The Euler-Poincaré equations and double bracket dissipation

Communications in Mathematical Physics

Volumn 175, Issue 1, 1996, Pages 1-42

  Bloch, Anthony ^a   Krishnaprasad, P S ^b   Marsden, Jerrold E ^c   Ratiu, Tudor S ^d  

^a UNIVERSITY OF MICHIGAN   (United States)

^b UNIVERSITY OF MARYLAND   (United States)

^c CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^d UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030539817     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02101622     Document Type: Article

References (83)

1. Abarbanel, H.D.I., Holm, D.D., Marsden, J.E., Ratiu, T.S. [1986]: Nonlinear stability analysis of stratified fluid equilibria. Phil. Trans. R. Soc. Lond. A 318, 349-409; also Phys. Rev. Lett. 52 2352-2355 [1984]
2. Abarbanel, H.D.I., Holm, D.D., Marsden, J.E., Ratiu, T.S. [1986]: Nonlinear stability analysis of stratified fluid equilibria. Phil. Trans. R. Soc. Lond. A 318, 349-409; also Phys. Rev. Lett. 52 2352-2355 [1984]
3. Abraham, R., Marsden, J.E. [1978]: Foundations of Mechanics. (2nd ed.) Reading, MA: Addison-Wesley
4. Alekseevski, D.V., Michor, P.W. [1993]: Characteristic classes and Cartan connections. Preprint
5. Arnold, V. [1988]: Dynamical Systems III. Encyclopedia of Mathematics, Berlin, Heidelberg, New York: Springer
6. Bloch, A.M., Brockett, R.W., Ratiu, T.S. [1992]: Completely integrable gradient flows. Commun. Math. Phys. 147, 57-74
7. Bloch, A.M., Crouch, P.E. [1994]: Reduction of Euler Lagrange problems for constrained variational problems and relation with optimal control problems. Proc. CDC 33, 2584-2590, IEEE
8. Bloch, A.M., Flaschka, H., Ratiu, T.S. [1990]: A convexity theorem for isospectral sets of Jacobi matrices in a compact Lie algebra. Duke Math. J. 61, 41-66
9. Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., Ratiu, T.S. [1991]: Asymptotic stability, instability, and stabilization of relative equilibria. Proc. of ACC., Boston IEEE, 1120-1125
10. Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., Ratiu, T.S. [1994]: Dissipation Induced Instabilities. Ann. Inst. H. Poincare, Analyse Nonlineare 11, 37-90
11. Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., Sánchez de Alvarez, G. [1992]: Stabilization of rigid body dynamics by internal and external torques. Automatica 28, 745-756
12. Brockett, R.W. [1973]: Lie theory and control systems defined on spheres. SIAM J. Appl. Math./ 23, 213-225
13. Brockett, R.W. [1988]: Dynamical systems that sort lists and solve linear programming problems. Proc. IEEE 27, 799-803 and Linear Algebra and its Appl. 146, (1991), 79-91
14. Brockett, R.W. [1988]: Dynamical systems that sort lists and solve linear programming problems. Proc. IEEE 27, 799-803 and Linear Algebra and its Appl. 146, (1991), 79-91
15. Brockett, R.W. [1993]: Differential geometry and the design of gradient algorithms. Proc. Symp. Pure Math., AMS 54, Part 1, 69-92
16. Brockett, R.W. [1994]: The double bracket equation as a solution of a variational problem. Fields Institute Comm. 3, 69-76, AMS
17. Chandrasekhar, K. [1977]: Ellipsoidal Figures of Equilibrium. Dover
18. Chern, S.J., Marsden, J.E. [1990]: A note on symmetry and stability for fluid flows. Geo. Astro. Fluid. Dyn. 51, 1-4
19. Chetaev, N.G. [1961]: The stability of Motion. Trans, by M. Nadler, New York: Pergamon Press
20. Crouch, P.E. [1981]: Geometric structures in systems theory. IEEE Proc, Part D, No 5 128
21. Ebin, D.G., Marsden, J.E. [1970]: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102-163
22. Giles, R., Patterson, G., Bagneres, A., Kotiuga, R., Humphrey, F., Mansuripur, M. [1991]: Micromagnetic simulations on the connection machine. Very Large Scale Computation in the 21st Century, J.P. Mesirov, ed., SIAM, 33-40
23. Grmela, M. [1993a]: Weakly nonlocal hydrodynamics. Phys. Rev. E 47, 351-365
24. Grmela, M. [1984]: Bracket formulation of dissipative fluid mechanics equations. Phys. Lett. A 102, 355-358
