Prerequisites for Landau and Lifshitz

Question

I am a first year maths student with a very strong interest in theoretical physics. I am fluent in Russian and I know that in Russian universities like MPTI Landau and Lifshitz's course in theoretical physics is still used to teach; what prerequisites would I need in order to fully read it, and complete the 'theoretical minimum'? My goal is to first confidently complete the 'Mathematics I' section of the theoretical minimum. Any advice on books, resources, or a list of topics I should confidently understand (and a corresponding order) would be much appreciated. The theoretical minimum I am referring to is Landau and Lifshitz's list, not Susskind's book.

Thank you!

Answer 1

Fair warning: Landau's books are NOT easy at all, and I honestly believe there's no point reading Landau until after (atleast midway through) your second year in mathematics, because otherwise you will be completely unprepared. I've only read volume 1 on Mechanics (this I read very closely) and I've read parts of Volume 2 (Classical Theory of Fields) and Volume 8 (Electrodynamics of Continuous Media), so I can only comment on these.

First of all, the Mathematics I syllabus mentions "integration, ODEs, vector algebra, tensor analysis". This is pretty vague, so let me elaborate on what is needed to understand Volume I of mechanics.

Volume 1 Mechanics.

You have to know basic vector algebra: dot products, cross products, vector addition, etc. Again, this is very easily learnt online or in a linear algebra course, and everything here is in 3-dimensions.

You need a very good understanding of single variable calculus: the meaning of differentiation, integration, all the rules of differentiation and integration, you need to know trigonometry, and at times a bit of the hyperbolic trig functions, and so on (so you have to be proficient with all the "basic" algebraic manipulations and trig identities). You also need to know the basic series expansions eg for $\sin,\cos,\arctan,\ln(1+x), (1+x)^{\alpha}$, Taylor series and Taylor's theorem etc. Also, you need to know about improper integrals.

Next, you need a very firm understanding of multivariable differential calculus, know how to calculate partial derivatives, chain rule, all that stuff. Towards the later chapters (eg chapter 7 on Hamiltonian mechanics) you will also need the knowledge of the inverse/implicit function theorems from multivariable calculus. You will also need to know how to carry out some surface/volume integrals in order to compute things like center of mass, moment of inertia etc.

I already mentioned this above, but you need to be proficient with indefinite integration. Also, you need to know how to solve ODEs (particularly second order linear ODEs). Actually, for ODEs I think just the basics of second order linear ODEs will do (eg the kind which you can read up online/learn from Khan Academy or wherever, which can be learnt in about 2-3 days), because I actually found that I learnt more about solving ODEs from Landau's book than from any of the books devoted purely to solving ODEs.

All of the material I've mentioned above is stuff you will cover in the first two years of a typical math degree. The one mathematically advanced topic you will need to even appreciate Landau and Lifshitz is the calculus of variations. They start off right away in the first few pages with the Lagrangian formalism. This is actually nothing but differential calculus when the domain is an infinite-dimensional vector space, so if you've learnt multivariable calculus from a decent book, this extra step shouldn't be too hard to get a hang of.

The above is the elaboration of topics (listed in order of prerequisites). The one tweak you can make is that ODEs can be learnt after some single-variable calculus; you don't need multivariable calculus for this. Of course, as you learn this mathematical material you should simultaneously learn the basics of Newtonian classical mechanics using calculus. Without this, I think it is impossible to appreciate the Lagrangian viewpoint.

Before I continue, here's my personal suggestion: don't be in a hurry to start Landau and Lifshitz ASAP. Of course, you should definitely stay curious and skim through the book to see what kinds of things are in there, but don't expect to learn much without the proper background. Also, since you're interested in theoretical physics, I think it is highly advisable that you learn mathematics as a pure math major would learn it. I can't tell you the number of times having a proper math foundation has helped me understand/fill in some of Landau's vague arguments (lol I call them vague for the lack of a better word... maybe Landau finds some things too trivial to even be elaborated on) and also to clarify some issues more rigorously.

Up to this point, there's no need at all for any tensor analysis. Volume 2 is essentially special relativity and electrodynamics. Note that this is NOT AT ALL a good place to start learning the material. You should definitely learn the basics of electromagnetism from a book like Griffiths (where you can also learn the basics of vector calculus). In this book, there is a lot more calculus of variations. At some points you will need things like Fourier Analysis, but this is the sort of thing which you can learn as you go along. One thing I should mention is that in this volume you will need to start learning tensor analysis, and that although Landau has a short 1 chapter introduction on it, I think it is absolutely terrible for actually learning the subject. If you're really serious about understanding some classical field theory and general relativity, you're better off spending some time to learn differential geometry as a mathematician would.

Finally, I should mention that Leonard Susskind has a couple of lecture series, including classical mechanics and another on special relativity and electromagnetism, and I think watching those before/concurrently would be very beneficial.

Answer 2

Well, there are like 10 books. I went through perhaps 6 of them with varying degrees of profundity, I'm most familiar with the first three ones, specially the course Mechanics, so I will answer based on what I know about the course. You won't really need analysis and other formal courses, but you must be comfortable with multivariable and vector calculus, and also series expansions (specially power series and Fourier), they'll be used a lot. If I recall correctly, there is also some complex analysis, with things like contour integration. It's a course on physics, so you will encounter partial and ordinary differential equations all the time, and the authors will sometimes trust you can solve them by yourself. There's also some group theory (both continuous and discrete groups appear) in the QM courses, and I think in other volumes as well. When a result from linear algebra is used, Landau will almost always assume you know it, they usually don't explain steps from LA, specially in calculations involving matrices or differential operators. Of course, "basic" math like plane geometry, trig, hyperbolic functions, etc is also needed. They will also use "special" functions like Elliptic Integrals, but I think it is easier to look them up as they appear.

Overall, the book is mostly self-contained though. They use results from many other areas of mathematics, like Fourier and functional analysis, but these things are always developed in their pages (this may be true for group theory as well, I think, although I rememember feeling the need to look for other sources on this topic), although not with a rigor that would come close to satisfy a mathematician. Technically, you don't need any physics, for they will start from the basics. But really, you should have had at least a basic course on the subject of the book you're currently reading. It will be very difficult to understand Lagrangian mechanics without Newton, for example, although it could be done in principle.