Variational problem

Many equations arising in physics are actually the Euler-Lagrange equation for some variational problem, thus the equation describes the critical point of some Lagrangian. In the case of relativistic equations (most notably nonlinear wave equations), the Lagrangian resembles the expression

$\int _{\mathbb {R} ^{1+d}}\partial _{\alpha }f\partial ^{\alpha }fdxdt.$

Unlike variational problems associated to elliptic equations, the Lagrangian here typically has no good convexity properties. In particular, critical points are extremely unlikely to be local extremizers of the Lagrangian and so it has not proven to be profitable to try to construct or analyze solutions by a minimization method.