Authors

Robert I. McLachlan, Klas Modin, Olivier Verdier, Matt Wilkins

Abstract

We show that symplectic Runge-Kutta methods provide effective symplectic integrators for Hamiltonian systems with index one constraints. These include the Hamiltonian description of variational problems subject to position and velocity constraints nondegenerate in the velocities, such as those arising in sub-Riemannian geometry and control theory.

Citation

  • Journal: SIAM Journal on Scientific Computing
  • Year: 2013
  • Volume: 35
  • Issue: 5
  • Pages: A2150–A2162
  • Publisher: Society for Industrial & Applied Mathematics (SIAM)
  • DOI: 10.1137/120885085

BibTeX

@article{McLachlan_2013,
  title={{Symplectic Integrators for Index 1 Constraints}},
  volume={35},
  ISSN={1095-7197},
  DOI={10.1137/120885085},
  number={5},
  journal={SIAM Journal on Scientific Computing},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={McLachlan, Robert I. and Modin, Klas and Verdier, Olivier and Wilkins, Matt},
  year={2013},
  pages={A2150--A2162}
}

Download the bib file

References

  • Benito R, Martín de Diego D (2005) Discrete vakonomic mechanics. Journal of Mathematical Physics 46(8):083521. https://doi.org/10.1063/1.2008214 – 10.1063/1.2008214
  • Bloch AM (2003) Nonholonomic Mechanics and Control. Springer New York, New York, NY – 10.1007/b97376
  • Burrage K, Butcher JC (1979) Stability Criteria for Implicit Runge–Kutta Methods. SIAM J Numer Anal 16(1):46–57. https://doi.org/10.1137/0716004 – 10.1137/0716004
  • Cortés Monforte J (2002) Geometric, Control and Numeric Aspects of Nonholonomic Systems. Springer Berlin Heidelberg, Berlin, Heidelberg – 10.1007/b84020
  • Fernandez OE, Bloch AM, Olver PJ (2012) Variational Integrators for Hamiltonizable Nonholonomic Systems. JGM 4(2):137–163. https://doi.org/10.3934/jgm.2012.4.137 – 10.3934/jgm.2012.4.137
  • HAIRER E, JAY L (1993) IMPLICIT RUNGE-KUTTA METHODS FOR HIGHER INDEX DIFFERENTIAL-ALGEBRAIC SYSTEMS. In: Contributions in Numerical Mathematics. WORLD SCIENTIFIC, pp 213–224 – 10.1142/9789812798886_0017
  • Donne E. Le, ETH Zürich (2009)
  • Leimkuhler B., Simulating Hamiltonian Dynamics (2004)
  • Martı́nez S, Cortés J, de León M (2000) The geometrical theory of constraints applied to the dynamics of vakonomic mechanical systems: The vakonomic bracket. Journal of Mathematical Physics 41(4):2090–2120. https://doi.org/10.1063/1.533229 – 10.1063/1.533229
  • McLachlan R. I., Foundations Comput. Math. (2013)
  • McLachlan R, Perlmutter M (2006) Integrators for Nonholonomic Mechanical Systems. J Nonlinear Sci 16(4):283–328. https://doi.org/10.1007/s00332-005-0698-1 – 10.1007/s00332-005-0698-1
  • Modin K, Verdier O (2014) Integrability of nonholonomically coupled oscillators. DCDS 34(3):1121–1130. https://doi.org/10.3934/dcds.2014.34.1121 – 10.3934/dcds.2014.34.1121
  • Montgomery R., Math. Surveys Monogr. 91 (2002)
  • Leimkuhler BJ, Skeel RD (1994) Symplectic Numerical Integrators in Constrained Hamiltonian Systems. Journal of Computational Physics 112(1):117–125. https://doi.org/10.1006/jcph.1994.1085 – 10.1006/jcph.1994.1085
  • Vershik A. M., Dynamical Systems VII, Encyclopaedia Math. Sci. 16 (1994)
  • Viterbo C (1995) Generating Functions, Symplectic Geometry, and Applications. In: Proceedings of the International Congress of Mathematicians. Birkhäuser Basel, Basel, pp 537–547 – 10.1007/978-3-0348-9078-6_47