# I Hamiltonian as the generator of time translations

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1. Sep 9, 2016

### Frank Castle

In literature I have read it is said that the Hamiltonian $H$ is the generator of time translations. Why is this the case? Where does this statement derive from?

Does it follow from the observation that, for a given function $F(q,p)$, $$\frac{dF}{dt}=\lbrace F,H\rbrace +\frac{\partial F}{\partial t}$$ In particular, if $F$ is not explicitly dependent on time, then $$\frac{dF}{dt}=\lbrace F,H\rbrace$$ Or is there more to it?

2. Sep 10, 2016

### pliep2000

Ah poisson brackets!

I think what you are looking for is Noethers theorem.

Susskinds classical mechanics course on youtube, i think lecture 4 (symmetries) and 8 (poisson) will help you.

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