Non-Existence of the Modified First Integral by Symplectic Integration Methods II: Kepler Problem |
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Authors: | Haruo Yoshida |
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Institution: | (1) National Astronomical Observatory of Japan, Mitaka, Tokyo, 181-8588, Japan |
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Abstract: | Symplectic integration methods conserve the Hamiltonian quite well because of the existence of the modified Hamiltonian as a formal conserved quantity. For a first integral of a given Hamiltonian system, the modified first integral is defined to be a formal first integral for the modified Hamiltonian. It is shown that the Runge-Lenz vector of the Kepler problem is not well conserved by symplectic methods, and that the corresponding modified first integral does not exist. This conclusion is given for a one-parameter family of symplectic methods including the symplectic Euler method and the Störmer/Verlet method. |
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Keywords: | symplectic integration method modified Hamiltonian modified first integral Kepler problem Runge-Lenz vector |
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