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Some Remarks on Geometric Mechanics

  • Jerrold E. Marsden

Abstract

This paper gives a few new developments in mechanics, as well as some remarks of a historical nature. To keep the discussion focussed, most of the paper is confined to equations of “rigid body”, or “hydrodynamic” type on Lie algebras or their duals. In particular, we will develop the variational structure of these equations and will relate it to the standard variational principle of Hamilton.

Keywords

Variational Principle Symplectic Form Poisson Structure Nonholonomic System Coadjoint Orbit 
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References

  1. Abraham, R. and J. Marsden (1978): Foundations of mechanics. Addison-Wesley Publishing Co., Reading, Mass.MATHGoogle Scholar
  2. Arms, J.M., J.E. Marsden and V. Moncrief (1981): Symmetry and bifurcations of momentum mappings. Comm. Math. Phys. 78, 455-478CrossRefMathSciNetGoogle Scholar
  3. Arnold, V.I. (1966): Sur la géometrie differentielle des groupes de Lie de dimenson infinie et ses applications à l’hydrodynamique des fluids parfaits. Ann. Inst. Fourier, Grenoble 16, 319-361CrossRefGoogle Scholar
  4. Arnold, V.I. (1988): Dynamical systems III. Encyclopaedia of mathematics, vol. 3. Springer, Berlin Heidelberg New YorkGoogle Scholar
  5. Arnold, V.I. (1989): Mathematical methods of classical mechanics, 2nd edn. Graduate Texts in Mathematics, vol. 60. Springer, Berlin Heidelberg New YorkGoogle Scholar
  6. Bloch, A.M., P.S. Krishnaprasad, J.E. Marsden and R. Murray (1993): Nonholonomic mechanical systems with symmetry (in preparation)Google Scholar
  7. Bloch, A.M., P.S. Krishnaprasad, J.E. Marsden and T.S. Ratiu (1993a): Dissipation induced instabilities. Ann. Inst. H. Poincaré, Analyse Nonlineaire (to appear)Google Scholar
  8. Bloch, A.M., P.S. Krishnaprasad, J.E. Marsden and T.S. Ratiu (1993b): The Euler-Poincaré equations and double bracket dissipation (preprint)Google Scholar
  9. 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, 745-756Google Scholar
  10. Cendra, H., A Ibort and J.E. Marsden (1987): Variational principles on principal fiber bundles: a geometric theory of Clebsch potentials and Lin constraints. J. Geom. Phys. 4, 183-206CrossRefMathSciNetGoogle Scholar
  11. Cendra, H. and J.E. Marsden (1987): Lin constraints, Clebsch potentials and variational principles. Physica D 27, 63-89CrossRefMathSciNetGoogle Scholar
  12. Chetayev, N.G. (1989): Theoretical Mechanics. Springer, New YorkGoogle Scholar
  13. Ebin, D.G. and J.E. Marsden (1970): Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102-163CrossRefMathSciNetGoogle Scholar
  14. Hamel, G (1904): Die Lagrange-Eulerschen Gleichungen der Mechanik. Z. Math. Phys. 50, 1-57Google Scholar
  15. Hamel, G (1949): Theoretische Mechanik. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  16. Koiller, J. (1992): Reduction of some classical nonholonomic systems with symmetry. Arch. Rat. Mech. Ann. 118, 113-148MathSciNetGoogle Scholar
  17. Kummer, M. (1981): On the construction of the reduced phase space of a Hamiltonian system with symmetry. Indiana Univ. Math. J. 30, 281-291CrossRefMathSciNetGoogle Scholar
  18. Lanczos, C. (1949): The variational principles of mechanics. University of Toronto PressMATHGoogle Scholar
  19. Lewis, D., T.S. Ratiu, J.C. Simo and J.E. Marsden (1992): The heavy top, a geometric treatment. Nonlinearity 5, 1-48CrossRefMathSciNetGoogle Scholar
  20. Lie, S. (1890): Theorie der Transformationsgruppen, three volumes. Teubner, Leipzig (reprinted by Chelsea, N.Y.)MATHGoogle Scholar
  21. Marsden, J.E. (1992): Lectures on Mechanics. London Mathematical Society Lecture note series, vol. 174. Cambridge University PressGoogle Scholar
  22. Marsden, J.E., R. Montgomery and T. Ratiu (1990): Reduction, symmetry, and phases in mechanics. Memoirs AMS vol. 436, p. 1-110Google Scholar
