On averaging,reduction, and symmetry in hamiltonian systems |
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Authors: | Richard C Churchill Martin Kummer David L Rod |
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Affiliation: | Department Mathematics, Hunter College, City University of New York, New York, New York 10021 USA;Department Mathematics, University of Toledo, Toledo, Ohio 43606 USA;Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada |
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Abstract: | The existence of periodic orbits for Hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged Hamiltonian on an associated “reduced space.” Alternatively, in classical (kinetic plus potential energy) Hamiltonians the existence of such orbits can often be established by elementary geometrical arguments. The present paper unifies the two approaches by exploiting discrete symmetries, including reversing diffeomorphisms, that occur in a given system. The symmetries are used to locate the periodic orbits in the averaged Hamiltonian, and thence in the original Hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries. In applications to the Hénon-Heiles Hamiltonian, it is illustrated how “higher order” averaging can sometimes be used to overcome degeneracies encountered at first order. |
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