Legendre Transform

$f^{\star }(x^{\star }):={\sup }_{x\in I}(x^{\star }x-f(x))$ for $x^{\star }\in I^{\star },x\in I$, where

• $I\subset \mathbb{R}$ is an interval in the real line
• $f:I\to \mathbb{R}$ is a Convex function
• $I^{\star }:=\{x^{\star }\in \mathbb{R}:f^{\star }(x^{\star })<\infty \}$
• $\sup$ is the supremum operator

The Legendre transform is an Involuntory Operator that converts Convex functions into Convex functions, as defined via the equation below. It is used to convert functions of one quantity into the conjugate quantity, such as position/momentum, pressure/volume, temperature/entropy, etc. It is unclear how the Legendre Transform should be visualised. Help me ❓

Converting between Lagrangian and Hamiltonian mechanics

We can use Legendre Transformations to convert between Lagrangian Mechanics and Hamiltonian mechanics. Since position and momentum are conjugate variables, we can convert the Lagrangian $\mathcal{L}(q,\dot{q},t)$ and the Hamiltonian $\mathcal{H}(q,p,t)$.

A detailed explanation on this conversion can be found on NYU’s lecture notes for mechanics, lecture 6.