Variationnels

Variationnels is a term used to refer to methods and principles based on the calculus of variations, a field in mathematics and theoretical physics concerned with finding functions that optimize functionals. In French-language scholarly writing, the adjective variationnel is often used to describe techniques, problems, and principles tied to variational methods and their applications.

The central idea of the variationnels is to determine a function that makes a functional stationary, typically a minimum, maximum, or saddle point. This involves considering small perturbations of the unknown function and requiring that the first variation of the functional vanishes. In classical form, this leads to Euler-Lagrange type equations, which provide differential equations and boundary conditions that the optimal function must satisfy. Variational problems may be constrained by endpoints or by integral conditions, leading to natural boundary conditions or Lagrange multipliers.

The scope of variationnels includes the calculus of variations, variational methods, variational principles, and variational inequalities. Key concepts include the action principle in physics, where the path of a system extremizes an action integral, and weak formulations in functional analysis used to study partial differential equations. Variational techniques also underpin numerical methods, such as finite element methods, and play a role in shape optimization and material science.

Applications of variationnels are widespread. In physics, variational principles underlie classical mechanics, optics, quantum mechanics, and general relativity. In engineering and economics, variational approaches appear in optimization problems, structural design, and inequality formulations that capture equilibrium states.

Historically, variationnels trace to contributions by Euler and Lagrange in the 18th and 19th centuries, with later developments by Hadamard, Hilbert, and many others enriching the theory and its methods.