Euler Paper

Euler Paper - James Westover PHY235 12/9/08 Isoperimetric...

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James Westover PHY235 12/9/08 Isoperimetric Problems and the Founding of Calculus of Variations Physics at the lowest levels is taught as though the field is static. The instructor always mentions it is a growing changing field. However I, as the student, wonder what advances there are left to be made. As I continue on into the higher reaches of undergraduate work that question remains open. Attempting to read journals is often like trying to read a Martian language. But noting this difficulty merely suggests that there is a rich history in the field. It seems to go unquestioned the complexity and unity of mathematical notation that professors use, but it too suggests the long history of physics and mathematics standardizing over time. And again as an undergraduate, one often wonderings why we solve the problems we do. The answer to that question is two-fold both technique and historical precedence. The subfield of physics, classical mechanics, and one of its main topics, calculus of variations, illustrate perfectly that problems we take for granted today were once up for debate and the methods now standard were once at the cutting edge of science. Calculus truly ushered in the new age of problem solving but not without a firm footing and the work of many great men. The entire body of problem solving techniques used on the general class of problems called isoperimetric problems had all previously been geometric. But, Pierre de Fermat, famous for his last theorem, was not quite able to enjoy the techniques from calculus which came after his work was completed. Fermat solved a problem involving light passing from one medium to another, previously solved by René Descartes. Fermat came to the same conclusion, as Descartes, about light rays bending as they pass from a less dense medium to a denser medium but under different assumptions about the nature
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of light. Descartes had proposed that light travel faster in more dense mediums and made no restriction as to the speed of light. Fermat suggested that light travel slower in denser materials and that it travels at a finite velocity. Fermat was shown later to be correct on both assumptions. however, denser not always the case instead the speed of light controlled more by relative permeability, both electric and magnetic, which is in general directly related to density. Fermat also made one other large step in the direction of generalization; he hypothesized that light travels in a path in such a way to minimize the time taken to get from one point to another, a principle now known as Fermat’s principle of least time. The results found
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This note was uploaded on 04/19/2008 for the course PHY 235W taught by Professor Cline during the Fall '07 term at Rochester.

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Euler Paper - James Westover PHY235 12/9/08 Isoperimetric...

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