Abstract
On the Geometry of Hamiltonian Symmetries
Highlights
Given a smooth manifold M a Poisson bracket on M assigns to each pair of smooth, real-valued functions F, H : M → R another smooth, real-valued function, which we denote by F, H
There are certain basic properties that such a bracket operation must satisfy in order to qualify as a Poisson bracket
Any distinguished function with respect to the Poisson bracket given by the matrix Jij(x) = {xi, xj} is the first integral of the Hamiltonian system (4)
Summary
Given a smooth manifold M a Poisson bracket on M assigns to each pair of smooth, real-valued functions F, H : M → R another smooth, real-valued function, which we denote by F, H. Let M be a Poisson manifold and let F, H : M → R be smooth functions with corresponding Hamiltonian vector fields XF , XH . Any distinguished function with respect to the Poisson bracket given by the matrix Jij(x) = {xi, xj} is the first integral of the Hamiltonian system (4).
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