. W. Burdick, A. D. Lewis and R. M. Murray [1995]: The mechanics of undulatory locomotion: The mixed kinematic and dynamic case. In: IEEE Intern. Conf. on Robotics and Automation, pp. 1945–1951

    Google Scholar 

  • Ostrowski, J., J. P. Desai and V. Kumar [1996]: Optimal gait selection for nonholonomic locomotion systems. In: IEEE Conf. on Robotics and Automation’, availablefrom http://www.cis.upenn.edu/~jpo/papers.html

    Google Scholar 

  • Otto, M. [1987]: A reduction scheme for phase spaces with almost Kähler symmetry. Regularity results for momentum level sets. .Geom. Phys. 4, 101–118

    Article  MATH  MathSciNet  Google Scholar 

  • Pauli, W. [1953]: On the Hamiltonian structure of non-local field theories. Il Nuovo CimentoX, 648–667

    Article  MATH  Google Scholar 

  • Pedroni, M. [1995]: Equivalence of the Drinfelćd-Sokolov reduction to a bi-Hamiltonian reduction. Lett. Math. Phys. 35, 291–302

    Article  MATH  MathSciNet  Google Scholar 

  • Poincaré, H. [1901a]: Sur une forme nouvelle des équations de la méchanique. C. R. Acad.Sci. 132, 369–371

    MATH  Google Scholar 

  • Poincaré, H. [1901b]: Sur la stabilité de l’équilibre des figures piriformes affectées par une masse fluide en rotation. Philosophical TransactionsA 198, 333–373

    Article  MATH  Google Scholar 

  • Poincaré, H. [1910]: Sur la precession des corps deformables. Bull. Astron. 27, 321–356

    MATH  Google Scholar 

  • Ratiu, T. S. [1980a]: The Euler-Poisson Equations and Integrability. Ph.D. Thesis, Univ. of Calif., Berkeley

    Google Scholar 

  • Ratiu, T. S. [1980b]: The motion of the free n-dimensional rigid body. Indiana Univ. Math. Journ. 29, 609–629

    Article  MATH  MathSciNet  Google Scholar 

  • Ratiu, T. S. [1981]: Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body. Proc. Natl. Acad. Sci., USA78, 1327–1328

    Article  MATH  MathSciNet  Google Scholar 

  • Ratiu, T. S. [1982a]: Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body. Amer. J. Math. 104, 409–448, 1337

    Article  MATH  MathSciNet  Google Scholar 

  • Ratiu, T. S. [1982b]: The Lie algebraic interpretation of the complete integrability of the Rosochatius system. In: Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems (La Jolla Institute, 1981), AIP Conference Proceedings, vol. 88, pp. 109–116

    MathSciNet  Google Scholar 

  • Rosenberg, R. M. [1977]: Analytical Dynamics of Discrete Systems. Plenum Press, New York

    Book  MATH  Google Scholar 

  • Routh, E. J. [1860]: Treatise on the Dynamics of a System of Rigid Bodies. MacMillan, London

    Google Scholar 

  • Routh, E.J. [1877]: Stability of a Given State of Motion. Halsted Press, New York [Reprinted in Stability of Motion (1975), A. T. Fuller, ed.]

    MATH  Google Scholar 

  • Routh, E.J. [1884]: Advanced Rigid Dynamics. MacMillian and Co., London

    Google Scholar 

  • Satzer, W. J. [1977]: Canonical reduction of mechanical systems invariant under Abelian group actions with an application to celestial mechanics. Ind. Univ. Math. J.26, 951–976

    Article  MATH  MathSciNet  Google Scholar 

  • Seliger, R. L. and G. B. Whitham [1968]: Variational principles in continuum mechanics. Proc. Roy. Soc. Lond. 305, 1–25

    Article  MATH  Google Scholar 

  • Simo, J. C., D. R. Lewis and J. E. Marsden [1991]: Stability of relative equilibria I: The reduced energy momentum method. Arch. Rational Mech. Anal. 115, 15–59

    Article  MATH  MathSciNet  Google Scholar 

  • Sjamaar, R. and E. Lerman [1991]: Stratified symplectic spaces and reduction. Ann. Math. 134, 375–422

    Article  MATH  MathSciNet  Google Scholar 

  • Smale, S. [1970]: Topology and Mechanics. Inv. Math. 10, 305–331; 11, 45–64

    Article  MATH  MathSciNet  Google Scholar 

  • Souriau, J. M. [1970]: Structure des Systemes Dynamiques. Dunod, Paris

    MATH  Google Scholar 

  • Sternberg, S. [1977]: Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang-Mills field. Proc. Nat. Acad. Sci. 74, 5253–5254

    Article  MATH  MathSciNet  Google Scholar 

  • Sumbatov, A. S. [1992]: Developments of some of Lagrange’s ideas in the works of Russian and Soviet mechanicians. In: La mécanique analytique de Lagrange et son héritage, 126, 169–200, Atti della Accademia delle Scienze di Torino

    MATH  MathSciNet  Google Scholar 

  • Sudarshan, E. C. G. and N. Mukunda [1974]: Classical Mechanics: A Modern Perspective.Wiley, New York; Second edition, Krieber, Melbourne-Florida, 1983

    MATH  Google Scholar 

  • Tsikiris, D. P. [1995]: Motion Control and Planning for Nonholonomic Kinematic Chains. PhD Thesis, University of Maryland

    Google Scholar 

  • Tulczyjew, W. M. [1977]: The Legendre transformation. Ann. Inst Poincaré27,101–114

    MATH  MathSciNet  Google Scholar 

  • Udwadia, F. E. and R. E. Kalaba [1996]: Analytical Dynamics: a New Perspective. Cambridge University Press

