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Variational discretization of constrained Birkhoffian systems

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Abstract

In this paper, we derive a variational characterization of constrained Birkhoffian dynamics in both continuous and discrete settings. When additional algebraic constraints appear, derivation of the necessary conditions under which the Pfaff action is extremized gives constrained Birkhoffian equations. Inspired by this continuous framework, we directly discretize the constraints as well as the Pfaff action and consequently formulate the discrete constrained Birkhoffian dynamics. Via this discrete variational approach which is parallel with the continuous case, the resulting discrete constrained Birkhoffian equations automatically preserve the intrinsic symplectic structure when identified as numerical algorithms. Considering that the obtained algorithms require not only the specification of an initial configuration but also a second configuration to operate, we present a natural, reasonable, and efficient method of initialization of simulations. While retaining the structure-preserving property, the obtained discrete schemes exhibit excellent numerical behaviors, demonstrated by numerical examples dealing with the mathematical pendulum and the 3D pendulum.

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Acknowledgments

The project was supported by the National Natural Science Foundation of China (Grant Nos. 10932002, 10972031, and 11272050).

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Authors and Affiliations

  1. College of Science, North China University of Technology, Beijing, 100144, China

    Xinlei Kong

  2. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Huibin Wu

  3. School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Fengxiang Mei

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  1. Xinlei Kong
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  2. Huibin Wu
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  3. Fengxiang Mei
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Correspondence to Xinlei Kong.

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Kong, X., Wu, H. & Mei, F. Variational discretization of constrained Birkhoffian systems. Nonlinear Dyn 78, 329–339 (2014). https://doi.org/10.1007/s11071-014-1442-x

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  • DOI: https://doi.org/10.1007/s11071-014-1442-x

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