Invariant variational problems on principal bundles and conservation laws. (English) Zbl 1265.49049

The author studies variational problems defined by \(G\)-invariant Lagrangians on the \(r\)-jet prolongation of a principal \(G\)-bundle \(P\). It is proved that a \(G\)-invariant Lagrangian on \(J^rP\) is a Lagrangian on the bundle of \(r\)-th order principal connections (\(G\)-invariant sections \(P\to J^rP\)). The relation between the Euler-Lagrange equation and the conservation law is given. These type of problems were studied by other authors only in the first order. General results for any finite order \(r\) are new. The paper is closely related to the previous papers by the same author ([Balkan J. Geom. Appl. 13, No. 1, 11–19 (2008; Zbl 1194.49068)], [Czech. Math. J. 61, No. 4, 1063–1076 (2011; Zbl 1249.53029)]).


49S05 Variational principles of physics
58A10 Differential forms in global analysis
58A20 Jets in global analysis
49Q99 Manifolds and measure-geometric topics