A setting for higher order differential equations fields and higher order Lagrange and Finsler spaces

TL;DR

This paper reformulates the inverse calculus of variations for higher order differential equations using the Fr"olicher-Nijenhuis formalism, and explores projective metrizability in higher order Finsler geometry.

Contribution

It introduces a new approach to the inverse problem using semi-basic 1-forms and characterizes higher order Finsler functions through projective metrizability conditions.

Findings

Necessary and sufficient conditions for higher order projective metrizability.

Characterization of higher order Finsler functions via semi-basic 1-forms.

Application of the Fr"olicher-Nijenhuis formalism to higher order differential equations.

Abstract

We use the Fr\"olicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss the projective metrizability problem for higher order differential equation fields. We provide necessary and sufficient conditions for higher order projectivpre-e metrizability in terms of homogeneous semi-basic 1-forms. Such a semi-basic 1-form is the Poincar\'e-Cartan 1-form of a higher order Finsler function, while the potential of such semi-basic 1-form is a higher order Finsler function.