  Background fields of electromagnetic and gravitational type emerge in the low
kinetic energy limit of any regular Lagrangian system and, in particular, in
the corresponding limit of any spacetime theory in which the free motion of
test particles is described by an unspecified regular Lagrangian.
Electromagnetic and gravitational type interactions are therefore a universal
feature of low kinetic energy Lagrangian systems. These background interactions
can be consistently turned into dynamic Einstein-Maxwell fields by promoting
the Lagrangian function to a dynamic scalar field on the tangent bundle of the
configuration space. Accordingly, Einstein-Maxwell theory can be deduced from
the assumption that the motion of elementary test particles in spacetime is
described by Lagrangian mechanics. For higher kinetic energy-type values,
identified with the square of the invariant mass of the particle, the
Lagrangian induces higher rank interactions that seem however too weak to have
been detected in spacetime physics, but which might prove relevant at the
Planck scale.
