Complementary extremum principles for isoperimetric optimization problems

Isoperimetric problems are of importance in engineering applications, where it is often desirable to maximize or minimize some physical variable by shape variation, subject to geometrical constraints, such as keeping an area or volume constant. The calculus of variations can offer a powerful tool for the solution of such problems where there is a governing variational minimum or maximum principle, e.g. Helmholtz's principle in slow viscous flow. In these problems the well-known Euler equations derived by the calculus of variations are supplemented with additional boundary conditions arising from the shape variation, as well as the usual physical boundary conditions. The exact solution of such unknown boundary problems can be difficult to find. A good approach then is to apply a complementary extremum principle that offers an algorithm for determining bounds on the exact extremal value of the original functional. This paper shows how this may be done in the case of the fundamental problem of the calculus of variations with variable endpoints. We apply this approach to a simple engineering problem of a stretched spring.