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Published in:
2020 | OriginalPaper | Chapter
8. Hamiltonian Classical Mechanics
Author : Gerardo F. Torres del Castillo
Published in: Differentiable Manifolds
Publisher: Springer International Publishing
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Abstract
In this chapter we start by showing that any finite-dimensional differentiable manifold M possesses an associated manifold, denoted by \(T^{*}M\), called the cotangent bundle of M, which has a naturally defined non-degenerate 2-form, which allows us to define an operation between real-valued functions defined on \(T^{*}M\), called Poisson bracket. We then apply this structure to classical mechanics and geometrical optics, emphasizing the applications of Lie groups and Riemannian geometry. Here we will have the opportunity of making use of all of the machinery introduced in the previous chapters.