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 denotes a scalar product in the function
space of the solution of the diﬀerential equation. Positive deﬁniteness of the
(Au, u) ≥ 0,
with zero attained only for the trivial (u = 0) solution. Let us consider the
diﬀerential equation of
Au = f,
where the operator obeys the above conditions and f is a known function. If
the diﬀerential equation has a solution, it corresponds to the minimum value
of the functional
I(u)=(Au, u)+2(u, f).
This may be proven by simply applying the appropriate Euler-Lagrange equa-
tion to this functional.