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Erschienen in:
2018 | OriginalPaper | Buchkapitel
18. Hamiltonian Dynamics
verfasst von : Antonio Romano, Addolorata Marasco
Erschienen in: Classical Mechanics with Mathematica®
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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.