Variational methods are mathematical techniques used to find the extrema (minimum or maximum) of functionals, which are mappings from a space of functions to real numbers. These methods are particularly useful in fields like calculus of variations, physics, and engineering, where they help solve problems involving optimization and approximation. In variational methods, one typically starts with a functional and seeks to determine the function that minimizes or maximizes it. This often involves using tools from differential equations and functional analysis to derive necessary conditions for optimality, leading to solutions that can describe physical systems or optimize performance in various applications.