Authors

Hiroaki Yoshimura, Jerrold E. Marsden

Abstract

This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space Q makes use of Dirac structures on T ∗ Q that are induced from a constraint distribution on Q as well as natural symplectomorphisms between the spaces T ∗ T Q , T T ∗ Q , and T ∗ T ∗ Q . Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane, and the second is an L–C circuit, a degenerate Lagrangian system with holonomic constraints.

Keywords

Dirac structures; Implicit Lagrangian systems; Nonholonomic systems

Citation

BibTeX

@article{Yoshimura_2006,
  title={{Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems}},
  volume={57},
  ISSN={0393-0440},
  DOI={10.1016/j.geomphys.2006.02.009},
  number={1},
  journal={Journal of Geometry and Physics},
  publisher={Elsevier BV},
  author={Yoshimura, Hiroaki and Marsden, Jerrold E.},
  year={2006},
  pages={133--156}
}

Download the bib file

References

  • Abraham, (1978)
  • Bates, L. & Śniatycki, J. Nonholonomic reduction. Reports on Mathematical Physics vol. 32 99–115 (1993) – 10.1016/0034-4877(93)90073-n
  • Birkhoff, G. Dynamical Systems. Colloquium Publications (1927) doi:10.1090/coll/009 – 10.1090/coll/009
  • Blankenstein, G. & van der Schaft, A. J. Reduction of implicit hamiltonian systems with symmetry. 1999 European Control Conference (ECC) 563–568 (1999) doi:10.23919/ecc.1999.7099364 – 10.23919/ecc.1999.7099364
  • Blankenstein, G. & van der Schaft, A. J. Symmetry and reduction in implicit generalized Hamiltonian systems. Reports on Mathematical Physics vol. 47 57–100 (2001) – 10.1016/s0034-4877(01)90006-0
  • Blankenstein, G. & Ratiu, T. S. Singular reduction of implicit Hamiltonian systems. Reports on Mathematical Physics vol. 53 211–260 (2004) – 10.1016/s0034-4877(04)90013-4
  • Bloch, Representations of Dirac structures on vector spaces and nonlinear L–C circuits. (1997)
  • Bloch, (2003)
  • Bloch, A. M., Krishnaprasad, P. S., Marsden, J. E. & Murray, R. M. Nonholonomic mechanical systems with symmetry. Archive for Rational Mechanics and Analysis vol. 136 21–99 (1996) – 10.1007/bf02199365
  • Brayton, R. K. & Moser, J. K. A theory of nonlinear networks. I. Quarterly of Applied Mathematics vol. 22 1–33 (1964) – 10.1090/qam/169746
  • Brayton, Nonlinear reciprocal networks. (1971)
  • Cendra, H., Holm, D. D., Hoyle, M. J. W. & Marsden, J. E. The Maxwell–Vlasov equations in Euler–Poincaré form. Journal of Mathematical Physics vol. 39 3138–3157 (1998) – 10.1063/1.532244
  • Cendra, (2001)
  • Chang, The equivalence of controlled Lagrangian and controlled Hamiltonian systems. Control Calc. Var. (2002)
  • Chang, D. E. & Marsden, J. E. Reduction of Controlled Lagrangian and Hamiltonian Systems with Symmetry. SIAM Journal on Control and Optimization vol. 43 277–300 (2004) – 10.1137/s0363012902412951
  • Chen, T. Critical manifolds and stability in Hamiltonian systems with non-holonomic constraints. Journal of Geometry and Physics vol. 49 418–462 (2004) – 10.1016/j.geomphys.2003.08.004
  • Chua, (1987)
  • Cortés, J., de León, M., de Diego, D. M. & Martínez, S. Geometric Description of Vakonomic and Nonholonomic Dynamics. Comparison of Solutions. SIAM Journal on Control and Optimization vol. 41 1389–1412 (2002) – 10.1137/s036301290036817x
  • Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
  • Courant, T. Tangent Dirac structures. Journal of Physics A: Mathematical and General vol. 23 5153–5168 (1990) – 10.1088/0305-4470/23/22/010
  • Courant, Beyond Poisson structures. (1988)
  • Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM Journal on Control and Optimization vol. 37 54–91 (1998)10.1137/s0363012996312039
  • Dirac, P. A. M. Generalized Hamiltonian Dynamics. Canadian Journal of Mathematics vol. 2 129–148 (1950) – 10.4153/cjm-1950-012-1
  • Dorfman, I. Ya. Dirac structures of integrable evolution equations. Physics Letters A vol. 125 240–246 (1987)10.1016/0375-9601(87)90201-5
  • Duinker, Traditors, a new class of non-energic non-linear network elements. Philips Res. Rep. (1959)
  • Duinker, Conjunctors, another new class of non-energic non-linear network elements. Philips Res. Rep. (1962)
  • Koiller, J. Reduction of some classical non-holonomic systems with symmetry. Archive for Rational Mechanics and Analysis vol. 118 113–148 (1992) – 10.1007/bf00375092
  • Koon, W. S. & Marsden, J. E. The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems. Reports on Mathematical Physics vol. 40 21–62 (1997) – 10.1016/s0034-4877(97)85617-0
  • Wang Sang Koon & Marsden, J. E. Poisson reduction for nonholonomic mechanical systems with symmetry. Reports on Mathematical Physics vol. 42 101–134 (1998) – 10.1016/s0034-4877(98)80007-4
  • Kron, (1939)
  • Kron, (1963)
  • Marsden, (1999)
  • Maschke, B. M., van der Schaft, A. J. & Breedveld, P. C. An intrinsic Hamiltonian formulation of the dynamics of LC-circuits. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications vol. 42 73–82 (1995) – 10.1109/81.372847
  • Moreau, L. & Aeyels, D. A Novel Variational Method for Deriving Lagrangian and Hamiltonian Models of Inductor-Capacitor Circuits. SIAM Review vol. 46 59–84 (2004) – 10.1137/s0036144502409020
  • Oster, G. F. & Perelson, A. S. Chemical reaction dynamics. Archive for Rational Mechanics and Analysis vol. 55 230–274 (1974) – 10.1007/bf00281751
  • Oliva, W. M. Lagrangian systems on manifolds, I. Celestial Mechanics vol. 1 491–511 (1970) – 10.1007/bf01231146
  • Perelson, A. S. & Oster, G. F. Chemical reaction dynamics part II: Reaction networks. Archive for Rational Mechanics and Analysis vol. 57 31–98 (1974) – 10.1007/bf00287096
  • Rowley, Variational integrators for point vortices. Proc. CDC (2002)
  • Sastry, S. & Desoer, C. Jump behavior of circuits and systems. IEEE Transactions on Circuits and Systems vol. 28 1109–1124 (1981) – 10.1109/tcs.1981.1084943
  • Skinner, R. & Rusk, R. Generalized Hamiltonian dynamics. I. Formulation on T*Q⊕ Q. Journal of Mathematical Physics vol. 24 2589–2594 (1983) – 10.1063/1.525654
  • Smale, S. On the mathematical foundations of electrical circuit theory. Journal of Differential Geometry vol. 7 (1972) – 10.4310/jdg/1214430827
  • Tulczyjew, The Legendre transformation. Ann. Inst. H. Poincaré A (1977)
  • Van Der Schaft, A. J. & Maschke, B. M. On the Hamiltonian formulation of nonholonomic mechanical systems. Reports on Mathematical Physics vol. 34 225–233 (1994) – 10.1016/0034-4877(94)90038-8
  • van der Schaft, The Hamiltonian formulation of energy conserving physical systems with external ports. Arch. Elektr. Übertrag. (1995)
  • van der Schaft, A. J. Implicit Hamiltonian systems with symmetry. Reports on Mathematical Physics vol. 41 203–221 (1998) – 10.1016/s0034-4877(98)80176-6
  • Vershik, Lagrangian mechanics in invariant form. Sel. Math. Sov. (1981)
  • Weber, R. W. Hamiltonian systems with constraints and their meaning in mechanics. Archive for Rational Mechanics and Analysis vol. 91 309–335 (1986) – 10.1007/bf00282337
  • Wyatt, J. L. & Chua, L. O. A theory of nonenergic N‐ports. International Journal of Circuit Theory and Applications vol. 5 181–208 (1977) – 10.1002/cta.4490050210