Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-06T13:51:48.134Z Has data issue: false hasContentIssue false
  • English
  • Français

On collective complete integrability according to the method of Thimm

Published online by Cambridge University Press:  19 September 2008

Victor Guillemin  and
Shlomo Sternberg
Victor Guillemin
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, U.S.A.
Shlomo Sternberg
Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts, 02138, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a Lie group acting in Hamiltonian fashion on a symplectic manifold M with moment map Φ:M → g*. A function of the form ƒ∘Φ where ƒ is a function on g* is called ‘collective’. We obtain necessary conditions on the G action for there to exist enough Poisson commuting functions on g* so that the corresponding collective functions on M form a completely integrable system. For the case G = O(n) or U(n) these conditions are sufficient. This explains Thimm's proof [17] of the complete integrability of the geodesic flow on the real and complex grassmanians. We also discuss related questions in the geometry of the moment map.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 3 , Issue 2 , June 1983 , pp. 219 - 230
Copyright
Copyright © Cambridge University Press 1983

References

REFERENCES

[1]Tkhi, Dao Chong. Integrability of Euler's equations on homogeneous symplectic manifolds. Math. Sb. 106 No. 2 (1978), 154–161.Google Scholar
[2]Duflo, M. & Vergne, M.. Une propriete de la representation coadjointe d'une algebre de Lie. CRAS Paris Set. A-B 268 (1969) A583–585.Google Scholar
[3]Guillemin, V. & Sternberg, S.. The moment map and collective motion. Ann. of Phys. 127 (1980), 220–253.Google Scholar
[4]Guillemin, V. & Sternberg, S.. Convexity properties of the moment map. Invenliones Math. 67 (1982) 491–513.CrossRefGoogle Scholar
[5]Guillemin, V. & Sternberg, . Moments and reductions. Conference on differential geometric methods in theoretical physics. World Scientific (Singapore) (1983) 87–102.Google Scholar
[6]Heckmann, G.. Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups. To appear.Google Scholar
[7]Jacobi, C.. Vorlesungen uber Mechanik. Gesammelte Abhandlungen.Google Scholar
[8]Kazhdan, D., Kostant, B. & Sternberg, S.. Hamiltonian group actions and dynamical systems of Calogero type. Comm. Pure and App. Math. 31 (1978), 481–507.CrossRefGoogle Scholar
[9]Kramer, M.. Sphärische Untergruppen in kompakten zusamenhängenden Liegruppen. Compositio Mathematica 38 (1979), 129–153.Google Scholar
[10]Marsden, J. & Weinstein, A.. Reduction of symplectic manifolds with symmetry. Reports on Math Phys. 5 (1974), 121–130.Google Scholar
[11]Mikitiuk, I. V.. In Dauk Akad. Nauk SSSR 265 (1982) 1074–1078.Google Scholar
[12]Mishchenko, A. S.. Integration of the geodesic flows on symmetric spaces. Matem. Zametki 31: 2 (1982), 257–262.Google Scholar
[13]Mishchenko, A. S. & Fomenko, A. T.. Euler equations on finite dimensional Lie groups, Itvestia 12 (1978).Google Scholar
[14]Moser, J.. Various aspects of integrable Hamiltonian systems. Proj. Math 8 Birkhauser, (1980).CrossRefGoogle Scholar
[15]Planchart, A.. Thesis, Berkeley, 1982.Google Scholar
[16]Sternberg, S.. Symplectic Homogeneous spaces. Trans. Amer. Math. Soc. 212 (1975) 113–130.CrossRefGoogle Scholar
[17]Thimm, A.. Integrable geodesic flows on homogeneous spaces. Ergod. Th. & Dynam. Sys. 1 (1981), 495–517.CrossRefGoogle Scholar