Variational integrators in discrete vakonomic mechanics

  1. García, Pedro L. 3
  2. Fernández, Antonio 1
  3. Rodrigo, César 2
  1. 1 Department of Applied Mathematics, University of Salamanca,
  2. 2 CINAMIL, Academia Militar, Amadora, Portugal
  3. 3 Department of Mathematics, University of Salamanca
Revista:
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas

ISSN: 1578-7303 1579-1505

Año de publicación: 2012

Volumen: 106

Número: 1

Páginas: 137-159

Tipo: Artículo

DOI: 10.1007/S13398-011-0030-X SCOPUS: 2-s2.0-84856860956 DIALNET GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas

Resumen

We introduce the discrete counterpart of the vakonomic method in Lagrangian mechanics with non-holonomic constraints. After defining the concepts of “admissible section” and “admissible infinitesimal variation” of a discrete vakonomic system, we aim to determinate those admissible sections that are critical for the Lagrangian of the system with respect to admissible infinitesimal variations. For sections that satisfy a certain regularity condition, we prove that critical sections are extremals of a variational problem without constraints canonically associated to the initial system (Lagrange multiplier rule). We introduce a notion of “constrained variational integrator”, which is characterized by a Cartan equation that ensures its simplecticity. Moreover, under certain regularity conditions we prove that these integrators can be locally constructed from a generating function of the second kind in the sense of symplectic geometry. Finally, the theory is illustrated with two elementary examples: an isoperimetric problem and an optimal control problem

Referencias bibliográficas

  • Arnol’d V.I., Kozlov V.V., Neĭshtadt A.I.: Dynamical Systems III. Encyclopaedia of Mathematical Sciences, vol. 3. Springer, Berlin (1988)
  • Benito R., Martín de Diego D.: Discrete vakonomic mechanics. J. Math. Phys. 46(8), 083521 (2005)
  • Bloch, A.M.: Nonholonomic Mechanics and Control, Springer Science Ed. (2003)
  • Cardin F., Favretti M.: On nonholonomic and vakonomic dynamics of mechanical systems with nonintegrable constraints. J. Geom. Phys. 18(4), 295–325 (1996)
  • Chen, J.-B., Guo, H.-Y., Wu, K.: Total variation and variational symplectic-energy-momentum integrators, preprint, hep-th/0109178 (2001)
  • Chen J.-B., Guo H.-Y., Wu K.: Discrete total variation calculus and Lee’s discrete mechanics. Appl. Math. Comput. 177(1), 226–234 (2006)
  • Cortés, J.: Geometric, Control and Numerical Aspects of Nonholonomic Systems. Lecture Notes in Mathematics, vol. 1793, Springer, Berlin (2002)
  • García, P.L., Rodrigo, C.: Cartan forms and second variation for constrained variational problems, Proceedings of the VII International Conference on Geometry, Integrability and Quantization (Varna, Bulgary), pp. 140–153. 7 Bulgarian Acad. Sci., Sofia (2006)
  • García P.L., García A., Rodrigo C.: Cartan forms for first order constrained variational problems. J. Geom. Phys. 56(4), 571–610 (2006)
  • Goldstein, H.: Classical mechanics, 2nd edn. In: Addison-Wesley Series in Physics. Addison-Wesley, Reading (1980)
  • Gràcia X., Marín Solano J., Muñoz Lecanda M.C.: Some geometric aspects of variational calculus in constrained systems. Rep. Math. Phys. 51(1), 127–148 (2003)
  • Guibout, V.M., Bloch, A.: Discrete variational principles and Hamilton-Jacobi theory for mechanical systems and optimal control problems e-print ccsd-00002863, version1-2004
  • Hsu L.: Calculus of variations via the Griffiths formalism. J. Differ. Geom. 36, 551–589 (1992)
  • Lee T.D.: Can time be a discrete dynamical variable?. Phys. Lett. B 122, 217–220 (1983)
  • de León M., Martínde Diego D., Santamaría Merino A.: Geometric integrators and nonholonomic mechanics. J. Math. Phys. 45(3), 1042–1064 (2004)
  • de León M., Martín de Diego D., Santamaría Merino A.: Discrete variational integrators and optimal control theory. Adv. Comput. Math. 26(1–3), 251–268 (2006)
  • de León M., Marrero J.C., Martín de Diego D.: Vakonomic mechanics versus non-holonomic mechanics: A unified geometrical approach. J. Geom. Phys. 35, 126–144 (2000)
  • Marsden J.E., Patrick G.W., Shkoller S.: Multisymplectic geometry, variational integrators and nonlinear PDEs. Commun. Math. Phys. 199(2), 351–398 (1998)
  • Marsden J.E., West M.: Discrete mechanics and variational integrators. Acta Numer. 10, 317–514 (2001)
  • Martínez S., Cortés J., de León M.: Symmetries in vakonomic dynamics: applications to optimal control. J. Geom. Phys. 38, 343–365 (2001)
  • Piccione P., Tausk D.V.: Lagrangian and Hamiltonian formalism for constrained variational problems. Proc. Roy. Soc. Edinb. Sect. A 132(6), 1417–1437 (2002)
  • West, M.: Variational Integrators Ph.D. thesis, California Institute of Technology, Pasadena (2004)
  • Vankerschaver J., Cantrijn F., de León M., Martínde Diego D.: Geometric aspects of nonholonomic field theories. Rep. Math. Phys 56(3), 387–411 (2005)