The buckling of an elastic rod is a classic problem in the calculus of variations, whose equilibrium solutions can be found by solving a boundary value problem in a system of ODEs. With a little work, you can extend this problem to investigate how things change when you consider contact, either with a fixed outside obstacle or with the rod itself (in the latter case, the equations become integro-differential equations). The classic calculus of variations theory also includes a method for determining the stability of solutions (in the sense of whether or not they minimize the energy) via the idea of conjugate points. However, the classic methods do not make it clear how to extend this idea of conjugate points to a wide range of problems, certainly not to the sort of contact problems which are far from the standard form in the calculus of variations. We will present a theory of conjugate points couched in functional analysis that shows the connection to finite-dimensional theory (the spectrum of the Hessian). This theory allows a relatively easy generalization of conjugate points to many problems, including the rod contact problems described above, for which we will show a few results.