A Noether's theorem for finite Hamiltonian systems

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Noether's theorem for finite Hamiltonian systems establishes a connection between symmetries and conserved quantities. To derive a first integral from a known symmetry, one can use the vector field generating the symmetry, denoted as ##\mathbf{w}##, and apply the relation ##\mathbf{w}(f) = \{ f, P \}## for an arbitrary function ##f##. The discussion highlights the process of deriving the Hamiltonian from the metric and emphasizes the necessity of identifying a symmetry to utilize Noether's theorem effectively. Participants express confusion about the steps involved in this derivation and the application of the theorem. Understanding these concepts is crucial for successfully applying Noether's theorem in finite Hamiltonian systems.
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How do I write a first integral knowing a symmetry?
The Noether's theorem for finite Hamiltonian systems says that:

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My question is: If I know a symmetry how can I write the first integral?
 
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If ##\mathbf{w}## is the vector field generating the symmetry then I think you can put ##\mathbf{w}(f) = \{ f, P \}## for an arbitrary function ##f##?
 
In fact my knowledge is very limited in this area, my situation is as follows...
I have the metric, with the metric I can write the Hamiltonian, with the Hamiltonian, can I write the field? if yes, after that do i have to look for a symmetry? Apparently once I have symmetry, Noether's theorem gives me the first integral (his proof in this case), but I'm not getting it.
 

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