Equivalence of variational problems of higher order

TL;DR

This paper demonstrates the fundamental equivalence among various higher-order variational problems, leading to new invariants and models, by linking geometric structures and differential equations through projective geometry.

Contribution

It establishes the equivalence of several higher-order variational and distribution problems and derives their invariants and maximally symmetric models using projective geometry.

Findings

Identifies the fundamental invariants for the equivalence problems.

Provides explicit descriptions of maximally symmetric models.

Connects the symmetry of problems to rational normal curves in projective space.

Abstract

We show that for n>2 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication by a nonzero constant, and modulo divergence; the equivalence problem for the special class of rank 2 distributions associated with underdetermined ODEs z'=f(x,y,y',..., y^{(n)}); the equivalence problem for variational ODEs of order 2n. This leads to new results such as the fundamental system of invariants for all these problems and the explicit description of the maximally symmetric models. The central role in all three equivalence problems is played by the geometry of self-dual curves in the projective space of odd dimension up to projective transformations via the linearization procedure (along the solutions of ODE or abnormal extremals of distributions). More precisely, we show that an object from one of the three equivalence problem is maximally symmetric if and only if all curves in projective spaces obtained by the linearization procedure are rational normal curves.