Introduction to Hamiltonian Dynamical Systems and the N-Body Problem by Kenneth R. Meyer & Daniel C. Offin

Author:Kenneth R. Meyer & Daniel C. Offin
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

A direct computation gives

The first identity shows that the standard inner product on induces an invariant inner product on and so the dual can be identified with using this inner product.

Let so the flow generated by ξ is (e ξ t q, e ξ t p) which is generated by the Hamiltonian − q T ξ p, so let S: ξ → −q T ξ p. The moment map is

where

Thus holding R q×p gives the same result as holding q × p fixed in the classical case.

A Hamiltonian H admits as a symmetry, or H is symmetric with respect to , if H(Φ(g, x)) = H(gx) = H(x) for all . This discussion leads to

Theorem 7.7.1 (Noether-Souriau).

Let the action be symplectic, the moment map be defined and a smooth Hamiltonian which admits the action Φ as a symmetry. Then S is constant along the solutions of dH ♯ .

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