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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.
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Abstract

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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.

Information

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

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