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I am trying to understand Hamiltonian mechanics, and the last time I've tried I got up to the Legendre-Fenchel transformation, and in the next chapter suddenly those commutators were used and I was completely lost where and why they came from. My guess is that I missed something about the Legendre-Fenchel transformation just before that.

As of my understanding, the Legendre-Fenchel transformation for a given function $f:\Bbb R\to\Bbb R^n$ is such:

Take the convex hull of the graph of $f$, call it $G$.

$G$ is the intersection of half-spaces $\{\vec x\; |\; \vec n\cdot \vec x \geq b\}$.

Then the Legendre-Fenchel transformation is a mapping $f^*: \vec n \mapsto b$.

The question is: is this correct or am I missing some important point or have some errors in logic or assumptions?

Gyro Gearloose
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    I wrote an explanation of how I think of it here: https://math.stackexchange.com/a/2226937/40119 – littleO Oct 04 '20 at 13:35
  • @littleO should I delete this question, ask for it to be marked a duplicate or change it to the next step of my questions, that would be: "why and how is this transformation useful"? – Gyro Gearloose Oct 04 '20 at 15:08
  • You're not obligated to delete this question, although you can if you feel that it really is a duplicate of other questions. For your next step question, I think it's better to post it as an entirely new question. – littleO Oct 04 '20 at 22:00

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