  In this paper, we study the relation between geodesic and harmonic mappings.
Harmonic mappings are defined between Riemannian manifolds as critical points
of the energy functional, on the other hand, geodesic mappings are defined in a
more general setting (manifolds with affine connections). Using the
well-established formalism of calculus of variations on fibred manifolds we
solve the weak inverse problem for the equation of geodesic mappings and get a
variational equation which is a consequence of the geodesic mappings equation.
For the connection on the target manifold we get the expected result, that it
is a metric connection. However, we find that the connection on the source
manifold need not be metric. The interesting result is that the metric which
induces the connection on the target manifold can change between fibres and
these changes are related to the connection on the source manifold. These
results hint onto a possibility for a more general structure on the fibred
manifold, than usually assumed.
