Abstract
In this chapter we introduce the phase space and its symplectic structure. Then, canonical coordinates and generating functions are defined. The fundamental symplectic 2-form is introduced to define an isomorphism between vector fields and differential forms. Further, Hamiltonian vector fields and Poisson’s brackets are presented. The relation between first integrals and symmetries is analyzed together with Poincaré’s absolute and relative integral invariants. Liouville’s theorem and Poincaré’s theorem are proved and the volume form on invariant submanifolds is defined. Time-dependent Hamiltonian mechanics and contact manifolds are introduced together with contact coordinates and locally and globally Hamiltonian fields.
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- 1.
For the contents of Chapters 18–20 see also [1, 3, 19, 30].
- 2.
The most general differential form on M\(_{2{n}}^{{*}}\) is
$$\begin{aligned} \varvec{\omega }= {X}_{h}(\mathbf {q},\mathbf {p})\mathbf {d}{q}^{h} + {X}^{h}(\mathbf {q},\mathbf {p})\mathbf {d}{p}_{ h}. \end{aligned}$$Then (18.1) is a particular nonexact differential form.
- 3.
See Exercise 7.
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Romano, A., Marasco, A. (2018). Hamiltonian Dynamics. In: Classical Mechanics with Mathematica®. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-77595-1_18
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DOI: https://doi.org/10.1007/978-3-319-77595-1_18
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