Mechanical systems with symmetry on homogeneous spaces

Lacomba, Ernesto A.

doi:10.1090/S0002-9947-1973-0331426-0

Mechanical systems with symmetry on homogeneous spaces
HTML articles powered by AMS MathViewer

by Ernesto A. Lacomba PDF

Trans. Amer. Math. Soc. 185 (1973), 477-491 Request permission

Abstract:

The geodesic flow on a homogeneous space with an invariant metric can be naturally considered within the framework of Smale’s mechanical systems with symmetry. In this way we have at our disposal the whole method of Smale for studying such systems. We prove that certain sets $\Sigma ’,\Sigma ,\operatorname {Im} ,\sigma$ and Re which play an important role in the global behavior of those systems, have a particularly simple structure in our case, and we also find some geometrical implications about the geodesics. The results obtained are especially powerful for the case of Lie groups, as in the rigid body problem.

References

Ralph Abraham, Jerrold E. Marsden, Al Kelley, and A. N. Kolmogorov, Foundations of mechanics. A mathematical exposition of classical mechanics with an introduction to the qualitative theory of dynamical systems and applications to the three-body problem, W. A. Benjamin, Inc., New York-Amsterdam, 1967. With the assistance of Jerrold E. Marsden; Four appendices, one by the author, two by Al Kelley, the fourth, a translation of an article by A. N. Kolmogorov. MR 0220467
Ralph Abraham and Joel Robbin, Transversal mappings and flows, W. A. Benjamin, Inc., New York-Amsterdam, 1967. An appendix by Al Kelley. MR 0240836
V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1966), no. fasc. 1, 319–361 (French). MR 202082
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
Robert Hermann, Differential geometry and the calculus of variations, Mathematics in Science and Engineering, Vol. 49, Academic Press, New York-London, 1968. MR 0233313
Andrei Iacob, Invariant manifolds in the motion of a rigid body about a fixed point, Rev. Roumaine Math. Pures Appl. 16 (1971), 1497–1521. MR 303801
Bertram Kostant, On differential geometry and homogeneous spaces. I, II, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 258–261, 354–357. MR 88017, DOI 10.1073/pnas.42.6.354
John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. MR 0239612
Katsumi Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33–65. MR 59050, DOI 10.2307/2372398

Relative equilibria in mechanical systems

S. Smale, Topology and mechanics. I, Invent. Math. 10 (1970), 305–331. MR 309153, DOI 10.1007/BF01418778
S. Smale, Topology and mechanics. II. The planar $n$-body problem, Invent. Math. 11 (1970), 45–64. MR 321138, DOI 10.1007/BF01389805