
5
The inverse problem of calculus of variations
It is often the case that the engineer starts from a differential equation with
certain boundary conditions, which is difficult to solve. Executing the inverse
of the Euler-Lagrange process and obtaining the variational formulation of
the boundary value problem may also be advantageous.
It is not necessarily easy, or may not even be possible to reconstruct the
variational problem from a differential equation. For differential equations,
partial or ordinary, containing a linear, self-adjoint, positive operator, the
task may be accomplished. Such an operator exhibits
(Au, v)=(u, Av),
where the parenthesis expression ...