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Calculus of variations finds stationary points of functionals, often among functions or regular curves.

The first variation of a functional $L$ at $y$ in the direction $\eta$ is

A global minimizer is a point attaining the infimum of an objective. For an extended-real functional, it is usual to require a finite objective value; this excludes points outside its effective domain. A convention allowing $+\infty$ as the attained value gives every point as a formal minimizer of the identically infinite function.

The Gross–Pitaevskii equation describes a dilute interacting condensate through a complex order parameter $\psi$, normalized so that $|\psi {|}^2$ is number density. Repulsive contact interactions have $g>0$. In a frame rotating at the condensate frequency, the operator includes $-\mu$, with chemical potential $\mu$. It is generated by a Hamiltonian functional; its homogeneous positive-density state has $n_0=\mu /g$ when $g,\mu >0$.

For the functionalthe vanishing of its first variation under fixed endpoint data gives