Advertisement

Journal of Nonlinear Science

, Volume 27, Issue 5, pp 1399–1434 | Cite as

Variational Integrators for Interconnected Lagrange–Dirac Systems

  • Helen Parks
  • Melvin Leok
Article

Abstract

Interconnected systems are an important class of mathematical models, as they allow for the construction of complex, hierarchical, multiphysics, and multiscale models by the interconnection of simpler subsystems. Lagrange–Dirac mechanical systems provide a broad category of mathematical models that are closed under interconnection, and in this paper, we develop a framework for the interconnection of discrete Lagrange–Dirac mechanical systems, with a view toward constructing geometric structure-preserving discretizations of interconnected systems. This work builds on previous work on the interconnection of continuous Lagrange–Dirac systems (Jacobs and Yoshimura in J Geom Mech 6(1):67–98, 2014) and discrete Dirac variational integrators (Leok and Ohsawa in Found Comput Math 11(5), 529–562, 2011). We test our results by simulating some of the continuous examples given in Jacobs and Yoshimura (2014).

Keywords

Interconnection Dirac structures Lagrange–Dirac systems Variational integrators Geometric integration Hamiltonian DAEs 

Mathematics Subject Classification

37J05 37J60 37N05 65P10 70F25 70G45 70H05 70H45 70Q05 93A30 93B27 

Notes

Acknowledgements

We gratefully acknowledge helpful comments and suggestions of the referee. HP has been supported by the NSF Graduate Research Fellowship Grant Number DGE-1144086. ML has been supported in part by NSF under Grants DMS-1010687, CMMI-1029445, DMS-1065972, CMMI-1334759, DMS-1411792, DMS-1345013.

References

  1. Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008)CrossRefzbMATHGoogle Scholar
  2. Bloch, A.M.: Nonholonomic Mechanics and Control. Springer, Berlin (2003)CrossRefGoogle Scholar
  3. Bou-Rabee, N., Marsden, J.E.: Hamilton–Pontryagin integrators on Lie groups. I. Introduction and structure-preserving properties. Found. Comput. Math. 9(2), 197–219 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bou-Rabee, N., Owhadi, H.: Stochastic variational integrators. IMA J. Numer. Anal. 29(2), 421–443 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bou-Rabee, N., Owhadi, H.: Long-run accuracy of variational integrators in the stochastic context. SIAM J. Numer. Anal. 48(1), 278–297 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Chang, D.-E.: On the method of interconnection and damping assignment passivity-based control for the stabilization of mechanical systems. Regul. Chaotic Dyn. 19(5), 556–575 (2014). doi: 10.1134/S1560354714050049 MathSciNetCrossRefzbMATHGoogle Scholar
  7. Cortés, J., Martínez, S.: Non-holonomic integrators. Nonlinearity 14(5), 1365–1392 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Duindam, V., Macchelli, A., Stramigioli, S., Bruyninckx, H.: Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
  9. Fedorov, Y.N., Zenkov, D.V.: Discrete nonholonomic LL systems on Lie groups. Nonlinearity 18(5), 2211–2241 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Fetecau, R., Marsden, J.E., Ortiz, M., West, M.: Nonsmooth Lagrangian mechanics and variational collision integrators. SIAM J. Appl. Dyn. Syst. 2(3), 381–416 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Gualtieri, M.: Generalized complex geometry. Ann. Math. 174, 75–123 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Jacobs, H.O., Yoshimura, H.: Tensor products of Dirac structures and interconnection in Lagrangian mechanics. J. Geom. Mech. 6(1), 67–98 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Lall, S., West, M.: Discrete variational Hamiltonian mechanics. J. Phys. A 39(19), 5509–5519 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Lee, T., Leok, M., McClamroch, N.H.: Lie group variational integrators for the full body problem. Comput. Methods Appl. Mech. Eng. 196(29–30), 2907–2924 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Lee, T., Leok, M., McClamroch, N.H.: Lagrangian mechanics and variational integrators on two-spheres. Int. J. Numer. Methods Eng. 79(9), 1147–1174 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Leok, M., Ohsawa, T.: Variational and geometric structures of discrete Dirac mechanics. Found. Comput. Math. 11(5), 529–562 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Leok, M., Zhang, J.: Discrete Hamiltonian variational integrators. IMA J. Numer. Anal. 31(4), 1497–1532 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. Lew, A., Marsden, J.E., Ortiz, M., West, M.: Asynchronous variational integrators. Arch. Ration. Mech. Anal. 167(2), 85–146 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Marsden, J.E., Patrick, G.W., Shkoller, S.: Multisymplectic geometry, variational integrators, and nonlinear PDEs. Commun. Math. Phys. 199(2), 351–395 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry, 2nd edn. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  21. Marsden, J.E., West, M.: Discrete mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. McLachlan, R., Perlmutter, M.: Integrators for nonholonomics mechanical systems. J. Nonlinear Sci. 16(4), 283–328 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Parks, Helen: Structured approaches to large-scale systems: variational integrators for interconnected Lagrange–Dirac systems and structured model reduction on Lie groups. PhD thesis, University of California San Diego (2015)Google Scholar
  24. Tulczyjew, W.M.: The Legendre transformation. Annales de l’Institute Henri Poincaré 27, 101–114 (1977)MathSciNetzbMATHGoogle Scholar
  25. van der Schaft, A.: Port-Hamiltonian systems: an introductory survey. In: Proceedings of the International Congress on Mathematicians, Madrid, Spain (2006)Google Scholar
  26. Yoshimura, H., Marsden, J.E.: Dirac structures in Lagrangian mechanics part I: implicit Lagrangian systems. J. Geom. Phys. 57(1), 133–156 (2006a)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Yoshimura, H., Marsden, J.E.: Dirac structures in Lagrangian mechanics part II: variational structures. J. Geom. Phys. 57(1), 209–250 (2006b)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, San DiegoLa JollaUSA

Personalised recommendations