Variationstheorien

Variationstheorien, or calculus of variations, is a branch of mathematical analysis that deals with finding functions that optimize (minimize or maximize) certain integrals. These integrals, often called functionals, depend on the function itself and its derivatives. The core problem in variation theory is to find a function or a set of functions that makes a given functional attain its extremum.

The simplest and most fundamental problem is to find a function y(x) that minimizes or maximizes the integral of a function L(x, y, y') over a fixed interval. The cornerstone of solving such problems is the Euler-Lagrange equation. This differential equation provides a necessary condition for a function to be an extremum of the functional. If the integrand L is known, solving the Euler-Lagrange equation can yield the candidate functions.

Historically, the calculus of variations arose from geometric problems, such as finding the curve of shortest distance between two points (a straight line) or the curve that a falling object traces under gravity that minimizes travel time (the brachistochrone problem). It has found widespread applications in physics, engineering, and economics. In physics, for instance, principles like the principle of least action in classical mechanics and quantum field theory are formulated using variation theory. The Euler-Lagrange equations are central to Lagrangian mechanics.

More advanced topics in variation theory include problems with higher-order derivatives, multiple dependent variables, constrained functionals (like isoperimetric problems), and boundary conditions other than fixed points. The existence and uniqueness of extremizing functions are also crucial aspects studied within this field, often employing techniques from functional analysis.