Lagrangian reduction by stages

Memoirs of the American Mathematical Society

Volumn , Issue 722, 2001, Pages 1-108

  Cendra, Hernán ^a   Marsden, Jerrold E ^a   Ratiu, Tudor S ^a  

^a NONE

Author keywords

Lagrangian reduction; Mechanical systems; Symmetry; Variational principles

Indexed keywords

EID: 0035402214     PISSN: 00659266     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (107)

1. Abraham, R. and J. E. Marsden [1978], Foundations of Mechanics, Addison-Wesley, Second edition.
2. Abraham, R., J. E. Marsden and T. S. Ratiu [1988], Manifolds, Tensor Analysis and Applications, Applied Mathematical Sciences, 75, Springer-Verlag, New York, Second edition.
3. Arnold, V. I. [1989], Mathematical Methods of Classical Mechanics, Graduate Texts in Math., 60, Springer-Verlag, Second edition.
4. Bates, L. and E. Lerman [1997], Proper group actions and symplectic stratified spaces, Pacific J. Math., 181, 201-229.
5. Blaom, A.D. [2000], Reconstruction phases via Poisson reduction, Diff. Geom. and Appl., 12, 231-252.
6. Bloch, A. M.; Crouch, P. E. [1993] Nonholonomic and vakonomic control systems on Riemannian manifolds. Dynamics and control of mechanical systems, Fields Inst. Commun. 1, 25-52.
7. Bloch, A. M. and P. E. Crouch [1994], Reduction of Euler-Lagrange problems for constrained variational problems and relation with optimal control problems, in Proc. CDC, 33, 2584-2590.
8. Bloch, A. M. and P. Crouch [1995], Nonholonomic control systems on Riemannian manifolds., SIAM J. on Control, 33, 126-148.
9. Bloch, A. M. and P. E. Crouch [1998], Newton's law and integrability of nonholonomic systems, SIAM J. Control Optim., 6, 2020-2039
10. Bloch, A. M. and P. E. Crouch [1999], Optimal control, optimization, and analytical mechanics, in Mathematical Control Theory, 268-321, J. Ballieul, Ed., Springer, New York.
11. 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.
12. 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.
13. 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.
14. Bloch, A. M., N. Leonard and J. E. Marsden, Controlled Lagrangians and the Stabilization of Mechanical Systems I: The First Matching Theorem, IEEE Trans. Automat. Control (to appear).
15. 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.
16. Bobenko, A. I. and Y. B. Suris [1999b], Discrete Lagrangian reduction, discrete Euler-Poincaré equations, and semidirect products, Lett. Math. Phys., 49, 79-93.
17. Bourbaki, N. [1983], Variétés differentielles et analytiqes. Fascicule de résultats, Diffusion C.C.L.S., Paris.
18. Bretherton, F. P. [1970], A note on Hamilton's principle for perfect fluids, J. Fluid Mech., 44, 19-31.
19. Cannas Da Silva, A. and A. Weinstein [1999], Geometric Models for Noncommutative Alebras, Amer. Math. Soc., Berkeley Mathematics Lecture Notes, 10.
20. Castrillón López, M., Ratiu, T. S. and Shkoller, S. [2000], Reduction in principal fiber bundles: Covariant Euler-Poincaré equations, Proc. Amer. Math. Soc., 128, 2155-2164.
21. 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.
22. 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.
23. 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.
24. Cendra, H. and J. E. Marsden [1987], Lin constraints, Clebsch potentials and variational principles, Physica D, 27, 63-89.
25. Cendra, H., J. E. Marsden and T. S. Ratiu [2000], Geometric Mechanics, Lagrangian Reduction and Nonholonomic Systems, in Mathematics Unlimited, Springer-Verlag, New York.
26. Courant, T. [1990], Dirac manifolds, Trans. Amer. Math. Soc., 319, 631-661.
27. De Azcárraga, J.E. and J.M. Izquierdo [1995] Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics. Cambridge University Press.
28. Ge, Z. and J. E. Marsden [1988], Lie-Poisson integrators and Lie-Poisson Hamilton-Jacobi theory, Phys. Lett. A, 133, 134-139.
29. Guillemin, V. and S. Sternberg [1980], The moment map and collective motion, Ann. of Phys., 1278, 220-253.
30. Guillemin, V. and S. Sternberg [1984], Symplectic Techniques in Physics, Cambridge University Press.
31. Hamel, G. [1904], Die Lagrange-Eulerschen Gleichungen der Mechanik, Z. für Mathematik u. Physik, 50, 1-57.
32. Holm, D. D. [2000], Euler-Poincaré Dynamics of Ideal Micropolar Complex Fluids, (preprint).
33. Holm, D. D., J. E. Marsden and T. S. Ratiu [1986], The Hamiltonian structure of continuum mechanics in material, spatial and convective representations, in Séminaire de Mathématiques supérieurs, 100, 11-122, Les Presses de L'Univ. de Montréal.
