Variational principles are fundamental concepts in physics and mathematics that provide a way to find the best solution to a problem by minimizing or maximizing a certain quantity. These principles often involve a functional, which is a mathematical expression that takes a function as input and produces a scalar value. The most famous example is the principle of least action, which states that the path taken by a system is the one that minimizes the action, a quantity that depends on the system's dynamics. In addition to physics, variational principles are also used in fields like calculus of variations and optimization. They help in solving problems where the goal is to find a function that minimizes or maximizes a specific functional. This approach is widely applied in areas such as engineering, economics, and machine learning, making it a versatile tool for researchers and practitioners alike.