Maximal degree variational principles and Liouville dynamics

Abstract

Let M be smooth n-dimensional manifold, fibered over a k-dimensional submanifold B as π :M→B , and ϑ∈ Λ k ( M); one can consider the functional on sections φ of the bundle π defined by ∫ Dφ ∗(ϑ) , with D a domain in B. We show that for k= n−2 the variational principle based on this functional identifies a unique (up to multiplication by a smooth function) nontrivial vector field in M, i.e., a system of ODEs. Conversely, any vector field X on M satisfying X⌟ dϑ=0 for some ϑ∈ Λ n−2 ( M) admits such a variational characterization. We consider the general case, and also the particular case M= P× R where one of the variables (the time) has a distinguished role; in this case our results imply that any Liouville (volume-preserving) vector field on the phase space P admits a variational principle of the kind considered here.