Skip to main content
SpringerLink
Log in

Variational and Geometric Structures of Discrete Dirac Mechanics

  • Published:
Foundations of Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we develop the theoretical foundations of discrete Dirac mechanics, that is, discrete mechanics of degenerate Lagrangian/Hamiltonian systems with constraints. We first construct discrete analogues of Tulczyjew’s triple and induced Dirac structures by considering the geometry of symplectic maps and their associated generating functions. We demonstrate that this framework provides a means of deriving discrete Lagrange–Dirac and nonholonomic Hamiltonian systems. In particular, this yields nonholonomic Lagrangian and Hamiltonian integrators. We also introduce discrete Lagrange–d’Alembert–Pontryagin and Hamilton–d’Alembert variational principles, which provide an alternative derivation of the same set of integration algorithms. The paper provides a unified treatment of discrete Lagrangian and Hamiltonian mechanics in the more general setting of discrete Dirac mechanics, as well as a generalization of symplectic and Poisson integrators to the broader category of Dirac integrators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Abraham, J.E. Marsden, Foundations of Mechanics, 2nd edn. (Addison–Wesley, Reading 1978).

    MATH  Google Scholar 

  2. P.-A. Absil, R. Mahony, R. Sepulchre, Optimization Algorithms on Matrix Manifolds (Princeton University Press, Princeton, 2008).

    MATH  Google Scholar 

  3. V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, Berlin, 1989).

    Google Scholar 

  4. L. Bates, J. Sniatycki, Nonholonomic reduction, Rep. Math. Phys. 32(1), 99–115 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Benettin, A.M. Cherubini, F. Fassò, A changing-chart symplectic algorithm for rigid bodies and other Hamiltonian systems on manifolds, SIAM J. Sci. Comput. 23(4), 1189–1203 (2001). ISSN 1064-8275.

    Article  MathSciNet  MATH  Google Scholar 

  6. A.M. Bloch, Nonholonomic Mechanics and Control (Springer, Berlin, 2003).

    Book  MATH  Google Scholar 

  7. A.M. Bloch, P.E. Crouch, Representations of Dirac structures on vector spaces and nonlinear L-C circuits, in Differential Geometry and Control Theory (American Mathematical Society, Providence, 1997), pp. 103–117.

    Google Scholar 

  8. N. Bou-Rabee, J.E. Marsden, Hamilton–Pontryagin integrators on Lie groups part I: introduction and structure-preserving properties, Found. Comput. Math. (2008).

  9. J.F. Cariñena, X. Gracia, G. Marmo, E. Martínez, M.C. Munõz Lecanda, N. Román-Roy, Geometric Hamilton–Jacobi theory for nonholonomic dynamical systems, Int. J. Geom. Methods Mod. Phys. 7(3), 431–454 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Cervera, A.J. van der Schaft, A. Baños, On composition of Dirac structures and its implications for control by interconnection, in Nonlinear and Adaptive Control. Lecture Notes in Control and Inform. Sci., vol. 281 (Springer, Berlin, 2003), pp. 55–63.

    Chapter  Google Scholar 

  11. J. Cortés, S. Martínez, Non-holonomic integrators, Nonlinearity 14(5), 1365–1392 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  12. T. Courant, Dirac manifolds, Trans. Am. Math. Soc. 319(2), 631–661 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  13. T. Courant, Tangent Dirac structures, J. Phys. A, Math. Gen. 23(22), 5153–5168 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Dalsmo, A.J. van der Schaft, On representations and integrability of mathematical structures in energy-conserving physical systems, SIAM J. Control Optim. 37(1), 54–91 (1998).

    Article  Google Scholar 

  15. M. de León, J.C. Marrero, D. Martín de Diego, Linear almost Poisson structures and Hamilton–Jacobi equation. Applications to nonholonomic mechanics, J. Geom. Mech. 2(2), 159–198 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  16. P.A.M. Dirac, Generalized Hamiltonian dynamics, Can. J. Math. 2, 129–148 (1950).

    Article  MathSciNet  MATH  Google Scholar 

  17. P.A.M. Dirac, Generalized Hamiltonian dynamics, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 246(1246), 326–332 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  18. P.A.M. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva University, New York, 1964).

    Google Scholar 

  19. H. Goldstein, C.P. Poole, J.L. Safko, Classical Mechanics, 3rd edn. (Addison–Wesley, Reading, 2001).

    Google Scholar 

  20. M.J. Gotay, J.M. Nester, Presymplectic Hamilton and Lagrange systems, gauge transformations and the Dirac theory of constraints, in Group Theoretical Methods in Physics, vol. 94 (Springer, Berlin, 1979), pp. 272–279.

    Chapter  Google Scholar 

  21. M.J. Gotay, J.M. Nester, Presymplectic Lagrangian systems. I: the constraint algorithm and the equivalence theorem, Ann. Inst. Henri Poincaré A 30(2), 129–142 (1979).

    MathSciNet  MATH  Google Scholar 

  22. M.J. Gotay, J.M. Nester, Presymplectic Lagrangian systems. II: the second-order equation problem, Ann. Inst. Henri Poincaré A 32(1), 1–13 (1980).

    MathSciNet  MATH  Google Scholar 

  23. E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations (Springer, Berlin, 2006).

    MATH  Google Scholar 

  24. D. Iglesias, J.C. Marrero, D.M. de Diego, E. Martínez, Discrete nonholonomic Lagrangian systems on Lie groupoids, J. Nonlinear Sci. 18(3), 221–276 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Iglesias-Ponte, M. de León, D.M. de Diego, Towards a Hamilton–Jacobi theory for nonholonomic mechanical systems. Journal of Physics A: Mathematical and Theoretical 41(1) (2008).

