Semi-basic and vertical differential forms in mechanics

Abstract

This chapter is first devoted to the discussion of semi-basic and vertical differential forms relative to a surjective submersion π: E → M. Intuitively, a semi-basic differential form on E is a linear combination of basic forms (i.e., of forms which are the pull-backs of forms on M), whose coefficients are parametrized by the fibers, while a vertical form is a family of differential forms on the fibers, parametrized by the base. Contrary to semi-basic forms, vertical forms are not differential forms in the usual sense. We can define the vertical differential of a differential form (in the usual sense), in particular of a function on E. In the case of a vector bundle, this leads to the Legendre transformation, which is a diffeomorphism from E onto its dual E*.