  We show that the theory of self-adjoint differential equations can be used to
provide a satisfactory solution of the inverse variational problem in classical
mechanics. A Newtonian equation when transformed to the self-adjoint form
allows one to find an appropriate Lagrangian representation (direct analytic
representation) for it. On the other hand, the same Newtonian equation in
conjunction with its adjoint provides a basis to construct a different
Lagrangian representation (indirect analytic representation) for the system. We
obtain the time-dependent Lagrangian of the damped Harmonic oscillator from the
self-adjoint form of the equation of motion and at the same time identify the
adjoint of the equation with the so called Bateman image equation with a view
to construct a time-independent indirect Lagrangian representation. We provide
a number of case studies to demonstrate the usefulness of the approach derived
by us. We also present similar results for a number of nonlinear differential
equations by using an integral representation of the Lagrangian function and
make some useful comments.
