  We discuss a recently proposed variational principle for deriving the
variational equations associated to any Lagrangian system. The principle gives
simultaneously the Lagrange and the variational equations of the system. We
define a new Lagrangian in an extended configuration space ---which we call
D'Alambert's--- comprising both the original coordinates and the compatible
``virtual displacements'' joining two solutions of the original system. The
variational principle is Hamilton's with the new Lagrangian. We use this
formulation to obtain constants of motion in the Jacobi equations of any
Lagrangian system with symmetries. These constants are related to constants in
the original system and so with symmetries of the original Lagrangian. We cast
our approach in an intrinsic coordinate free formulation. Our results can be of
interest for reducing the dimensions of the equations that characterize
perturbations in a Lagrangian control system.
