The geometry of systems of third order differential equations induced by second order Lagrangians

TL;DR

This paper explores the geometric structure of third order differential equations derived from second order Lagrangians, establishing unique semisprays, nonlinear connections, and covariant derivatives that preserve the Lagrangian metric.

Contribution

It introduces a novel geometric framework linking second order Lagrangians to third order systems via semisprays and nonlinear connections, with unique determination and metric preservation.

Findings

Unique semispray determined by two Cartan-Poincaré forms

Construction of a nonlinear connection as a Lagrangian subbundle

Second order dynamical derivative of the metric tensor vanishes

Abstract

A dynamical system on the total space of the fibre bundle of second order accelerations, $T^2M$, is defined as a third order vector field $S$ on $T^2M$, called semispray, which is mapped by the second order tangent structure into one of the Liouville vector field. For a regular Lagrangian of second order we prove that this semispray is uniquely determined by two associated Cartan-Poincar\'e one-forms. To study the geometry of this semispray we construct a nonlinear connection, which is a Lagrangian subbundle for the presymplectic structure. Using this semispray and the associated nonlinear connection we define covariant derivatives of first and second order. With respect to this, the second order dynamical derivative of the Lagrangian metric tensor vanishes.