Question:
Why are there not many books on the calculus of variations? Can the concept
of variation be extended to any set and has it been done? In differential
calculus one seems to vary a variable and in calc of variations one varies
functions. Has research stopped in the subject or is there a subject which
encompasses it?
Replies:
I assume differential topology and theories of differentiable manifolds are
essentially generalizations in the sense you seem to be after, of the idea
of differentiation to essentially arbitrary "sets" (really topological
spaces of various sorts). But in generalizing that way, something has
perhaps been lost. I know I found variational calculus to have a very
different feel from any of the topology stuff I took (it has been a long
time though.). The real downfall of variational calculus though has
probably been the computer, since just about any variational problem can be
discretized and put as a simple minimization problem (for example expanding
the function in Fourier series) for which there are all sorts of computa-
tional approaches.
asmith It has gone to physics. Much of physics is based upon minimizing a function
called "the action." You will find a lot of literature in advanced books on
mechanics. Look for the terms "Lagrangian" and "Hamiltonian". Another
relevant topic is "Quantum Field Theory."
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