Spring 14: Math 534 Calculus of Variations
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This announces the course offering Math 534 (Calculus of Variations)
for Spring 2014.
Typically this course runs every 3-4 years, so if you want to ever
take it, it's wise to do so now.
Also if you plan to sign up, please sign up soon, to make sure we meet the
quota to run the course.
The course is perfectly appropriate for Honors undergraduates,
in particulare those who plan to choose or have chosen 447-48 as their
honors sequence. It requires a certain mathematical maturity, which
is usually a given for students in the Math Honors concentration,
but does not have any significant contents prerequisites.
Last time I taught the course, an honors student wrote an honors thesis
based on material from the course.
Feel free to contact me for any questions you may have
Details below
Jochen Denzler
Contents:
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Calculus of Variations is about minimization problems in which the variable
is a function. A classical problem is the brachystochrone problem: On which
curve between two points A,B would a mass point under the influence of
gravity slide fastest from A to B? The quantity to be minimized is typically
an integral involving the function and its derivative.
The subject comes in two flavors: classical theory that is mainly
concerned with single variable integrals, and modern, that is more
concerned with `direct methods' that involve abstract existence proofs
(and more, like saddle point methods).
In the absence of heavy analysis prerequisites, the course will focus
to maybe 2/3 on the classical theory, but there will be enough of the
modern theory in it to give insight into the usage of calculus of
variations for existence proofs in differential equations, even though
technical lemmas will be stated without proof in that case; this should
give the gist of the idea to those not specializing in the area, and a
motivating introduction to those that may pursue it in more depth later.
The classical results interlace well with theoretical physics (especially
the Lagrangian variant of Newtonian mechanics) and also add a geometric
flavor
to the questions (geodesics: shortest connecting curves on surfaces).
Viewing
classical hands-on examples also in more modern terms will allow to
motivate and illustrate core concepts/ideas of analysis in a Banach space
setting from a practical point of view.
Scheduling:
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The course is scheduled TBA. Based on signup by mid-November, I'll
then find a mutually convenient slot.
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