Invariant volume forms and first integrals for geodesically equivalent Finsler metrics

TL;DR

This paper explores invariant volume forms and first integrals for geodesically equivalent Finsler metrics, providing a practical method to compute these invariants via characteristic polynomial coefficients.

Contribution

It introduces a new approach to compute invariant volume forms and first integrals for geodesically equivalent Finsler metrics using characteristic polynomials.

Findings

Invariant volume forms are determined by proportionality factors that are geodesically invariant.

First integrals are 0-homogeneous functions common to the entire projective class.

A practical method for computing these first integrals is provided.

Abstract

Two geodesically (projectively) equivalent Finsler metrics determine a set of invariant volume forms on the projective sphere bundle. Their proportionality factors are geodesically invariant functions and hence they are first integrals. Being 0-homogeneous functions, the first integrals are common for the entire projective class. In Theorem 1.1 we provide a practical and easy way of computing these first integrals as the coefficients of a characteristic polynomial.