25. Grmela, M. [1993b]: Thermodynamics of driven systems. Phys. Rev. E 48, 919-930
26. Guckenheimer, J., Mahalov, A. [1992]: Instability induced by symmetry reduction. Phys. Rev. Lett. 68, 2257-2260
27. Hahn, W. [1967]: Stability of Motion. Springer, Berlin, New York: Heidelberg
28. Haller, G. [1992]: Gyroscopic stability and its loss in systems with two essential coordinates. Int. J. Nonlinear Mech. 27, 113-127
29. Helman, J.S., Braun, H.B., Broz, J.S., Baltensperger, W. [1991]: General solution to the Landau-Lifschitz-Gilbert equations linearized around a Bloch wall. Phys. Rev. B 43, 5908-5914
30. Holm, D.D., Marsden, J.E., Ratiu, T.S., Weinstein, A. [1985]: Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1-116
31. Kammer, D.C., Gray, G.L. [1993]: A nonlinear control design for energy sink simulation in the Euler-Poinsot problem. J. Astron. Sci. 41, 53-72
32. Kandrup, H.E. [1991]: The secular instability of axisymmetric collisionless star cluster. Astrophy. J. 380, 511-514
33. Kandrup, H.E., Morrison, P. [1992]: Hamiltonian structure of the Vlasov-Einstein system and the problem of stability for spherical relativistic star clusters. Preprint.
34. Kaufman, A.N. [1984]: Dissipative Hamiltonian systems: A unifying principle. Physics Letters A 100, 419-422
35. Kaufman, A.N. [1985]: Lorentz-covariant dissipative Lagrangian systems. Physics Letters A 109, 87-89
36. Kolář, I., Michor, P.W., Slovák, J. [1993]: Natural Operations in Differential Geometry. Berlin, Heidelberg, New York: Springer
37. Knobloch, E., Mahalov, Marsden, J.E. [1994], Normal forms for three-dimensional parametric instabilities in ideal hydrodynamics. Physica D 73, 49-81
38. Krein, M.G. [1950]: A generalization of some investigations of linear differential equations with periodic coefficients. Doklady Akad. Nauk SSSR N.S. 73, 445-448
39. Krishnaprasad, P.S. [1985]: Lie-Poisson structures, dual-spin spacecraft and asymptotic stability. Nonl. An. Th. Meth. and Appl. 9, 1011-1035
40. LaSalle, J.P., Lefschetz, S. [1963]: Stability by Lyapunov's direct method. New York: Academic Press
41. Lewis, D.K. [1992]: Lagrangian block diagonalization. Dyn. Diff. Eqn's. 4 1-42
42. Lewis, D., Ratiu, T.S., Simo, J.C., Marsden, J.E. [1992]: The heavy top, a geometric treatment. Nonlinearity 5, 1-48
43. Lewis, D.K., Simo, J.C. [1990]: Nonlinear stability of rotating pseudo-rigid bodies. Proc. Roy. Soc. Lon. A 427, 281-319
44. MacKay, R. [1991]: Movement of eigenvalues of Hamiltonian equilibria under non-Hamiltonian perturbation. Phys. Lett. A155, 266-268
45. Marsden, J.E. [1992]: Lectures on Mechanics. London Mathematical Society Lecture note series, 174, Cambridge: Cambridge University Press
46. Marsden, J.E., Ratiu, T.S. [1992]: Symmetry and Mechanics. Texts in Applied Mathematics, 17, Berlin, Heilderberg, New York: Springer
47. Marsden, J.E., Ratiu, T.S., Raugel, G. [1991]: Symplectic connections and the linearization of Hamiltonian systems. Proc. Roy. Soc. Ed. A 117, 329-380
48. Marsden, J.E., Ratiu, T.S., Weinstein, A. [1984]: Semi-direct products and reduction in mechanics. Trans. Am. Math. Soc. 281, 147-177
49. Marsden, J.E., Scheurle, J. [1993a]: Lagrangian reduction and the double spherical pendulum. ZAMP 44, 17-43
50. Marsden, J.E., Scheurle, J. [1993b]: The reduced Euler-Lagrange equations. Fields Institute Comm. 1, 139-164
51. Marsden, J.E., Weinstein, A. [1982]: The Hamiltonian structure of the Maxwell-Vlasov equations. Physica D 4, 394-406
52. Morrison, P.J. [1980]: The Maxwell-Vlasov equations as a continuous Hamiltonian system. Phys. Lett. A 80, 383-386
53. Morrison, P.J. [1982]: Poisson Brackets for fluids and plasmas. In: Mathematical Methods in Hydrodynamics and Integrability in Related Dynamical Systems. AIP Conf. Proc. 88, M. Tabor and Y.M. Treve (eds.), La Jolla, CA
54. Morrison, P.J. [1986]: A paradigm for joined Hamiltonian and dissipative systems. Physica D 18, 410-419
