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Name: existing
Status: N/A
Age: N/A
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Date: Around 1995

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?

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.


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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