Preservation of adiabatic invariants for disturbed Hamiltonian systems under variational discretization

  • Li-Li XiaEmail author
  • Long Bai
Original Paper


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.



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.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this article.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of ScienceBeijing Information Science and Technology UniversityBeijingChina
  2. 2.Mechanical and Electrical Engineering SchoolBeijing Information Science and Technology UniversityBeijingChina

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