55. Morrison, P.J., Kotschenreuther, M. [1989]: The free energy principle, negative energy modes and stability Proc 4th Int. Workshop on Nonlinear and Turbulent Processes in Physics, Singapore: World Scientific Press.
56. Newcomb, W.A. [1962]: Lagrangian and Hamiltonian methods in Magnetohydrodynamics. Nuc. Fusion Suppl., part 2, 451-463
57. O'Dell, T.H. [1981]: Ferromagnetodynamics. New York: John Wiley
58. O'Reilly, O, Malhotra, N.K., Namamchehivaya, N.S. [1993]: Destabilization of the equilibria of reversible dynamical systems (preprint)
59. Pego, R.L., Weinstein, M.I. [1992]: Eigenvalues and instabilities of solitary waves. Phil. Trans. Roy. Soc. Lon. 340, 47-94
60. Poincaré, H. [1885]: Sur l'équilibre d'une masse fluide animée d'un mouvement de rotation. Acta. Math. 7, 259
61. Poincaré, H. [1892]: Les formes d'équilibre d'une masse fluide en rotation. Revue Générale des Sciences 3, 809-815
62. Poincaré, H. [1901]: Sur la stabilité de l'équilibre des figures piriformes affectées par une masse fluide en rotation. Philosophical Transactions A 198, 333-373
63. Poincaré, H. [1910]: Sur la precession des corps deformables. Bull Astron 27, 321-356
64. Posbergh, T.A. [1994]: The damped vibration absorber as a feedback control problem. Preprint
65. Posbergh, T.A., Zhao, R. [1993]: Stabilization of the uniform rotation of a rigid body by the energy-momentum method. Fields Inst. Comm. 1, 263-280
66. Riemann, B. [1860]: Untersuchungen über die Bewegung eines flüssigen gleichartigen Ellipsoides. Abh. d. Königl. Gesell, der Wiss. zu Göttingen 9, 3-36
67. Routh, E.J. [1877]: Stability of a given state of motion. Reprinted in Stability of Motion, ed. A.T. Fuller, New York: Halsted Press, 1975
68. Seliger, R.L., Whitham, G.B. [1968]: Variational principles in continuum mechanics. Proc. Roy. Soc. Lond. 305, 1-25
69. Shepherd, T.G. [1992]: Extremal properties and Hamiltonian structure of the Euler equations. Topological Aspects of the Dynamics of Fluids and Plasmas, Moffatt, H.K. et. al., eds, Dordrecht. Kluwer, 275-292
70. Simo, J.C., Lewis, D.K., Marsden, J.E. [1991]: Stability of relative equilibria I: The reduced energy momentum method. Arch. Rat. Mech. Anal. 115, 15-59
71. Simo, J.C., Posbergh, T.A., Marsden, J.E. [1990]: Stability of coupled rigid body and geometrically exact rods: Block diagonalization and the energy-momentum method. Physics Reports 193, 280-360
72. Simo, J.C., Posbergh, T.A., Marsden, J.E. [1991]: Stability of relative equilibria II: Three dimensional elasticity. Arch. Rat. Mech. Anal. 115, 61-100
73. Sri Namachchivaya, N., Ariaratnam, S.T. [1985]: On the dynamic stability of gyroscopic systems. SM Archives 10, 313-355
74. Sternberg, S. [1963]: Lectures on Differential Geometry. New York: Prentice-Hall (Reprinted by Chelsea, 1983)
75. Taussky, O. [1961]: A Generalization of a Theorem of Lyapunov. SIAM J. Appl. Math 9, 640-643
76. Thomson, L., Tait, P.G. [1879]: Treatise on Natural Philosophy. Cambridge: Cambridge Univ. Press
77. Touma, J., Wisdom, J. [1992]: Lie-Poisson integrators for rigid body dynamics in the solar system. Preprint
78. Turski, L.A., Kaufman, A.N. [1987]: Canonical-dissipative formulation of relativistic plasma kinetic theory with self-consistent Maxwell field. Physics Lett. A 120, 331-333
79. Vallis, G.K., Carnevale, G.F., Young, W.R. [1989]: Extremal energy properties and construction of stable solutions of the Euler equations. J. Fluid Mech. 207, 133-152
80. van Gils, S.A., Krupa, M., Langford, W.F. [1990]: Hopf bifurcation with non-semisimple 1:1 resonance. Nonlinearity 3, 825-830
81. Vershik, A.M., Faddeev, L.D. [1981]: Lagrangian mechanics in invariant form. Sel. Math. Sov. 1, 339-350
82. Wang, L.S., Krishnaprasad, P.S. [1992]: Gyroscopic control and stabilization. J. Nonlinear Sci. 2, 367-415
83. Ziegler, H. [1956]: On the concept of elastic stability. Adv. Appl. Mech. 4, 351-403