Constants of Motion and Lie Group Actions

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Abstract
A classical Hamiltonian dynamical system with 2N-dimensional phase space is studied in the case when a Lie algebra of constants of the motion exists which contains 2N −- 1 functionally independent elements and when each constant of motion generates a complete integral curve. It is proved that a connected (global) Lie group G acts on the phase space and acts transitively on each connected component of each surface of constant energy. When G is compact, each component of the space of time orbits corresponding to a fixed energy is shown to be a (2N − 2)-dimensional compact symplectic manifold diffeomorphic to an orbit of G in the dual of the adjoint representation. It is shown that a (global) Lie group does act in the case of the harmonic oscillator, but does not act in the case of the negative energy Kepler problem.

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