ABSTRACT
Many physical concepts may be formulated as variational principles, which are conditions on the functions describing a physical system such that some given quantity is stationary. A large number of physics problems deal with the optimisation of different functions and functionals, that is, the search for extreme values. In many optimisation problems for functions and functionals, the arguments are not arbitrary but subjected to constraints of different types. The chapter deals with functionals that depend on several functions, the problem at hand may be simpler if it can be reformulated in a different way by using a different set of functions. Variational methods also have some interesting applications to Sturm—Liouville problems. The chapter considers the problem of finding the eigenvalues of the Laplace operator on some arbitrary domain with homogeneous Dirichlet boundary conditions. In many optimisation problems for functions and functionals, the arguments are not arbitrary but subjected to constraints of different types.
