Legendre transform

The Legendre transform is a mathematical tool that converts one function into another, highlighting different aspects of the original. Think of it like changing perspectives: if a function describes a system's energy based on a variable, the Legendre transform shifts the focus to how the system responds to changes, like forces or slopes. It's commonly used in physics and calculus to switch between different descriptions—such as from position-based to force-based views—making complex problems easier to analyze. Essentially, it provides a new function that captures the original's core information but in a form better suited for particular applications.