34. Holm, D. D., J. E. Marsden and T. S. Ratiu [1998a], The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137, 1-8.
35. Holm, D. D., J. E. Marsden and T. S. Ratiu [1998b], Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 349, 4173-4177.
36. Jalnapurkar, S. M., M. Leok, J. E. Marsden and M. West [2000], Discrete Routh reduction; (preprint).
37. Jalnapurkar, S. M. and J. E. Marsden [1999], Stabilization of Relative Equilibria II, Regul. Chaotic Dyn., 3, 161-179.
38. Jalnapurkar, S. M. and J. E. Marsden [2000], Reduction of Hamilton's Variational Principle, Dynam. Stability Systems, 15, 287-318.
39. 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., 49, 1295-1325.
40. Kobayashi, S. and K. Nomizu [1963], Foundations of Differential Geometry, Wiley.
41. Kolář, I., P.W. Michor, and J. Slovák [1993] Natural Operators in Differential Geometry. Springer-Verlag.
42. 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.
43. Koon, W. S. and J. E. Marsden [1997b], The geometric structure of nonholonomic mechanics, Proc. CDC, 36, 4856-4862.
44. 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.
45. Koon, W. S. and J. E. Marsden [1998], The Poisson reduction of nonholonomic mechanical systems, Reports on Math. Phys., 42, 101-134.
46. Lagrange, J. L. [1788], Mécanique Analytique, Chez la Veuve Desaint.
47. Leonard, N. E. and J. E. Marsden [1997], Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry, Physica D, 105, 130-162.
48. Lewis, A. D. [1996] The geometry of the Gibbs-Appell equations and Gauss' principle of least constraint. Rep. Math. Phys. 38, 11-28.
49. Lewis, A. D. [2000] Towards F = ma in a general setting for Lagrangian mechanics (To appear in Annales Henri Poincaré).
50. Libermann, P. and C. M. Marle [1987], Symplectic Geometry and Analytical Mechanics, Kluwer Academic Publishers.
51. Mackenzie, K. [1987] Lie Groupoids and Lie Algebroids in Differntial Geometry. London Math. Soc. Lect. Note Series 124, Cambrigde Univ. Press.
52. Marsden, J. E. [1992], Lectures on Mechanics, London Math. Soc. Lecture Note Ser., 174, Cambridge University Press.
53. Marsden, J., G. Misiolek, M. Perlmutter and T. Ratiu [1998], Symplectic reduction for semidirect products and central extensions, Diff. Geom. and its Appl., 9, 173-212.
54. Marsden, J. E., G. Misiolek, M. Perlmutter and T. S. Ratiu [2000], Reduction by stages and group extensions, Preprint.
55. Marsden, J. E., R. Montgomery, P. J. Morrison and W. B. Thompson [1986], Covariant Poisson brackets for classical fields, Annals of Physics, 169, 29-48.
56. Marsden, J. E., P. J. Morrison and A. Weinstein, The Hamiltonian structure of the BBGKY hierarchy equations, Contemp. Math., AMS, 28, 115-124.
57. Marsden, J. E., G. W. Patrick and S. Shkoller [1998], Mulltisymplectic Geometry, Variational Integrators and Nonlinear PDEs, Comm. Math. Phys., 199, 351-395.
58. Marsden, J. E., S. Pekarsky and S. Shkoller [1999], Discrete Euler-Poincaré and Lie-Poisson equations, Nonlinearity, 12, 1647-1662.
59. Marsden, J. E., S. Pekarsky and S. Shkoller [2000], Symmetry reduction of discrete Lagrangian mechanics on Lie groups, J. Geom. and Phys., 36, 140-151.
60. Marsden, J. E. and M. Perlmutter [2000], The Orbit Bundle Picture of Cotangent Bundle Reduction, C. R. Math. Rep. Acad. Sci. Canada, 22, 33-54.
61. Marsden, J. E. and T. Ratiu [1986], Reduction of Poisson Manifolds, Lett. in Math. Phys., 11, 161-170.
62. Marsden, J. E. and T. S. Ratiu [1999], Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, 17; Second Edition, 1999.
63. Marsden, J. E., T. S. Ratiu and J. Scheurle [2000], Reduction theory and the Lagrange-Routh equations, J. Math. Phys., 41, 3379-3429.
64. Marsden, J. E., T. Ratiu and S. Shkoller [2000], The geometry and analysis of the averaged Euler equations and a new diffeomorphism group, Geom. Fund. Anal., 10, 582-599.
65. Marsden, J. E., T. S. Ratiu and A. Weinstein [1984], Semi-direct products and reduction in mechanics, Trans. Amer. Math. Soc., 281, 147-177.
66. Marsden, J. E., T. S. Ratiu and A. Weinstein [1984], Reduction and Hamiltonian structures on duals of semidirect product Lie Algebras, Amer. Math. Soc., Providence, RI, Contemp. Math., 28, 55-100.
67. Marsden, J. E. and J. Scheurle [1993], Lagrangian reduction and the double spherical pendulum, ZAMP, 44, 17-43.
68. Marsden, J. E. and J. Scheurle [1993], The reduced Euler-Lagrange equations, Fields Inst. Commun., 1, 139-164.