    Book  MATH  Google Scholar 

  • Vanhaecke, P. [1996]: Integrable Systems in the Realm of Algebraic Geometry. Lecture Notes in Mathematics, vol. 1638. Springer-Verlag, Berlin Heidelberg

    Book  MATH  Google Scholar 

  • van der Schaft, A.J. and B.M. Maschke [1994]: On the Hamiltonian formulation of nonholonomic mechanical systems. Rep. Math. Phys. 34, 225–233

    Article  MATH  MathSciNet  Google Scholar 

  • Vershik, A.M. and Faddeev [1981]: Lagrangian mechanics in invariant form. Sel. Math.Sov. 1, 339–350

    MATH  Google Scholar 

  • Vershik, A.M. and V. Ya Gershkovich [1994]: Non-holonomic Riemannian manifolds.In: Dynamical Systems 7. Encyclopaedia of Mathematics, vol. 16. Springer

    Google Scholar 

  • Veselov, A. P. [1988]: Integrable discrete-time systems and difference operators. Funct.An. and Appl. 22, 83–94

    Article  MATH  Google Scholar 

  • Veselov, A. P. [1991]: Integrable Lagrangian correspondences and the factorization of matrix polynomials. Funct. An. and Appl. 25, 112–123

    Article  MATH  Google Scholar 

  • Vierkandt, A. [1892]: Über gleitende und rollende Bewegung. Monatshefte der Math.und Phys. III, 31–54

    Article  MATH  MathSciNet  Google Scholar 

  • Vinogradov, A.M. and B. A. Kupershmidt [1977]: The structures of Hamiltonian mechanics.Russ. Math. Surv. 32, 177–243

    Article  MATH  MathSciNet  Google Scholar 

  • Wang, L. S. and P. S. Krishnaprasad [1992]: Gyroscopic control and stabilization. J.Nonlinear Sci. 2, 367–415

    Article  MATH  MathSciNet  Google Scholar 

  • Walker, G.T. [1896]: On a dynamical top. Quart. J. Pure Appl. Math. 28, 175–184

    MATH  Google Scholar 

  • Weber, R. W. [1986]: Hamiltonian systems with constraints and their meaning in mechanics.ARMA91, 309–335

    Article  MATH  MathSciNet  Google Scholar 

  • Weinstein, A. [1977]: Lectures on symplectic manifolds. CBMS Regional Conf. Ser. in Math., 29. Amer. Math. Soc., Providence, RI

    Google Scholar 

  • Weinstein, A. [1978]: A universal phase space for particles in Yang-Mills fields. Lett.Math. Phys. 2, 417–420

    Article  MATH  MathSciNet  Google Scholar 

  • Weinstein, A. [1978]: Bifurcations and Hamilton’s principle. Math. Zeit. 159, 235–248

    Article  MATH  MathSciNet  Google Scholar 

  • Weinstein, A. [1983a]: Sophus Lie and symplectic geometry. Expo. Math. 1, 95–96J. Diff. Geom. 18,523–557

    Article  MATH  Google Scholar 

  • Weinstein, A. [1996]: Lagrangian mechanics and groupoids. Fields Inst. Commun. 7,207–231

    MATH  MathSciNet  Google Scholar 

  • Wendlandt, J. M. and J. E. Marsden [1997]: Mechanical integrators derived from a discrete variational principle. Physica D 106, 223–246

    Article  MATH  MathSciNet  Google Scholar 

  • Whittaker, E. T. [1907]: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, [1938] 4th edition; [Reprinted by Dover 1944 and Cambridge University 1988]

    Google Scholar 

  • Wong, S. K. [1970]: Field and particle equations for the classical Yang-Mills field and particles with isotopic spin. Il Nuovo CimentoLXV, 689–694

    Article  Google Scholar 

  • Yang, R., P. S. Krishnaprasad and W. Dayawansa [1993]: Chaplygin dynamics and Lagrangian reduction. In: Proc. 2nd Int. Cong, on Nonlinear Mechanics, W-Z. Chien and Z. H. Guo and Y. Z. Guo, eds. Peking University Press, pp. 745–749

    Google Scholar 

  • Zaalani, N. [1999]: Phase space reduction and Poisson structure. J. Math. Phys. 40,3431–3438

    Article  MATH  MathSciNet  Google Scholar 

  • Zenkov, D. V., A. M. Bloch and J. E. Marsden [1998]: The energy momentum method for the stability of nonholonomic systems. Dyn. Stab. Systems13, 123–166

    Article  MATH  Google Scholar 

  • Zenkov, D. V. [1995]: The geometry of the Routh problem. J. Nonlinear Sci. 5, 503–519

    Article  MATH  MathSciNet  Google Scholar 

Download references

Authors
  1. Hernán Cendra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jerrold E. Marsden
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Tudor S. Ratiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Center for Parallel Computers, Royal Institute of Technology (KTH), Lindstedtsvägen 25, 10044, Stockholm, Sweden

    Björn Engquist

  2. Department of Mathematics, University of California at Los Angeles, 3148 Murphy Hall, 7619A MSB, 90095-1555, Los Angeles, CA, USA

    Björn Engquist

  3. Department of Mathematics, Harvard University, 02138-2901, Cambridge, MA, USA

    Wilfried Schmid

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Cendra, H., Marsden, J.E., Ratiu, T.S. (2001). Geometric Mechanics, Lagrangian Reduction, and Nonholonomic Systems. In: Engquist, B., Schmid, W. (eds) Mathematics Unlimited — 2001 and Beyond. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56478-9_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-56478-9_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63114-6

  • Online ISBN: 978-3-642-56478-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation