  We consider non-autonomous calculus of variations problems with a state
constraint represented by a given closed set. We prove that if the interior of
the Clarke tangent cone of the state constraint set is non-empty (this is the
constraint qualification that we suggest here), then the necessary optimality
conditions apply in the normal form. We establish normality results for (weak)
local minimizers and global minimizers, employing two different approaches and
invoking slightly diverse assumptions. More precisely, for the local minimizers
result, the Lagrangian is supposed to be Lipschitz with respect to the state
variable, and just lower semicontinuous in its third variable. On the other
hand, the approach for the global minimizers result (which is simpler) requires
the Lagrangian to be convex with respect to its third variable, but the
Lipschitz constant of the Lagrangian with respect to the state variable might
now depend on time.

Optimization and Control