  This work addresses the occupation measure relaxation of calculus of
variations problems, which is an infinite-dimensional linear programming
relaxation amenable to numerical approximation by a hierarchy of semidefinite
optimization problems. We address the problem of equivalence of this relaxation
to the original problem. Our main result provides sufficient conditions for
this equivalence. These conditions, revolving around the convexity of the data,
are simple and apply in very general settings that may be of arbitrary
dimensions and may include pointwise and integral constraints, thereby
considerably strengthening the existing results. Our conditions are also
extended to optimal control problems. In addition, we demonstrate how these
results can be applied in non-convex settings, showing that the occupation
measure relaxation is at least as strong as the convexification using the
convex envelope; in doing so, we prove that a certain weakening of the
occupation measure relaxation is equivalent to the convex envelope. This opens
the way to application of the occupation measure relaxation in situations where
the convex envelope relaxation is known to be equivalent to the original
problem, which includes problems in magnetism and elasticity.