69. Marsden, J. E. and S. Shkoller [1999], Multisymplectic geometry, covariant Hamiltonians and water waves, Math. Proc. Camb. Phil. Soc., 125, 553-575.
70. Marsden, J. E. and A. Weinstein [1974], Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5, 121-130.
71. Meyer, K. R. [1973], Symmetries and integrals in mechanics, in Dynamical Systems, M. Peixoto, ed., 259-273, Academic Press.
72. Marsden, J. E., R. Montgomery and T. S. Ratiu [1990], Reduction, symmetry and phases in mechanics, Amer. Math. Soc., Providence, RI Memoirs, 436.
73. Montgomery, R. [1984], Canonical formulations of a particle in a Yang-Mills field, Lett. Math. Phys., 8, 59-67.
74. Montgomery, R. [1986], The Bundle Picture in Mechanics, Ph.D. Thesis, University of California Berkeley.
75. Montgomery, R. [1990], Isoholonomic problems and some applications, Comm. Math Phys., 128, 565-592.
76. Montgomery, R. [1993], Gauge theory of the falling cat, Fields Inst. Commun., 1, 193-218.
77. Montgomery, R., J. E. Marsden and T. S. Ratiu [1984], Gauged Lie-Poisson structures, Contemp. Math., 28, 101-114, Amer. Math. Soc., Providence, RI.
78. 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.
79. Ortega, J.-P. [1998], Symmetry, Reduction and Stability in Hamiltonian Systems, Ph.D. Thesis, University of California Santa Cruz.
80. 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.
81. Ortega, J.-p. and T. S. Ratiu [2001], Hamiltonian Singular Reduction, Birkhäuser, Progress in Math.; (to appear).
82. Palais, R. S. [1961a], Equivalence of nearby differentiable actions of a compact group, Bull. Amer. Math. Soc., 67, 362-364.
83. Palais, R. S. [1961b], On the existence of slices for actions of non-compact Lie groups, Ann. of Math., 73, 295-323.
84. Poincaré, H. [1901a], Sur une forme nouvelle des équations de la méchanique, C. R. Acad. Sci., 132, 369-371.
85. Poincaré, H. [1901b], Sur la stabilité de l'équilibre des figures piriformes affectées par une masse fluide en rotation, Philosophical Transactions A, 198, 333-373.
86. Ratiu, T. S. [1980a], The Euler-Poisson Equations and Integrability, Ph.D. Thesis, University of California at Berkeley.
87. Ratiu, T. S. [1980b], The motion of the free n-dimensional rigid body, Indiana Univ. Math. Journ., 29, 609-629.
88. Ratiu, T. S. [1980c], Involution theorems, in Geometric Methods in Mathematical Physics, G. Kaiser and J. Marsden, eds., Springer Lecture Notes, 775, 219-257.
89. Ratiu, T. S. [1981], Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body, Proc. Natl. Acad. Sci., USA, 78, 1327-1328.
90. Ratiu, T. S. [1982a], Euler-Poisson equations on Lie algebras and the N-dimensional heavy rigid body, Amer. J. Math., 104, 409-448, 1337.
91. 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, 88, 109-116.
92. 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.
93. Sjamaar, R. and E. Lerman [1991], Stratified symplectic spaces and reduction, Ann. of Math., 134, 375-422.
94. 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.
95. Tulczyjew, W. M. [1977], The Legendre transformation, Ann. Inst. Poincaré, 27, 101-114.
96. Vershik, A. M.; Gershkovich, V. Ya. Nonholonomic problems and the theory of distributions. Acta Appl. Math. 12, 181-209.
97. Vershik, A. M. and V. Ya Gershkovich [1994], Non-holonomic Riemannian manifolds, in Dynamical Systems 7, Springer, Encyclopaedia of Mathematics, 16.
98. Veselov, A. P. [1988], Integrable discrete-time systems and difference operator's, Fund. An. and Appl., 22, 83-94.
99. Veselov, A. P. [1991], Integrable Lagrangian correspondences and the factorization of matrix polynomials, Fund. An. and Appl., 25, 112-123.
100. Weinstein, A. [1978], A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys., 2, 417-420.
101. Weinstein, A. [1978], Bifurcations and Hamilton's principle, Math. Zeit., 159, 235-248.
102. Weinstein, A. [1983a], The local structure of Poisson manifolds, J. of Diff. Geom., 18, 523-557.
103. Weinstein, A. [1983b], Sophus Lie and symplectic geometry, Expo. Math., 1, 95-96.
104. Weinstein, A. [1996], Lagrangian Mechanics and Groupoids, Fields Inst. Commun., 7, 207-231.
105. Weinstein, A. [1998] Poisson geometry, Diff. Geom. Appl. 9, 213-238.
106. Wendlandt, J. M. and J. E. Marsden [1997], Mechanical integrators derived from a discrete variational principle, Physica D, 106, 223-246.
107. Wong, S. K. [1970], Field and particle equations for the classical Yang-Mills field and particles with isotopic spin, Il Nuovo Cimento, LXV, 689-694.