Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Calculus of variations
Name: existing
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995


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."

jlu



Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (help@newton.dep.anl.gov), or at Argonne's Educational Programs

NEWTON AND ASK A SCIENTIST
Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory