![]() ![]() In this paper the authors give a proof for general Lie groups. In the case of matrix groups this result has already been obtained. Some extra conditions must be imposed, namely, the vanishing of the curvature of the critical sections. In this framework the Euler-Poincaré equations do not suffice to reconstruct the Euler-Lagrange equations. The authors study the compatibility conditions needed for reconstruction. The reduced variational problem has a nice geometrical interpretation in terms of connections. It is known that the quotient manifold (J1P)/G can be identified with the bundle of connections of π:P→M. Let l:(J1P)/G→R be the reduced Lagrangian. The Lagrangian is assumed to be invariant under the natural action of G on J1P. 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. These equations are known as the Euler-Poincaré equations. 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. Then L induces a function l:(TG)/G≅g→R called the reduced Lagrangian, g being the Lie algebra of G. 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. ![]()
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January 2023
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