Two Proofs of Graves's Theorem

2. PROOF BY MECHANICS. An isolated system with at least one degree of freedom tends to lower its energy. Deviation from a current state is possible if and only if the system goes to a state of lower energy. Therefore, if an isolated system is in static equilibrium over a continuous range of configurations, its energy must remain constant over this range. In order to apply this principle to the situation in the Graves theorem, we make a mechanical interpretation of the problem. In Figure 2, suppose that ?2 is a solid elliptical plate with a grooved circumference and that ' is an elliptical wire, confocal with ?2, on which a small bead P can slide freely. Both ?2 and 4T are fixed to a frame and an elastic loop is strung around ?2 and joined to 4' at the bead P. Using an elastic loop whose length, when unstretched, is less than the perimeter of ?2 ensures that it is always under tension for any position of the bead P. Being taut,