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*.
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© 1987 D. Reidel Publishing Company, Dordrecht, Holland
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Libermann, P., Marle, CM. (1987). Semi-basic and vertical differential forms in mechanics. In: Symplectic Geometry and Analytical Mechanics. Mathematics and Its Applications, vol 35. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3807-6_2
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DOI: https://doi.org/10.1007/978-94-009-3807-6_2
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-277-2439-7
Online ISBN: 978-94-009-3807-6
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