Maximal degree variational principles and Liouville dynamics (Q1877586)

Let \(M\) be a smooth \(n\)-dimensional manifold, fibered over a \(k\)-dimensional submanifold \(B\). The main result of the paper establishes that if \(k=n-2\), then the variational principle identifies a unique (up to multiplication by a smooth function) nontrivial vector field in \(M\), that is, a system of ordinary differential equations. The framework includes the general case, and also the particular case \(M=P\times {\mathbb R}\), where one of the variables (the time) has a distinguished role. The abstract results of the paper imply that any Liouville (volume-preserving) vector field on the phase space \(P\) admits a variational principle of the kind considered in the paper.