Variational calculus on Lie algebroids
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 2, pp. 356-380.

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.

DOI : 10.1051/cocv:2007056
Classification : 49S05, 49K15, 58D15, 70H25, 17B66, 22A22
Mots-clés : variational calculus, lagrangian mechanics, Lie algebroids, reduction of dynamical systems, Euler-Poincaré equations, Lagrange-Poincaré equations 
@article{COCV_2008__14_2_356_0,
     author = {Mart{\'\i}nez, Eduardo},
     title = {Variational calculus on {Lie} algebroids},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {356--380},
     publisher = {EDP-Sciences},
     volume = {14},
     number = {2},
     year = {2008},
     doi = {10.1051/cocv:2007056},
     zbl = {1135.49027},
     mrnumber = {2394515},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2007056/}
}
TY  - JOUR
AU  - Martínez, Eduardo
TI  - Variational calculus on Lie algebroids
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2008
SP  - 356
EP  - 380
VL  - 14
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2007056/
DO  - 10.1051/cocv:2007056
LA  - en
ID  - COCV_2008__14_2_356_0
ER  - 
%0 Journal Article
%A Martínez, Eduardo
%T Variational calculus on Lie algebroids
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2008
%P 356-380
%V 14
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2007056/
%R 10.1051/cocv:2007056
%G en
%F COCV_2008__14_2_356_0
Martínez, Eduardo. Variational calculus on Lie algebroids. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 2, pp. 356-380. doi : 10.1051/cocv:2007056. http://www.numdam.org/articles/10.1051/cocv:2007056/

[1] R. Abraham, J.E. Marsden and T.S. Ratiu Manifolds, tensor analysis and applications Addison-Wesley, (1983) | MR | Zbl

[2] 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 | Numdam | MR | Zbl

[3] V.I. Arnold Dynamical Systems III Springer-Verlag (1988) | MR | Zbl

[4] A. Cannas Da Silva and A. Weinstein Geometric models for noncommutative algebras Amer. Math. Soc., Providence, RI (1999) xiv + 184 pp | MR | Zbl

[5] 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 | MR | Zbl

[6] 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 | Zbl

[7] H. Cendra, J.E. Marsden and T.S. Ratiu Lagrangian reduction by stages Mem. Amer. Math. Soc 152 (2001) x + 108 pp | MR

[8] 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 | MR | Zbl

[9] J. Cortés, M. De León, J.C. Marrero and E. Martínez Nonholonomic Lagrangian systems on Lie algebroids Preprint 2005, arXiv:math-ph/0512003

[10] 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 | MR | Zbl

[11] M. Crainic and R.L. Fernandes Integrability of Lie brackets Ann. Math 157 (2003) 575-620 | MR | Zbl

[12] M. Crampin Tangent bundle geometry for Lagrangian dynamics J. Phys. A: Math. Gen 16 (1983) 3755-3772 | MR | Zbl

[13] 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 | MR

[14] K. Grabowska, J. Grabowski and P. Urbanski Geometrical Mechanics on algebroids Int. Jour. Geom. Meth. Math. Phys 3 (2006) 559-576 | MR

[15] 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 | MR | Zbl

[16] J. Klein Espaces variationnels et mécanique Ann. Inst. Fourier 12 (1962) 1-124 | Numdam | MR | Zbl

[17] S. Lang Differential manifolds Springer-Verlag, New-York (1972) | MR | Zbl

[18] C. López Variational calculus, symmetries and reduction Int. J. Geom. Meth. Math. Phys 3 (2006) 577-590 | MR | Zbl

[19] K.C.H. Mackenzie General Theory of Lie Groupoids and Lie Algebroids Cambridge University Press (2005) | MR | Zbl

[20] J.E. Marsden and T.S. Ratiu Introduction to Mechanics and symmetry Springer-Verlag, 1999 | MR | Zbl

[21] E. Martínez Lagrangian Mechanics on Lie algebroids Acta Appl. Math 67 (2001) 295-320 | MR | Zbl

[22] 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 | MR | Zbl

[23] E. Martínez Reduction in optimal control theory Rep. Math. Phys 53 (2004) 79-90 | MR | Zbl

[24] E. Martínez Classical field theory on Lie algebroids: Multisymplectic formalism Preprint 2004, arXiv:math.DG/0411352

[25] E. Martínez Classical Field Theory on Lie algebroids: Variational aspects J. Phys. A: Mat. Gen 38 (2005) 7145-7160 | MR | Zbl

[26] E. Martínez, T. Mestdag and W. Sarlet Lie algebroid structures and Lagrangian systems on affine bundles J. Geom. Phys 44 (2002) 70-95 | MR | Zbl

[27] P. Michor Topics in differential geometry Book on the internet | MR

[28] J.P. Ortega and T.S. Ratiu Momentum maps and Hamiltonian Reduction Birkhäuser (2004) | MR

[29] P. Piccione and D. Tausk Lagrangian and Hamiltonian formalism for constrained variational problems Proc. Roy. Soc.Edinburgh Sect. A 132 (2002) 1417-1437 | MR | Zbl

[30] 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 | MR | Zbl

[31] A. Weinstein Lagrangian Mechanics and groupoids Fields Inst. Comm 7 (1996) 207-231 | MR | Zbl

Cité par Sources :