  23. Marsden, J.E., T.S. Ratiu and G. Raugel (1991): Symplectic connections and the linearization of Hamiltonian systems. Proc. Roy. Soc. Ed. A 117, 329-380MathSciNetGoogle Scholar
  24. Marsden, J.E. and J. Scheurle (1993): Lagrangian reduction and the double spherical pendulum, ZAMP 44, 17-43CrossRefMathSciNetGoogle Scholar
  25. Marsden, J.E. and J. Scheurle (1993): The reduced Euler-Lagrange equations. Fields Institute Communications 1, 139-164MathSciNetGoogle Scholar
  26. Marsden, J.E., and A. Weinstein (1974): Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5, 121-130CrossRefMathSciNetGoogle Scholar
  27. Marsden, J.E. and A. Weinstein (1982): The Hamiltonian structure of the Maxwell-Vlasov equations. Physica D 4, 394-406CrossRefMathSciNetGoogle Scholar
  28. Marsden, J.E. and A. Weinstein (1983): Coadjoint orbits, vortices and Clebsch variables for incompressible fluids. Physica D 7, 305-323CrossRefMathSciNetGoogle Scholar
  29. Marsden, J.E., A. Weinstein, T.S. Ratiu, R. Schmid and R.G. Spencer (1983): Hamiltonian systems with symmetry, coadjoint orbits and plasma physics. In: Proc. IUTAM-ISlMM Symposium on Modem Developments in Analytical Mechanics, Torino 1982. Atti della Acad. della Sc. di Torino 117, 289-340Google Scholar
  30. Martin, J.L. (1959): Generalized classical dynamics and the “classical analogue” of a Fermi oscillation. Proc. Roy. Soc. A 251, 536CrossRefMATHGoogle Scholar
  31. Montgomery, R. (1984): Canonical formulations of a particle in a Yang-Mills field. Lett. Math. Phys. 8, 59-67CrossRefGoogle Scholar
  32. Montgomery, R., J.E. Marsden and T.S. Ratiu (1984): Gauged Lie-Poisson structures. Cont. Math. AMS 28, 101-114MathSciNetGoogle Scholar
  33. Naimark, Ju. I. and N.A. Fufaev (1972): Dynamics of nonholonomic systems. Translations of Mathematical Monographs, AMS, vol. 33Google Scholar
  34. Nambu, Y. (1973): Generalized Hamiltonian dynamics. Phys. Rev. D 7, 2405-2412CrossRefMathSciNetGoogle Scholar
  35. Pauli, W. (1953): On the Hamiltonian structure of non-local field theories. I1 Nuovo Cimento X, 648-667Google Scholar
  36. Poincaré, H. (1901): Sur une forme nouvelle des equations de la mecanique. Compt. Rend. Acad. Sci. 132, 369-371Google Scholar
  37. Poincaré, H. (1910): Sur la precession des corps deformables. Bull Astron.Google Scholar
  38. Routh, E.J. (1877): Stability of a given state of motion. Reprinted in Stability of Motion, ed. A.T. Fuller. Halsted Press, New York 1975Google Scholar
  39. Seliger, R.L. G.B. Whitham (1968): Variational principles in continuum mechanics. Proc. Roy. Soc. Lond. 305, 1-25Google Scholar
  40. Simo, J.C., D. Lewis and J.E. Marsden (1991): Stability of relative equilibria I: The reduced energy momentum method. Arch. Rat. Mech. Anal. 115, 15-59MathSciNetGoogle Scholar
  41. Sjamaar, R. and E. Lerman (1991): Stratified symplectic spaces and reduction. Ann. Math. 134, 375-422CrossRefMathSciNetGoogle Scholar
  42. Smale, S. (1970): Topology and Mechanics. Invent. Math. 10, 305–331, 11, 45-64CrossRefMATHMathSciNetGoogle Scholar
  43. Souriau, J.M. (1970): Structure des Systemes Dynamiques. Dunod, ParisMATHGoogle Scholar
  44. Sudarshan, E.C.G. and N. Mukunda (1974): Classical mechanics: a modem perspective. Wiley, New York 1974; 2nd edn. Krieber, Melbourne (Florida) 1983Google Scholar
  45. Weinstein, A. (1983): Sophus Lie and symplectic geometry. Expo. Math. 1, 95–96MATHGoogle Scholar
  46. Weinstein, A. (1981): Symplectic geometry. Bulletin A.M.S. (new series) 5, 1-13Google Scholar
  47. Weinstein, A. (1993): Lagrangian mechanics and groupoids. PreprintGoogle Scholar
  48. Wintner, A. (1941): The Analytical Foundations of Celestial Mechanics. Princeton University PressGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Jerrold E. Marsden

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