The Classical Problems of the Calculus of Variations
Charles R. MacCluer
Department of Mathematics
Michigan State University
One of the earliest uses of the calculus was to attack "variational problems," where the objective is to minimize certain path integrals. These first problems were proposed by Johann Bernoulli, Newton, von Leibniz, and others --- in certain cases as challenges to smoke out their competition. We will tour (but not solve) a collection of these early problems on least time, geodesics, bluff bodies, isoperimetric problems, hanging cable, etc, as well as the modern Nobel-winning Mirrlees formulation of optimal tax structure.
If time permits, we will also sketch the derivation of the Euler-Lagrange equation for solving variational problems and its application to conservative mechanical systems. Finally, we will formulate a representative optimal control problem.