  26. L. Kharevych, W. Yang, Y. Tong, E. Kanso, J.E. Marsden, P. Schröder, M. Desbrun, Geometric, variational integrators for computer animation, in ACM/EG Symposium on Computer Animation (2006), pp. 43–51.

    Google Scholar 

  27. W.S. Koon, J.E. Marsden, The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems, Rep. Math. Phys. 40(1), 21–62 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  28. H.P. Künzle, Degenerate Lagrangean systems, Ann. Inst. Henri Poincaré A 11(4), 393–414 (1969).

    MATH  Google Scholar 

  29. S. Lall, M. West, Discrete variational Hamiltonian mechanics, J. Phys. A, Math. Gen. 39(19), 5509–5519 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  30. B. Leimkuhler, S. Reich, Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol. 14 (Cambridge University Press, Cambridge, 2004).

    MATH  Google Scholar 

  31. M. Leok, T. Ohsawa, Discrete Dirac structures and implicit discrete Lagrangian and Hamiltonian systems, in XVIII International Fall Workshop on Geometry and Physics, vol. 1260 (AIP, New York, 2010), pp. 91–102.

    Google Scholar 

  32. M. Leok, T. Ohsawa, D. Sosa, Hamilton–Jacobi theory for degenerate Lagrangian systems with constraints (in preparation).

  33. A. Lew, J.E. Marsden, M. Ortiz, M. West, An overview of variational integrators, in Finite Element Methods: 1970’s and Beyond (CIMNE 2003).

  34. J.C. Marrero, D. Martín de Diego, E. Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity 19(6), 1313–1348 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  35. J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry (Springer, Berlin, 1999).

    MATH  Google Scholar 

  36. J.E. Marsden, M. West, Discrete mechanics and variational integrators, in Acta Numerica (2001), pp. 357–514.

    Google Scholar 

  37. J.E. Marsden, S. Pekarsky, S. Shkoller, Discrete Euler–Poincaré and Lie–Poisson equations, Nonlinearity 12(6), 1647–1662 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  38. R. McLachlan, M. Perlmutter, Integrators for nonholonomic mechanical systems, J. Nonlinear Sci. 16(4), 283–328 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  39. T. Ohsawa, A.M. Bloch, Nonholonomic Hamilton–Jacobi equation and integrability, J. Geom. Mech. 1(4), 461–481 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  40. T. Ohsawa, O.E. Fernandez, A.M. Bloch, D.V. Zenkov, Nonholonomic Hamilton–Jacobi theory via Chaplygin Hamiltonization, J. Geom. Phys. 61(8), 1263–1291 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Stern, Discrete Hamilton–Pontryagin mechanics and generating functions on Lie groupoids, J. Symplectic Geom. 8(2), 225–238 (2010).

    MathSciNet  MATH  Google Scholar 

  42. W.M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sci. Paris 283, 15–18 (1976).

    MathSciNet  MATH  Google Scholar 

  43. W.M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris 283, 675–678 (1976).

    MathSciNet  MATH  Google Scholar 

  44. A.J. van der Schaft, Implicit Hamiltonian systems with symmetry, Rep. Math. Phys. 41(2), 203–221 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  45. A.J. van der Schaft, Port-Hamiltonian systems: an introductory survey, in Proceedings of the International Congress of Mathematicians, vol. 3 (2006), pp. 1339–1365.

    Google Scholar 

  46. A.J. van der Schaft, B.M. Maschke, On the Hamiltonian formulation of nonholonomic mechanical systems, Rep. Math. Phys. 34(2), 225–233 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  47. J. Vankerschaver, H. Yoshimura, J.E. Marsden, Multi-Dirac structures and Hamilton–Pontryagin principles for Lagrange–Dirac field theories. Preprint, arXiv:1008.0252 (2010).

  48. V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations (Springer, New York, 1984).

    MATH  Google Scholar 

  49. A. Weinstein, Lagrangian mechanics and groupoids, Fields Inst. Commun. 7, 207–231 (1996).

    Google Scholar 

  50. H. Yoshimura, J.E. Marsden, Dirac structures in Lagrangian mechanics Part I: implicit Lagrangian systems, J. Geom. Phys. 57(1), 133–156 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  51. H. Yoshimura, J.E. Marsden, Dirac structures in Lagrangian mechanics Part II: variational structures, J. Geom. Phys. 57(1), 209–250 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  52. H. Yoshimura, J.E. Marsden, Reduction of Dirac structures and the Hamilton–Pontryagin principle, Rep. Math. Phys. 60(3), 381–426 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  53. H. Yoshimura, J.E. Marsden, Dirac structures and the Legendre transformation for implicit Lagrangian and Hamiltonian systems, in Lagrangian and Hamiltonian Methods for Nonlinear Control 2006 (2007), pp. 233–247.

    Chapter  Google Scholar 

  54. H. Yoshimura, J.E. Marsden, Dirac cotangent bundle reduction, J. Geom. Mech. 1(1), 87–158 (2009).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA, USA

    Melvin Leok & Tomoki Ohsawa

Authors
  1. Melvin Leok
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tomoki Ohsawa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Melvin Leok.

Additional information

Communicated by Arieh Iserles.

Dedicated to the memory of Jerrold E. Marsden.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Leok, M., Ohsawa, T. Variational and Geometric Structures of Discrete Dirac Mechanics. Found Comput Math 11, 529–562 (2011). https://doi.org/10.1007/s10208-011-9096-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10208-011-9096-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation