The Euler-Lagrange equation is a fundamental equation in the field of calculus of variations, which helps find the path or function that minimizes or maximizes a certain quantity. It is derived from the principle of least action, stating that the actual path taken by a system is the one that minimizes the action, a quantity defined as the integral of the Lagrangian function over time. In mathematical terms, the Euler-Lagrange equation relates the derivatives of a function to its variations. It is expressed as a differential equation, where the Lagrangian is a function of the variables and their derivatives. Solving this equation provides critical insights into the behavior of physical systems, such as those described by classical mechanics.