Invariants for the Lagrangian Equivalence Problem

Abstract

Let $M$ be a connected smooth manifold, let $\mathrm{Aut}(p)$ be the group automorphisms of the bundle $p:\mathbb{R}\times M\to \mathbb{R}$, and let $q:J^1(\mathbb{R},M)\times \mathbb{R}\to J^1(\mathbb{R},M)$ be the canonical projection. Invariant functions on $J^r(q)$ under the natural action of $\mathrm{Aut}(p)$ are discussed in relationship with the Lagrangian equivalence problem. The second-order invariants are determined geometrically as well as some other higher-order invariants for $\dim M\ge 2$.