  Tools of the intrinsic analysis on manifolds, helpful in solving the
invariant inverse problem of the calculus of variations are being presented
comprising a combined approach which consists in the simultaneous imposition of
symmetry principles and the inverse variational problem considerations in terms
of vector-valued differential forms. In three-dimensional space-time we obtain
a unique (covector) third-order Poincar\'e-invariant variational equation,
which then is identified with the motion of a free relativistic top in flat
three-dimensional space-time.
