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Symplectic connections and the linearisation of Hamiltonian systems

Published online by Cambridge University Press:  14 November 2011

J. E. Marsden ,
T. Ratiu  and
G. Raugel
J. E. Marsden
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720, U.S.A. and Cornell University, Ithaca, NY 14853-7901, U.S.A.
T. Ratiu
Affiliation:
Department of Mathematics, University of California, Santa Cruz, CA 95064, and MSRI, 1000 Centennial Drive, Berkeley, CA 94720, U.S.A.
G. Raugel
Affiliation:
Laboratoire d'Analyse Numérique (Unité Associée au CNRS D760) Bâtiment 425, Université de Paris-Sud, 91405 Orsay Cedex, France
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Synopsis

This paper uses symplectic connections to give a Hamiltonian structure to the first variation equation for a Hamiltonian system along a given dynamic solution. This structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the Hamiltonian (plus Casimir) as energy. This structure is different from the well-known and elementary tangent space construction. Our results are applied to systems with symmetry and to Lie–Poisson systems in particular.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 117 , Issue 3-4 , 1991 , pp. 329 - 380
Copyright
Copyright © Royal Society of Edinburgh 1991

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References

1Abarbanel, H. D. I., Holm, D. D., Marsden, J. E. and Ratiu, T. S.. Nonlinear stability analysis of stratified fluid equilibria. Philos Trans. Roy. Soc. London, Ser. A 318 (1986). 349–409.Google Scholar
2Abraham, R. and Marsden, J.. Foundations of Mechanics, 2nd edn. (Reading, Mass.: Addison-Wesley, 1978).Google Scholar
3Abraham, R., Marsden, J. and Ratiu, T.. Manifolds, Tensor Analysis, and Applications, 2nd edn. (New York: Springer, 1988).CrossRefGoogle Scholar
4Arms, J. M.. The structure of the solution set for the Yang–Mills equations. Math. Proc. Cambridge. Philos. Soc. 90 (1981), 361–372.Google Scholar
5Arms, J. M., Marsden, J. E. and Moncrief, V.. The structure of the space solutions of Einstein's equations II: Several Killings fields and the Einstein–Yang–Mills equations. Ann. of Phys. 144 (1982), 81–106.CrossRefGoogle Scholar
6Arnold, V. I.. Sur la géometrie differentielle des groupes de Lie de dimensional infinie et ses applications a l'hydrodynamique des fluids parfaits. Ann. Inst. Fourier. Grenoble 16 (1966), 319–361.CrossRefGoogle Scholar
7Arnold, V. I.. Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics 60 (Berlin: Springer, 1978).Google Scholar
8Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A. and Sternheimer, D.. Deformation theory and quantization I. Ann. of Phys. 111 (1978), 61–110.CrossRefGoogle Scholar
9Cendra, H. and Marsden, J.. Lin Constraints, Clebsch potentials and variational prinicples. Physica D 27 (1987), 63–89.CrossRefGoogle Scholar
10Chernoff, P. and Marsden, J. E.. Properties of Infinite Dimensional Hamiltonian Systems, Springer Lecture Notes in Mathematics 425 (Berlin: Springer, 1974).CrossRefGoogle Scholar
11Eardley, D. and Moncrief, V.. Global Existence of Yang–Mills Higgs Fields in Four Dimensional Minkowski Space. Comm. Math. Phys. 83 (1981), 171–211.CrossRefGoogle Scholar
12Fischer, A. E., Marsden, J. E. and Moncrief, V.. The structure of the space of solutions of Einstein's equations I: One Killing field. Ann. Inst. H. Poincaré 33 (1980), 147–194.Google Scholar
13Gimmsy, G.. Momentum Maps and Classical Relativistic Fields (in prep.)Google Scholar
14Greene, J. M. and Kim, J.-S.. Introduction of a metric tensor into linearized evolution equations. Physica D 36 (1989), 83–91.Google Scholar
15Hess, H.. Connections on symplectic manifolds and geometric quantization. Springer Lecture Notes in Mathematics 836, Differential Geometrical Methods in Mathematical Physics, Proc. Aix-en-Provence and Salamanca, 1979, eds. Perez-Rendon, A. and Souriau, J. M., 153–166 (Berlin: Springer, 1980).Google Scholar
16Hess, H.. Symplectic connections in geometric quantization and factor orderings (Ph.D. Thesis, Physics, Freie Universität Berlin, 1981).Google Scholar
17Holm, D. D., Marsden, J. E., Ratiu, T. S. and Weinstein, A.. Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123 (1985), 1–116.CrossRefGoogle Scholar
18Kobayashi, S. and Nomizu, K.. Foundations of Differential Geometry, Vol. 1 (New York: Interscience Publishers, 1963).Google Scholar
19Kostant, B.. Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold. Trans. Amer. Math. Soc. 80 (1955), 528–542.CrossRefGoogle Scholar
20Lichnerowicz, A.. Connexions symplectiques et *-produits invariants. C.R. Acad. Sci. Paris 291 (1980), 413–417.Google Scholar
21Marsden, J. E. and Weinstein, A.. Coadjoint orbits, vortices and Clebsch variables for incompressible fluids. Physica D 7 (1983), 305–323.Google Scholar
22Marsden, J. E., Weinstein, A., Ratiu, T., Schmid, R. and Spencer, R. G.. Hamiltonian systems with symmetry, coadjoint orbits and plasma physics. Proc. IUTAM-ISIMM Symposium on “Modern Developments in Analytical Mechanics,” Torino, June 7–11, 1982. Atti della Academia delta Scienze di Torino 117 (1983), 289–340.Google Scholar
23Poor, W.. Differential Geometric Structures (New York: McGraw-Hill, 1981).Google Scholar
24Alvarez, G. Sanchez de. Geometric methods of classical mechanics applied to control theory (Ph. D. Thesis, University of California, Berkeley, 1986).Google Scholar
25Spivak, M.. Differential Geometry, Vols. 1–5 (Waltham, Mass.: Publish or Perish, 1979).Google Scholar
26Tondeur, Ph.. Affine Zusammenhänge auf Mannigfaltigkeiten mit fast-symplektischer Struktur. Comment. Math. Helv. 36 (1961), 234–243.CrossRefGoogle Scholar
27Vey, J.. Déformation due crochet de Poisson sur une variété symplectique. Comment. Math. Helv. 50 (1975), 421–454.Google Scholar
28Vaisman, I.. Symplectic Twistor Spaces. J. Geom. Phys. 3 (1986), 507–524.Google Scholar
29Mardsen, J. E., Montgomery, R. and Ratiu, T.[1990] Reduction, symmetry, and phases in mechanics. Memoirs Amer. Math. Soc. 436, 1–110.Google Scholar