I want to learn about the calculus of variations...

[eLiu]

So it struck me that this is something I ought to know about given my field of interest... so could somebody recommend a good text for learning/reference?

Thanks,
-Eric

 

[CycloWizard]

I've been trying to pick this up on my own as well... The book that gave me the most insight so far was "The variational principles of mechanics" by Cornelius Lanczos. It gives a good historical background in the preface that really helped me understand what it was all about. Once I understood what was actually going on in a general sense, the rest seems relatively straightforward and probably any book would do. I'm still probably going to take a class in it this fall, since I've heard such a class is 'good for the soul'.

 

[eLiu]

Good for the soul eh? haha

I don't think my school offers a class specifically discussing this topic... and if it does, the course descriptions don't make it obvious at all, doh. I know it is surely taught as a tool in some more advanced courses, but I haven't the time to deal with graduate classes just yet

I do have some vague idea of how the calculus of variations works as I've run across it in Whitham's 'Linear and Nonlinear Waves' but I wanna know moooorrrrreeee, lol. I'll see if our library has Lanczos.

sidenote: is this the same Cornelius Lanczos responsible for the lanczos algorithm of numerical linear algebra?

 

[CycloWizard]

Originally posted by: eLiu
Good for the soul eh? haha

I don't think my school offers a class specifically discussing this topic... and if it does, the course descriptions don't make it obvious at all, doh. I know it is surely taught as a tool in some more advanced courses, but I haven't the time to deal with graduate classes just yet

I do have some vague idea of how the calculus of variations works as I've run across it in Whitham's 'Linear and Nonlinear Waves' but I wanna know moooorrrrreeee, lol. I'll see if our library has Lanczos.

sidenote: is this the same Cornelius Lanczos responsible for the lanczos algorithm of numerical linear algebra?

Probably, though I'm not really familiar with that one. I can't imagine there were too many Lanczos guys running around in this time period.

 

[imported_inspire]

Calculus of Variations? What exactly is that? My first instinct was Theoretical Inference, but that doesn't seem right...

 

[eLiu]

wiki!

I think that article is pretty nice. It introduces the main ideas and gives examples that you have probably already seen presented w/o the accompanying math. So there shouldn't be anything too weird/bizarre there.

 

[imported_inspire]

thx. I had never heard of that, but it makes sense. What level of education are you at where such a class is taught?

 

[eLiu]

I'm about to be a junior in college. As far as I can tell, variational methods only really come up in graduate courses at my school (both theoretical & applied classes), though I'm sure that they pop up in undergrad courses at a more elementary level. I'm interested in learning it b/c I have an interest in PDEs (CFD in particular), and this topic is apparently a powerful method of solution.

Also for those interested, Hildebrand's "Advanced Calculus for Engineers" has a gentle (but rather short) introduction to this topic. (And it's an amazing reference to boot--fills a lot of gaps if your fundamentals weren't well taught.)

 

[imported_inspire]

Ah, PDE's. Ew. That makes sense as I always stayed far within the womb of pure math - Better you than me is all I can say.

Thanks for the references.

 

[eLiu]

What part of pure math? Some areas are rife (sp) with PDEs--differential analysis/geometry, some aspects of measure theory, and so on. If real analysis wasn't so d*mn hard for me, I might've gone down that path instead.

 

[imported_inspire]

Analysis was just meh for me. I enjoyed Topology - that was challenging. But I really liked Number Theory, Linear Algebra, and Abstract Algebra as an undergrad - Number Theory was my favorite.

 

[CycloWizard]

Originally posted by: inspire
Calculus of Variations? What exactly is that? My first instinct was Theoretical Inference, but that doesn't seem right...

As eLiu implied, it's useful for solving very complicated equations or series of equations, such as nonlinear coupled PDEs and other things that would make me cry if I had to try to solve them analytically. Instead, one need only identify a suitable optimization criterion for the system at hand, subject to constraints (i.e. boundary conditions). Once the optimization criterion is determined and a working model formulated, the calculus of variations may be used to optimize the solution by changing the solution parameters.

In practice, it is most widely used in solving PDE systems such as the Navier-Stokes equations for momentum, mass, and heat transfer and elasticity theory. For example, the finite element method provides a systematic solution technique that allows the solution of impossibly complex problems. What's more - if the solution space is constructed correctly, the solution determined using this method will be equal to the exact solution of the working model. For simpler problems, one can actually use the solver in Excel to perform the optimization and solution. I've spent the last couple months solving various problems using both techniques and it's pretty incredible what you can do with a relatively simple spreadsheet or finite element software package.

 

[eLiu]

I went to my school library and semi-arbitrarily grabbed a few texts to look at. I tried to go for the ones that weren't overly theoretical (e.g. some people approach it starting with L-measure) & got one with some examples. Hopefully I'll post back in a few days if I liked them.

Btw if you're interested in PDE and haven't read this, you should totally, totally check out "Linear and Nonlinear Waves" by G.B. Whitham. It's awesome.

 

[CP5670]

I got this one a while ago, mainly just because of the price. It seems to be well written and is more accessible than other books on this topic (seems to be geared for physics and engineering students), although I haven't gotten around to really looking through it carefully.

I'm mostly into complex analysis and some related areas myself and will probably end up going into that later on.

 

[darthsidious]

Originally posted by: eLiu
wiki!

I think that article is pretty nice. It introduces the main ideas and gives examples that you have probably already seen presented w/o the accompanying math. So there shouldn't be anything too weird/bizarre there.

So you can wiki link that, but not search for Cornelius Lanczos....

Cornelius Lanczos