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Syllabus

• Shortest path between two points
• Geodesics on sphere (shortest path on a sphere between two points)
• Path followed by light in nonhomogeneous media and derivation of Snell's law from Fermat's principle
• Catenary
• Brachistochrone
• Minimal surfaces of revolution (catenoid)
• Harmonic functions
• The isoperimetric problem

Necessary conditions for a minimum include the vanishing of the first variation of the functional being studied, which translates into a differential equation (Euler's equation). We will derive the particular Euler equation for some of the paradigms quoted above.

Sufficient conditions and other means to prove minimality (such as calibrations) will de discussed next. However one can explicitely determine minima only in very particular cases (for instance the straight line segment joining two points is the shortest path, and on a sphere the geodesics are arcs of great circle). In general we must content ourselves with some qualitive description of minimizers (starting with their existence).

The need for developing a proper existence (of minima) theory will be illustrated by two famous examples where a minimizer does not exist. First we will discuss Weierstrass' example designed to point out a difficulty in Riemann's statement of Dirichlet's principle (related to the existence of harmonic functions). Next we will describe Besicovitch's surprising solution of Kakeya's problem. This problem asks for a region (in the plane) of least area in which a needle (say a line segment of length 1) can be turned 360 degrees by means of rotations and translations.

The existence theory that will be discussed applies to one dimensional problems and involves the definition of length of a rectifiable curve (a continuous curve of bounded variation), the arc length representation, and the Ascoli compactness theorem.

If time permits we will also introduce enough mathematical machinery to completely solve the isoperimetric problem in the plane (and to prove existence of a minimum for modified versions of the problem). This asks for the region in the plane of fixed area with the least perimeter.

References

• Charles Fox, An introduction to the calculus of variations, Dover
• George Ewing, Calculus of variations with applications, Dover
• Giuseppe Butazzo, Mariano Giaquinta, Stefan Hildebrandt, One-dimensional Variational Problems, Oxford Science Publications
• Handouts

Lecture Notes (PDF file)

Homeworks

HW #1 due Jan 18th 2007 (the following numbers refer to the notes of Jan. 10th 2007)

• 1.3.1
• 1.3.2
• 1.3.4
• 1.4.1
• 1.4.3
• 1.4.4
• 1.4.5

HW #2 due Jan 25th 2007 (the following numbers refer to the notes of Jan. 18th 2007)

• 1.2.5
• 1.3.14
• 1.3.15
• 1.4.16
• 1.4.19

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