Legendre transformation

The Legendre transformation is a mathematical method used to switch between different descriptions of a system, especially in physics and optimization. Imagine you have a function describing a system's energy based on certain variables, like position. The Legendre transform allows you to change that perspective to focus instead on related variables, such as momentum, by converting the original function into a new one. This is useful for simplifying problems and understanding different aspects of a system's behavior, especially in fields like thermodynamics and classical mechanics, by providing a complementary view that highlights different quantities of interest.