energifunktional

An energy functional is a functional that assigns a single real number, interpreted as the energy, to a field configuration. In mathematics and theoretical physics it plays a central role in variational problems and in describing static or dynamic systems at equilibrium.

For a field φ defined on a domain Ω, the energy functional E[φ] typically has the form E[φ] = ∫Ω L(x, φ(x), ∇φ(x)) dx, where L is an energy density. The exact form of L depends on the model and may include the field, its gradient, and position. The domain is usually subject to boundary conditions or constraints.

Equilibrium configurations are often stationary points of E, found by minimizing E or by solving the Euler–Lagrange equations obtained from the first variation δE = 0. Existence and uniqueness of minimizers depend on mathematical properties such as convexity, coercivity, and lower semicontinuity.

Examples include elasticity, where E[u] = ∫Ω W(ε(u)) dx with ε(u) the strain; electrostatics, where E[φ] = (ε0/2) ∫Ω |∇φ|^2 dx for a given charge distribution; and magnetostatics, where E[A] = (1/2μ0) ∫Ω |∇ × A|^2 dx. In quantum mechanics and materials science, density functional theory uses an energy functional E[n] = Ts[n] + VH[n] + Exc[n] + ∫ v_ext n dx, combining kinetic, Hartree, exchange–correlation and external potential terms.

Applications range from continuum mechanics and phase transitions to quantum chemistry and image processing. Related topics include calculus of variations, Euler–Lagrange equations, and variational methods.