Calculus Gelfand & Fomin vs. Lanczos to learn Calculus of Variations

[ian_dsouza]

I am learning the Lagrangian formalism from Landau & Lifshitz but I'm not very familiar with variational calculus. Landau assumes its knowledge and uses it directly. Although the equations look analogous to what you'd do with ordinary calculus, I'd like to understand the foundation and ideas behind variational calculus before I continue with Landau's book.

I am looking for a book on the Calculus of Variation and have searched this forum a bit. I have boiled it down to Gelfand & Fomin and Lanczos. I want to work towards a strong mathematical foundation to continue with Landau's treatment of the principle of least action. Which one would you recommend? Also, any good video lectures on the topic?

 

[Dr.D]

I have both books, have used them both, and don't really like either one very well. I think I would lean slightly in favor of Gelfand & Fomin.

One of the best presentations, at least for starters, is an old, 2 vol set, by Doherty & Keller, written for the GE advanced course in the 1930s. The title was roughly "Advanced Engr Mathematics," but I'm not too sure on the exact title. This is where I would recommend you start.

 

[verty]

I'd like to understand the foundation and ideas behind variational calculus before I continue with Landau's book.

This book looks nice and is very cheap indeed, probably worth getting anyway: https://www.amazon.com/dp/0486450341/?tag=pfamazon01-20. Certainly it seems to cover what the ideas are or what the purpose is.

I want to work towards a strong mathematical foundation to continue with Landau's treatment of the principle of least action.

I think that book would help because it is an optimization problem and if you take the point of view that an integral is the name of a problem, which I always found to be most sensible, this book is giving you the necessary background.

 

[tom26]

The benefit of Gelfand and Fomin is that everything is rigorous. It is presented clearly, and there's really very little uncertainty in what they are saying. You'll read the first 100 pages and know all you need to know about the Calculus of Variations. It is a lovely book.

I read Lanczos a few years ago, and I found that, while he does develop the calculus of variations, that's more of a side-goal for him. I had the feeling while reading that the book demanded more physics maturity than I had at the time (where I define physics maturity as the ability to fill in the mathematical details from a physical argument.) There's lots of subtleties in classical mechanics (and the calculus of variations for that matter) and I feel that (perhaps counter-intuitively) these should first be explained rigorously (so you know where you stand), and only later should the simpler, intuitive ways of thinking about them be introduced (which you find in Lanczos in spades).

So if all you want is CoV, get G&F. But for a more historical and intuitive perspective, you could Lanczos too (they're both cheap!)