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Variational Integrators in Holonomic Mechanics

Author

Listed:
  • Shumin Man

    (State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116023, China)

  • Qiang Gao

    (State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116023, China)

  • Wanxie Zhong

    (State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116023, China)

Abstract

Variational integrators for dynamic systems with holonomic constraints are proposed based on Hamilton’s principle. The variational principle is discretized by approximating the generalized coordinates and Lagrange multipliers by Lagrange polynomials, by approximating the integrals by quadrature rules. Meanwhile, constraint points are defined in order to discrete the holonomic constraints. The functional of the variational principle is divided into two parts, i.e., the action of the unconstrained term and the constrained term and the actions of the unconstrained term and the constrained term are integrated separately using different numerical quadrature rules. The influence of interpolation points, quadrature rule and constraint points on the accuracy of the algorithms is analyzed exhaustively. Properties of the proposed algorithms are investigated using examples. Numerical results show that the proposed algorithms have arbitrary high order, satisfy the holonomic constraints with high precision and provide good performance for long-time integration.

Suggested Citation

  • Shumin Man & Qiang Gao & Wanxie Zhong, 2020. "Variational Integrators in Holonomic Mechanics," Mathematics, MDPI, vol. 8(8), pages 1-19, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1358-:d:398483
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    References listed on IDEAS

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    1. C. L. Hwang & L. T. Fan, 1967. "A Discrete Version of Pontryagin's Maximum Principle," Operations Research, INFORMS, vol. 15(1), pages 139-146, February.
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    Cited by:

    1. Kong, Xinlei & Wang, Zhongxin & Wu, Huibin, 2022. "Variational integrators for forced Lagrangian systems based on the local path fitting technique," Applied Mathematics and Computation, Elsevier, vol. 416(C).

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    1. Man, Shumin & Gao, Qiang & Zhong, Wanxie, 2023. "High order symmetric algorithms for nonlinear dynamical systems with non-holonomic constraints," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 524-547.

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