Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations

Details Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations

2009-03-06

arxiv.org/abs/0903.1169v1

Journal of Geometric Mechanics, vol. 1, no. 2, June 2009, 159--180.

Mathematics

Differential Geometry

Scientific paper

10.3934/jgm.2009.1.159

We use Fr\"olicher-Nijenhuis theory to obtain global Helmholtz conditions, expressed in terms of a semi-basic 1-form, that characterize when a semispray is locally Lagrangian. We also discuss the relation between these Helmholtz conditions and their classic formulation written using a multiplier matrix. When the semi-basic 1-form is 1-homogeneous (0-homogeneous) we show that two (one) of the Helmholtz conditions are consequences of the other ones. These two special cases correspond to two inverse problems in the calculus of variation: Finsler metrizability for a spray, and projective metrizability for a spray.

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