Narrow Results

In this paper, we present geometrical interpretations of the Lie derivative and the Lie-invariance for an affine connection, based on two geometrical constructions: the former is given on the tangent bundle and relies on the notion of linear Ehresmann connection; the latter is developed on the frame bundle and is based on the more common notion of principal connection.

Let P → M be a principal G-bundle over a pseudo-Riemannian manifold (M, g). If G is semisimple, the Euler-Lagrange and the Hamilton-Cartan equations of the Yang-Mills Lagrangian defined by g are proved to remain unchanged if the Cartan-Killing metric is replaced by any other non-degenerate, adjoint-invariant bilinear form on the Lie algebra 𝔤. Moreover, it is shown that the Hamilton-Cartan equations for any gauge invariant problem are essentially the same as their Euler-Lagrange equations if some simple regularity assumption is made. Actually, the set of solutions to Hamilton equations trivially fibers over the set of extremals of the variational problem.