Variationsrechnung

Variationsrechnung, or calculus of variations, is a branch of mathematical analysis focused on optimizing functionals, which assign a number to a function. The typical problem is to find a function y defined on an interval [a,b] that extremizes a functional of the form F[y] = ∫_{a}^{b} L(x, y(x), y'(x)) dx, where L is called the Lagrangian. Solutions must satisfy prescribed boundary conditions such as y(a) = α and y(b) = β, though sometimes endpoints or the time interval may be free.

A central result is the Euler–Lagrange equation. If y is sufficiently smooth and extremal, it satisfies d/dx (∂L/∂y') - ∂L/∂y = 0 for all x in [a,b]. This yields a differential equation for y, which, together with boundary conditions, determines candidates for extrema.

Beyond first-order conditions, the second variation and related criteria (Legendre condition, Jacobi fields) give information about whether an extremal is a minimum or maximum. For many problems, additional constraints lead to isoperimetric or variational inequalities, and sometimes transversality conditions for free boundaries.

Examples include the problem of the shortest path between two points (L = 1) giving a straight line, the brachistochrone problem (L = sqrt(1 + y'^2) with gravity) and geodesics in curved spaces. The calculus of variations also underpins physics through the principle of least action and the Lagrangian formalism.

Modern developments connect to optimal control, numerical methods, and Noether's theorem, which links symmetries to conserved quantities. The field spans pure mathematics and applied disciplines in engineering, economics, and physics.