Extremal

An extremal in the calculus of variations is an admissible curve or function that makes the first variation of a functional vanish for every admissible perturbation. For a sufficiently smooth integral functional with fixed endpoint values, it satisfies the Euler-Lagrange differential equation. An extremal need not give a minimum or maximum (Liberzon 2012).

In a simply connected plane region, a field of extremals is a one-parameter family of extremals with exactly one curve passing through each point of the region (Gelfand and Fomin 1963, pp. 133 and 140). Weierstrass introduced the concept of a field of extremals (Kreyszig 1994).