Euler-Poincaré reduction on principal bundles

Abstract

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.