Maximal degree variational principles

TL;DR

This paper develops a variational principle for identifying certain vector fields on manifolds, especially volume-preserving ones, using maximal degree differential forms, linking geometric structures to variational calculus.

Contribution

It introduces a variational framework for vector fields via maximal degree forms, providing a new characterization of volume-preserving fields and their relation to differential forms.

Findings

Unique vector fields correspond to variational principles for k=n-2.

Any vector field satisfying i_X(dθ)=0 admits a variational characterization.

Liouville vector fields on phase space have associated variational principles.

Abstract

Let $M$ be smooth $n$-dimensional manifold, fibered over a $k$-dimensional submanifold $B$ as $\pi:M \to B$, and $\vartheta \in \Lambda^k (M)$; one can consider the functional on sections $\phi$ of the bundle $\pi$ defined by $\int_D \phi^* (\vartheta)$, 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 $i_X ({\rm d} \vartheta) = 0$ for some $\vartheta \in \Lambda^{n-2} (M)$ admits such a variational characterization. We consider the general case, and also the particular case $M = P \times R$ 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.