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Preservation of adiabatic invariants for disturbed Hamiltonian systems under variational discretization

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Abstract

The perturbation to conformal invariance and the numerical algorithm of Hamiltonian systems with disturbed forces under variational discretization are studied in this paper. Based on the discrete difference variational principles, the discrete Hamiltonian equations (variational integrators) for dynamical systems are obtained in the undisturbed and the disturbed cases, respectively. The determining equations of perturbation to conformal invariance are established for disturbed Hamiltonian systems. The exact invariants of Noether type led by conformal invariance for an undisturbed Hamiltonian system are derived. For disturbed discrete Hamiltonian systems, the condition of perturbation to conformal invariance leading to adiabatic invariants is proposed. Two examples are considered: a simple harmonic oscillator and the Kepler problem. The dynamical analysis is given by using the numerical results.

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Acknowledgements

This work is partly supported by National Natural Science Foundation of China (Grand Nos.11502071 and 11732005), the spatial research project and Qin Xin Talents Cultivation Program of Beijing Information Science and Technology University.

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Authors and Affiliations

  1. College of Science, Beijing Information Science and Technology University, Beijing, 100192, China

    Li-Li Xia

  2. Mechanical and Electrical Engineering School, Beijing Information Science and Technology University, Beijing, 100192, China

    Long Bai

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  1. Li-Li Xia
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Correspondence to Li-Li Xia.

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Xia, LL., Bai, L. Preservation of adiabatic invariants for disturbed Hamiltonian systems under variational discretization. Acta Mech 231, 783–793 (2020). https://doi.org/10.1007/s00707-019-02571-z

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  • DOI: https://doi.org/10.1007/s00707-019-02571-z

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