Albion College

Mathematics and Computer Science

Mathematics and Computer Science

COLLOQUIUM

The Classical Problems of the Calculus of Variations

Charles R. MacCluer

Professor

Department of Mathematics

Michigan State University

Professor

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.

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.