A SETTING FOR HIGHER ORDER DIFFERENTIAL EQUATION FIELDS AND HIGHER ORDER LAGRANGE AND FINSLER SPACES

We use the Frolicher-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 [15] to formulate and discuss the projective metrizability problem for higher order differential equation fields. We provide necessary and sufficient conditions for higher order projective metrizability in terms of homogeneous semi-basic 1-forms. Such a semi-basic 1-form is the Poincare-Cartan 1-form of a higher order Finsler function, while the potential of such semi-basic 1-form is a higher order Finsler function.