Symmetry and integrable canonical flows

Following the method of a group theoretic formulation of rigid body dynamics, we construct an elementary proof that f commuting generators of symmetries of an f degree-of-freedom Hamiltonian system yield integrability of the dynamics in the form of f independent translations in phase space. The integrability of the dynamical system follows directly from the trivial integrability of a particular set of group parameter velocities that are nonintegrable in the absence of symmetry, and does not rely at all upon any assumption of separability of the Hamiltonian-Jacobi partial differential equation. Our method relies upon Hamel's explanation of when one can and cannot choose group parameters as generalized coordinates, and uses the Poisson bracket formulation of mechanics that is familiar to physicists.

We formally extend Euler's theorem on rigid body motions to other transformation groups for Hamiltonian flows in phase space, and also note the analogy between nonholonomic coordinates in classical mechanics and uncertainty principles in quantum mechanics.