Publication: Euler-Poincaré reduction on principal bundles
Full text at PDC
Advisors (or tutors)
Let G be a Lie group and let L:TG→R be a Lagrangian invariant under the natural action of G on its tangent bundle. Then L induces a function l:(TG)/G≅g→R called the reduced Lagrangian, g being the Lie algebra of G. As is well known, the Euler-Lagrange equations defined by L for curves on G are equivalent to a new kind of equation for l for the reduced curves in the Lie algebra g. These equations are known as the Euler-Poincaré equations. In the paper under review, the authors extend the idea of the Euler-Poincaré reduction to a Lagrangian L:J1P→R defined on the first jet bundle of an arbitrary principal bundle π:P→M with structure group G. The Lagrangian is assumed to be invariant under the natural action of G on J1P. Let l:(J1P)/G→R be the reduced Lagrangian. It is known that the quotient manifold (J1P)/G can be identified with the bundle of connections of π:P→M. The reduced variational problem has a nice geometrical interpretation in terms of connections. The authors study the compatibility conditions needed for reconstruction. In this framework the Euler-Poincaré equations do not suffice to reconstruct the Euler-Lagrange equations. Some extra conditions must be imposed, namely, the vanishing of the curvature of the critical sections. In the case of matrix groups this result has already been obtained [M. Castrillón López, T. S. Ratiu and S. Shkoller, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2155–2164;]. In this paper the authors give a proof for general Lie groups. Moreover, they point out several facts concerning the reduced variational problem: its relation with the variational calculus with constraints, Noether's theorem for reduced symmetries, and the second variation formula.
Arnold, V. I.: Dynamical Systems III, Encyclop. Math. 3, Springer, New York. Castrillón López, M. and Mu~noz Masqué, J.: The geometry of the bundle of connections, Math. Z. 236 (9) (2001), 797–811. Castrillón López, M., Ratiu, T. S. and Shkoller, S.: Reduction in principal fiber bundles: covariant Euler–Poincaré equations, Proc. Amer. Math. Soc. 128 (7), (2000), 2155–2164. Eck, D. J.: Gauge-natural bundles and generalized gauge theories, Mem. Amer. Math. Soc. 247 (1981). García, P. L.: Gauge algebras, curvature and symplectic structure, J. Differential Geom. 12 (1977), 209–227. García, P. L.: The Poincaré—Cartan invariant in the calculus of variations, Symposia Math. 14 (1974), 219–246. Giachetta, G., Mangiarotti, L. and Sarnanashvily, G.: New Lagrangian and Hamiltonian Methods in Field Theory, World Scientific, Singapore, 1997. MR2001723 (2004g:70049) Goldschmidt, H. and Sternberg, S.: The Hamiltonian–Cartan formalism in the calculus of variations, Ann. Inst. Fourier Grenoble 23 (1), (1973), 203–267. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Wiley, New York, Volume I, 1963; Volume II, 1969. Marsden, J. E. and Ratiu, T. S.: Introduction to Mechanics and Symmetry, Text Appl. Math. 17, Springer, New York, 1999. Pluzhnikov, A. I.: Some properties of harmonic mappings in the case of spheres and Lie groups, Soviet Math. Dokl. 27 (1983) 246–248. Urakawa, H.: Calculus of Variations and Harmonic Maps, Transl. Amer. Math. Soc., Providence, 1993. Varadarajan, V. S. Lie Groups, Lie Algebras, and their Representations, Springer, New York, 1984.