  Given a smooth family of unparameterized curves such that through every point
in every direction there passes exactly one curve, does there exist a
Lagrangian with extremals being precisely this family? It is known that in
dimension 2 the answer is positive. In dimension 3, it follows from the work of
Douglas that the answer is, in general, negative. We generalise this result to
all higher dimensions and show that the answer is actually negative for almost
every such a family of curves, also known as path structure or path geometry.
On the other hand, we consider path geometries possessing infinitesimal
symmetries and show that path and projective structures with submaximal
symmetry dimensions are variational. Note that the projective structure with
the submaximal symmetry algebra, the so-called Egorov structure, is not
pseudo-Riemannian metrizable; we show that it is metrizable in the class of
Kropina pseudo-metrics and explicitly construct the corresponding Kropina
Lagrangian.
