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Consider a convex function $f(x)$ and the line $y=px$. Let $x_p$ be the value of $x$ that maximizes $xp-f(x)$. This allows us to define the Legendre transform of $f$ as a function of $p$:

$g(p) = x_pp - f(x_p)$

This definition is so nice and simple, and many important properties of the Legendre transform can be deduced from it (for example, that it is an involution).

I'm trying to understand how this simply defined transformation of a function can be viewed as sending a function on the tangent bundle of a manifold to it's cotangent bundle. I've been studying this all in the context of mechanics, where the Legendre transformation turns Lagrangian systems to a Hamiltonian systems.

https://en.wikipedia.org/wiki/Legendre_transformation#Legendre_transformation_on_manifolds

Maybe it would help if I could first clear up this basic misunderstanding of mine: In the context of mechanics, I understand why it's natural that the vectors of the tangent bundle represent velocity vectors, but I don't understand why the covectors in the cotangent bundle should represent momentum vectors. A covector sends a vector to a scalar, and so momentum sends velocities to a scalar? How do this make sense.

PhysicsIsHard
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  • The first step is to understand how this works for a function $f: X \rightarrow \mathbb{R}$, where $X$ is a real vector space. In that case, the Legendre transform of $f$ is defined to be the function $$f^: X^ \rightarrow \mathbb{R}, $$ where $X^$ is the dual of $X$, given by $$ f^(\xi) = \sup { \langle\xi,x\rangle -f(x)\ :\ x \in X }. $$ – Deane Mar 18 '24 at 20:23
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    https://math.stackexchange.com/a/2805892/10584 – Deane Mar 18 '24 at 21:40

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