  A variant of the usual Lagrangian scheme is developed which describes both
the equations of motion and the variational equations of a system. The required
(prolonged) Lagrangian is defined in an extended configuration space comprising
both the original configurations of the system and all the virtual
displacements joining any two integral curves. Our main result establishes that
both the Euler-Lagrange equations and the corresponding variational equations
of the original system can be viewed as the Lagrangian vector field associated
with the first prolongation of the original LagrangianAfter discussing certain
features of the formulation, we introduce the so-called inherited constants of
the motion and relate them to the Noether constants of the extended system.
