Reddit reviews The Variational Principles of Mechanics (Dover Books on Physics)

I recommend the following book on the subject: The Variational Principle Of Mechanics which elaborates on the relationship between the two views much more effectively than I can.

Yes, it is definitely focussed on variational calculus, but I still found it highly readable. It is also far from out of print: it's available as a Dover paperback:
http://www.amazon.com/Variational-Principles-Mechanics-Dover-Physics/dp/0486650677

Anyone had any first hand experience with Lancazos' Variational Principles of Mechanics?. I'm almost through Landau's Mechanics and was interested in learning more about the action principle, although I don't have any background in the calculus of variations and such.

The Variational Principles of Mechanics by Lanczos is an amazing book for understanding calculus of variations. The majority of it covers ODEs rather than PDEs / field equations, but to be honest the book is so good that the generalization to field theory is almost obvious. It does have a chapter or two on fields though. The book has the most beautiful economy of words I've ever seen in a textbook, concise and yet crystal clear. Also, the book is cheap! Just $16 at Amazon right now. It's definitely written for physicists, it's not a math book at all.

I can't say enough good things about this book. Reading it was the first time I understood calculus of variations. He actually explains what you are doing conceptually when you vary a path, whereas I feel like most physics books introduce it solely as a mathematical manipulation. I finally gained a good intuition for it.

My introduction to calculus of variations in field theory came through classical electrodynamics in Landau & Lifshitz and Jackson. I agree that those books don't tell you at all how it works; they just start performing manipulations and you just follow what they do.

I can't recommend much in the way of math books, but I can give you some more hints on what you should be looking for and reading about.

The specific problem that you've asked about isn't quite undergrad level material, unfortunately. Here's a document that introduces most of the relevant topics:

http://personal.lse.ac.uk/sasane/ma305.pdf

I can't guarantee that it will be helpful, but everything in that document is relevant to what you're looking for. You might use the table of contents or the introductory section to prime your wikipedia searches.

Basically, you specifically want to learn about optimization theory (which is what a lot of control theory is about).

Optimization, at its most basic, is not hard. An example of an optimization problem is to find the value of x that minimizes (or maximizes) some function f(x).

This is something that's covered in basic calculus. If f(x) has only one minimum/maximum (aka an extremum), then you can solve the problem easy by solving the equation df(x)/dx = 0.

Things get harder when you have constraints - maybe you want the value of x such that a<x<b, that minimizes f(x). In that case you use things like lagrange multipliers and the KKT conditions, which allow you to deal with constraints on your solutions.

I linked to wikipedia there, but the wiki pages aren't necessarily the best resources for learning this stuff. If you search for those things, though, you'll find a lot of good resources, because many people are in the same boat as you about this stuff.

Your problem is a bit trickier than basic optimization, though - in your case, you're trying to minimize a functional, which is a function that takes another function as an argument, and returns a number. The solution to that problem is a function. So, instead of using regular calculus and finding where the derivative is zero, you use variational calculus and find where the derivative is zero.

I can't recommend a specific book about this subject, but I can recommend a book that's very closely related: The Variational Principles of Mechanics. It's an excellent physics book that explains things in terms of variational calculus. The principle of least action is one way of solving physics problems, and it's very similar to the problem that you're asking about.

There seems to often be this sort of tragedy of the commons with the elementary courses in mathematics. Basically the issue is that the subject has too much utility. Be assured that it is very rich in mathematical aesthetic, but courses, specifically those aimed at teaching tools to people who are not in the field, tend to lose that charm. It is quite a shame that it's not taught with all the beautiful geometric interpretations that underlie the theory.

As far as texts, if you like physics, I can not recommend highly enough this book by Lanczos. On the surface it's about classical mechanics(some physics background will be needed), but at its heart it's a course on dynamical systems, Diff EQs, and variational principles. The nice thing about the physics perspective is that you're almost always working with a physically interpretable picture in mind. That is, when you are trying to describe the motion of a physical system, you can always visualize that system in your mind's eye (at least in classical mechanics).

I've also read through some of this book and found it to be very well written. It's highly regarded, and from what I read it did a very good job touching on the stuff that's normally brushed over. But it is a long read for sure.

I don't want to top post since you've pretty much answered this. I'd like to add a book suggestion on the topic for OP or any others who would like to better understand what you're describing. It's cheap, and explains things quite well. I'm halfway through it myself.

You might appreciate this book https://www.amazon.com/Variational-Principles-Mechanics-Dover-Physics/dp/0486650677

Any suggestions on how to approach high-level physics without a formal math background?

I am an engineer with an academic concentration in signals processing and a minor in physics, so I do have a strong quantitative background. However, my training was heavily slanted towards ad-hoc problem solving rather than rigorous analysis, so I find myself lost as I tackle topics grounded in formal mathematics.

Specifically, I have been reading Lanczos' The Variational Principles of Mechanics, a popular analytical mechanics text, with great difficulty.

Is it worth reading a pure math book on differential geometry or something similar? How do most graduate students study advanced physics, when an undergraduate physics education doesn't use much math beyond basic PDEs?

If you look on Amazon, there are a lot of inexpensive physics texts put out by Dover - for example, analytical mechanics or E&M. They're so cheap that I usually pick these up to supplement whatever text is recommended for each of my courses.

This poster is also baller as shit.

This should be at the top, as it really helps to rephrase the question. To anyone else interested: energy conservation is a direct consequence of an "action principle" combined with a symmetry, so the question should instead be,

"Why is nature so well-described by action principles, and why is physics invariant if you shift the time coordinate?"

This is stuff that physicists still debate and try to understand, so you're not going to get a definitive answer.

But you can get some intuition about why these things should be true. For the second question, it's pretty natural to think of all times as being "equivalent" in the eyes of physical law. Then it makes sense that shifting in time should be a symmetry.

Unfortunately, the former question is a little harder to dig if you're five. If anybody is interested: one way to understand it is to get a physical intuition for D'Alembert's principle. Doing some work then gets you to Hamilton's principle, which is what the first question is talking about; see e.g. chapter 5.1 of this book for a derivation.