The Remarkable Isoperimetric Problem and the Euler-Lagrange Equation Revisited

4:00 pm Wednesday, April 7, 2010
Richard Tapia (Rice CAAM department)


In contrast to other disciplines in mathematics, problems in optimization are usually quite easy to state and to understand-even for those with limited mathematical sophistication. Writing circa 200 BC, the Greek mathematician Zenodorus considered the so-called isoperimetric problem: Determine, from all simple closed planar curves of the same perimeter, the one that encloses the greatest area. Even the mathematically uninitiated guess correctly that the solution is the circle.

The isoperimetric problem has been the most influential mathematics problem of all time. It played a major role in motivating the calculus of variations activity credited to the Bernoullis, Newton, Euler, and Lagrange in the late 1600's and early 1700's. Yet a complete solution of the isoperimetric problem eluded these early pioneers. Indeed, it was Weierstrass who first gave a complete proof more than a century later. . In this talk the speaker will argue that Euler and later Lagrange were one direct observation away from deriving a sufficiency condition that would have given a straightforward resolution of the isoperimetric problem. The missing ingredient was function convexity. We then ask rhetorically: was convexity of functions not known to the great mathematicians of that time. Set convexity was known to the early Greeks.

Moreover, the derivation of the Euler-Lagrange equation presented by Euler and Lagrange is well known to be flawed. A correct derivation was given by du Bois-Reymond some 150 years later. We argue quite surprisingly that the du Bois-Reymond's derivation can be viewed as presenting the Euler-Lagrange equation as a multiplier rule. As such, it would be the world’s first multiplier rule and would precede the very notion of Lagrange multiplier rules.

 

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