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.

If the Lagrangian associated to a relativistic equation is geometric (covariant) in nature, Noether's theorem will generate a divergence-free stress-energy tensor $T^{ab}$, which (when the underlying spacetime manifold has a time translation symmetry, which is for instance true for Minkowski space) in turn leads to a conserved Hamiltonian $E(f)$. on constant time slices (and other spacelike surfaces). There are a few other conserved quantites such as momentum and angular momentum, but these are rarely useful in the well-posedness theory. It is often worthwhile to study the behaviour of $E(Df)$ where $D$ is some differentiation operator of order one or greater, preferably corresponding to one or more Killing or conformal Killing vector fields. These are particularly useful in investigating the decay of energy at a point, or the distribution of energy for large times.