Variationales

Variationales is a term encountered in certain branches of mathematics and physics to denote a family of functionals that organize the criteria to be optimized in a variational problem. It emphasizes the functional, or action-like, nature of the quantity being extremized, rather than the equations that result from it.

Definition: A variationale J maps an admissible field configuration φ, defined on a domain Ω with prescribed boundary conditions, to a real number J[φ]. The collection of such functionals, together with their parameterizations, is called a variationales in some texts. A primary objective is to find φ that makes J[φ] stationary, typically a minimum or maximum.

Mathematical framework: Analysis proceeds by computing the first variation δJ and setting it to zero, yielding Euler–Lagrange type equations. The second variation δ^2 J provides information about local stability and convexity of J at the extremal. In multi-parameter problems, a family of variationales may be studied simultaneously to understand trade-offs between competing objectives.

Examples: Classical mechanics uses the action S[ q ] = ∫ L(q, q̇, t) dt as a variationale. In elasticity, the elastic energy functional minimizes deformation energy. In image processing, energy functionals combine fidelity and smoothness terms and are treated as variationales.

History: The term variationale appears in a subset of variational analysis literature, intended to highlight the functional basis of optimization problems. It is not universally adopted in standard texts, where "functional" and "variational principle" are more common.

See also: variational principle, functional, calculus of variations, Euler–Lagrange equation, optimization.