A StructurePreserving Algorithm for Hamiltonian Systems With Nonholonomic Constraints

MAN Shumin; GAO Qiang; ZHONG Wanxie

doi:10.21656/1000-0887.400375

Volume 41 Issue 6

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MAN Shumin, GAO Qiang, ZHONG Wanxie. A StructurePreserving Algorithm for Hamiltonian Systems With Nonholonomic Constraints[J]. Applied Mathematics and Mechanics, 2020, 41(6): 581-590. doi: 10.21656/1000-0887.400375

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A StructurePreserving Algorithm for Hamiltonian Systems With Nonholonomic Constraints

doi: 10.21656/1000-0887.400375

State Key Laboratory of Structural Analysis for Industrial Equipment; Department of Engineering Mechanics, Dalian University of Technology, Dalian, Liaoning 116023, P.R.China

Funds: The National Natural Science Foundation of China（11972107;91748203）

Received Date: 2019-12-23
Publish Date: 2020-06-01

Abstract

Abstract

Based on the concept of variational integrator and the Lagrange-d’Alembert principle with dual variables, a high-order structure-preserving algorithm for Hamiltonian systems with nonholonomic constraints was proposed. Based on the variational integrator, a discretization form of the Lagrange-d’Alembert principle with dual variables was obtained by means of appropriate polynomials and quadrature rules. On the basis of this discretization form, a numerical integration method was given with displacements at both ends of the integral interval as independent variables. Meanwhile, the nonholonomic constraints were strictly met at the endpoints of the integral interval and the control points within the interval. The symmetric property of the proposed algorithm was proved. Numerical examples show that, the proposed algorithm has a high convergence order, strictly meets the nonholonomic constraints and has good long-time behaviors.
- nonholonomic constraint,
- variational integrator,
- structure-preserving algorithm,
- dual variables

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References(17)

References

[1]	HERTZ A, GARBASSO A. Die prinzipien der mechanik in neuem zusammenhang dargestellt[J]. Il Nuovo Cimento 〖STBX〗（1895—1900）,1895,1(1): 40-59.
[2]	APPELL P. Traité de Mécanique Rationnelle [M]. Gauthier-Villars, 1924.
[3]	HAMEL G. Theoretische Mechanik [M]. Berlin: Springer, 1978.
[4]	BLOCH A M. Nonholonomic Mechanics and Control [M]. Berlin: Springer, 2015.
[5]	MARSDEN J E, WEST M. Discrete mechanics and variational integrators[J]. Acta Numerica,2001,10(1): 357-514.
[6]	LEW A, MARSDEN J E, ORTIZ M, et al. Variational time integrators[J]. International Journal for Numerical Methods in Engineering,2010,60(1): 153-212.
[7]	HAIRER E, WANNER G, LUBICH C. Geometric Numerical Integration [M]. Berlin: Springer, 2002.
[8]	CORTS J, MARTNEZ S. Non-holonomic integrators[J]. Nonlinearity,2001,14(5): 1365-1392.
[9]	CORTS J. Energy conserving nonholonomic integrators[J]. Discrete & Continuous Dynamical Systems,2003,2003(S): 189-199.
[10]	DE LEN M, DE DIEGO D M, SANTAMARIA-MERINO A. Geometric integrators and nonholonomic mechanics[J]. Journal of Mathematical Physics,2002,45(3): 1042.
[11]	DE LEN M, DE DIEGO D M, SANTAMARIA-MERINO A. Geometric numerical integration of nonholonomic systems and optimal control problems[J]. European Journal of Control,2004,10(5): 515-521.
[12]	MCLACHLAN R, PERLMUTTER M. Integrators for nonholonomic mechanical systems[J]. Journal of Nonlinear Science,2006,16(4): 283-328.
[13]	KOBILAROV M, MARSDEN J E, SUKHATME G S. Geometric discretization of nonholonomic systems with symmetries[J]. Discrete and Continuous Dynamical Systems(Series S),2010,3(1): 61-84.
[14]	GAO Q, TAN S J, ZHANG H W, et al. Symplectic algorithms based on the principle of least action and generating functions[J]. International Journal for Numerical Methods in Engineering,2012,89(4): 438-508.
[15]	HALL J, LEOK M. Spectral variational integrators[J]. Numerische Mathematik,2012,130(4): 681-740.
[16]	HAIRER E, LUBICH C, WANNER G. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations [M]. Berlin: Springer, 2006.
[17]	OSTROWSKI J, LEWIS A, MURRAY R, et al. Nonholonomic mechanics and locomotion: the snakeboard example[C]// Paper Presented at the Proceedings of the 1994 IEEE International Conference on Robotics and Automation.Piscataway, USA, 1994.