Two Conservation Laws in Theoretical Economics

Summary

A released body falling to the earth travels a distance proportional to the second rather than first power of its terminal velocity. This fact, belatedly recognized by Galileo, denied by Descartes, established by Huygens and Newton, culminated in the law of conservation of (mechanical) energy. Mathematically, this law is based upon a property of the calculus of variations according to which the optimizing solution to

$$0 = \delta \smallint L(q,\dot q)dt = \delta \smallint [\Sigma \Sigma {a_{ij}}({q_1}, \ldots )\dot q - \dot V({q_1}, \ldots )]dt$$

has, because the integrand does not involve time explicitly, an “energy” integral of the form

$$L - \sum\limits_{i = 1}^n {{{\dot q}_i}\partial L/\partial {{\dot q}_i} = cons\tan t = kinetic\;energy + potentional\;energy} $$

.¹

I owe thanks to the National Science Foundation. In Chapter 2, I prove similar relations without considering the minimum-time formulations.