An important class of variational problems concerns properties of geometric curves (such as length, curvature, etc.) that are independent of the parametrization. When formulated in terms of an arbitrary parameter, many of the familiar tools appear to fail for these problems. For example, the associated Euler–Lagrange equations are not independent, the process of solving for the Hamiltonian breaks down, and, as a consequence, the form of the Hamilton–Jacobi equation is obscured. Several alternative methods have previously been developed specifically for the solution of problems of this form, but these relatively sophisticated techniques have no direct links to the methods familiar from the context of classical mechanics. It is shown here that it is possible to solve such problems by using the conventional tools of mechanics. In particular, the integration of the Euler–Lagrange equations is realized in terms of a specific parameter that is completely determined save for a crucial, arbitrary scale factor. By considering an unusual Legendre transform, a direct analogue of Hamilton’s canonical equations is shown to emerge. More importantly, a unique form for the analogue of the Hamilton–Jacobi equation is derived from the conventional Hamilton–Jacobi equations for members of a family of auxiliary variational problems. Fermat’s principle is discussed in some detail to illustrate the methods introduced here.