An extremum principle is basically a mathematical concept that relies on some physical law. In mechanics, extremum principles such as the principle of minimum total potential energy and minimum total complementary energy form an important base of knowledge that has provided the means for obtaining approximate solutions to a variety of problems in engineering. This is particularly the case with the theory of elasticity. The celebrated principles of least work attributed to Alberto Castigliano, are also in the realm of extremum principles that have been used extensively in the solution of problems in classical structural mechanics dealing with elastic materials. In general, extremum principles and for that matter variational principles start with the basic premise that the solution to a problem can be represented as a class of functions that would satisfy some but not all of the equations governing the exact solution. It is then shown that a certain functional expression, usually composed of scalar quantities such as the total potential energy, strain energy, energy dissipation rate, etc., that have physical interpretations associated with them and are defined through the use of this class of functions, will yield an extremum (i.e. either a maximum or a minimum) for that function. Moreover, the extremum will satisfy the remaining equations required for the complete solution. For example, the principle of minimum total potential energy states that of all the kinematically admissible displacement fields in an elastic body, which also satisfy the governing constitutive equations, only those that satisfy the equations of equilibrium will give rise to a total potential energy that has a stationary value or an extremum.