Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-04-30T16:53:03.276Z Has data issue: false hasContentIssue false
  • English
  • Français

Variational calculus on Lie algebroids

Published online by Cambridge University Press:  20 March 2008

Eduardo Martínez
Eduardo Martínez*
Affiliation:
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain; emf@unizar.es
Get access

Abstract

It is shown that the Lagrange's equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of Lagrangian reduction and the relation with the method of Lagrange multipliers are also studied.

Keywords

Variational calculusLagrangian mechanicsLie algebroidsreduction of dynamical systemsEuler-Poincaré equationsLagrange-Poincaré equations
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 2 , April 2008 , pp. 356 - 380
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

R. Abraham, J.E. Marsden and T.S. Ratiu Manifolds, tensor analysis and applications Addison-Wesley, (1983)
C. Altafini Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric ESAIM: COCV 10 (2004) 526–548
V.I. Arnold Dynamical Systems III Springer-Verlag (1988)
A. Cannas da Silva and A. Weinstein Geometric models for noncommutative algebras Amer. Math. Soc., Providence, RI (1999) xiv + 184 pp
J.F. Cariñena and E. Martínez Lie algebroid generalization of geometric mechanics in Lie Algebroids and related topics in differential geometry (Warsaw 2000), Banach Center Publications 54 (2001) 201
H. Cendra, A. Ibort and J.E. Marsden Variational principal fiber bundles: a geometric theory of Clebsch potentials and Lin constraints J. Geom. Phys 4 (1987) 183–206
H. Cendra, J.E. Marsden and T.S. Ratiu Lagrangian reduction by stages Mem. Amer. Math. Soc 152 (2001) x + 108 pp
H. Cendra, J.E. Marsden, S. Pekarsky and T.S. Ratiu Variational principles for Lie-Poisson and Hamilton-Poincaré equations Moscow Math. J 3 (2003) 833–867
J. Cortés, M. de León, J.C. Marrero and E. Martínez Nonholonomic Lagrangian systems on Lie algebroids Preprint 2005, arXiv:math-ph/0512003
J. Cortés, M. de León, J.C. Marrero, D. Martín de Diego and E. Martínez A survey of Lagrangian mechanics and control on Lie algebroids and groupoids Int. J. Geom. Meth. Math. Phys 3 (2006) 509–558
M. Crainic and R.L. Fernandes Integrability of Lie brackets Ann. Math 157 (2003) 575–620
M. Crampin Tangent bundle geometry for Lagrangian dynamics J. Phys. A: Math. Gen 16 (1983) 3755–3772
M. de León, J.C. Marrero and E. Martínez Lagrangian submanifolds and dynamics on Lie algebroids J. Phys. A: Math. Gen 38 (2005) R241–R308
K. Grabowska, J. Grabowski and P. Urbanski Geometrical Mechanics on algebroids Int. Jour. Geom. Meth. Math. Phys 3 (2006) 559–576
D.D. Holm, J.E. Marsden and T.S. Ratiu The Euler-Poincaré equations and semidirect products with applications to continuum theories Adv. Math 137 (1998) 1–81
J. Klein Espaces variationnels et mécanique Ann. Inst. Fourier 12 (1962) 1–124
S. Lang Differential manifolds Springer-Verlag, New-York (1972)
C. López Variational calculus, symmetries and reduction Int. J. Geom. Meth. Math. Phys 3 (2006) 577–590
K.C.H. Mackenzie General Theory of Lie Groupoids and Lie Algebroids Cambridge University Press (2005)
J.E. Marsden and T.S. Ratiu Introduction to Mechanics and symmetry Springer-Verlag, 1999
E. Martínez Lagrangian Mechanics on Lie algebroids Acta Appl. Math 67 (2001) 295–320
E. Martínez Geometric formulation of Mechanics on Lie algebroids, in Proceedings of the VIII Fall Workshop on Geometry and Physics, Medina del Campo, 1999, Publicaciones de la RSME 2 (2001) 209–222
E. Martínez Reduction in optimal control theory Rep. Math. Phys 53 (2004) 79–90
E. Martínez Classical field theory on Lie algebroids: Multisymplectic formalism Preprint 2004, arXiv:math.DG/0411352
E. Martínez Classical Field Theory on Lie algebroids: Variational aspects J. Phys. A: Mat. Gen 38 (2005) 7145–7160
E. Martínez, T. Mestdag and W. Sarlet Lie algebroid structures and Lagrangian systems on affine bundles J. Geom. Phys 44 (2002) 70–95
P. Michor Topics in differential geometry Book on the internet. http://www.mat.univie.ac.at/~michor/dgbook.pdf
J.P. Ortega and T.S. Ratiu Momentum maps and Hamiltonian Reduction Birkhäuser (2004)
P. Piccione and D. Tausk Lagrangian and Hamiltonian formalism for constrained variational problems Proc. Roy. Soc.Edinburgh Sect. A 132 (2002) 1417–1437
W. Sarlet, T. Mestdag and E. Martínez Lagrangian equations on affine Lie algebroids Differential Geometry and its Applications, in Proc. 8th Int. Conf. (Opava 2001), D. Krupka et al Eds
A. Weinstein Lagrangian Mechanics and groupoids Fields Inst. Comm 7 (1996) 207–231