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Noether's theorem

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Translate Noether's theorem into Hamiltonian mechanics. That is, define a symmetry for Hamiltonian mechanics (by translating the Lagrangian-definition), and prove that symmetries give rise to conserved observables.

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Noether's theorem, in its original version, applies to theories described by a Lagrangian. There is also a version which applies to theories described by a Hamiltonian. Suppose there is a particle moving on a line with Lagrangian L(q,q'), where q is its position and q' = dq/dt is its velocity. The momentum of this particle is defined to be p = dL/dq'.
The force on it is defined to be F = ...

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