I was so exciting to find a series of lecture videos on YouTube about Classical Physics by Professor Alexander Polyakov early this year. From the course website here or here, you can find the description of this course:

There are 22 lectures, each about 1 hour 20 minutes. The course can be regarded as a second course on Classical Mechanics. I started working on this course from April and I must say ** I have had an amazing experience by now, finishing about half of them**. I highly recommend this lecture series to anyone who is looking for a second, more advance discussion about classical mechanics.

I list some additional resources about this course I found online or I made myself,

- Descriptions of each lecture in this old website of Princeton (may not work properly) or this pdf file I made for my personal convenience.
- My personal notes (keep updating…).

## My personal history leading to this course

My previous knowledge on this topic is mostly from John Taylor’s textbook, Classical Mechanics, which is a very good for introduction. Later, I jumped right into advanced course on quantum field theory (QFT), quantum many-body physics, statistical field theory, and conformal field theory, etc., after coming here in UTokyo pursuing my PhD in theoretical physics.

I had much difficulty understanding and grasping these advanced courses. After much failure, I realized I should go back to more basic stuffs, as indicated in the above classic lines in Jiang Wen’s movie, Let the Bullets Fly. The first thing I realized is that although I hear and speak the words “Hamiltonian” and “Lagrangian” all the time, I almost know nothing deep about these concepts. I know I should learn this in a classical mechanics course, but the introductory one like Taylor’s textbook does not go deep enough. Last year, I finished a short PSI lecture, see my previous post or my 2022 summary on physics. The course talks much advanced formalism in the mathematical language called differential geometry. A complementary physical approach would be perfect. As a result, when I saw the course description of Polyakov’s course, I immediately know I probably met the right one!

## First half of this lecture series: Polyakov's style

Setting first aside the content of the course, Polyakov’s teaching style is just so unique and matches my preference perfectly! I believe his style is very close to * Socratic questioning *approach. An analogy would be that attending ordinary lecturers are like watching a movie, but

*Socratic-questioning-style lectures bring you into the movie*! I can very intensely feel Polyakov’s passion and joy on the content he is teaching! I’m not sure whether this style comes from Landau School. I suspect it is, since I feel the same kind of passion in this 2012 prose by Alexender Migdal, Polyakov’s best friend.

This style of teaching is like a unicorn. Recently, during a summer school in Sweden (see my previous post), Professor Artur Ekert is such a unicorn. Around early 2019, I watched a series of PSI lectures by Carl Bender about asymptotic expansion, he is also such a unicorn.

## Course contents are even more exciting and interesting

Polyakov is a great physicist. He is one of the pioneers of conformal field theory, string theory, critical phenomena, and so on. In this lecture series, he shared lots of key ideas and techniques that permeate all modern theoretical physics, including

- Symmetry of dynamics,
- Action principle (or more broadly, variational principle) as a unifying framework
- Hamilton’s description of dynamics: its motivation and geometry view
- Low-energy effective description: from discrete to continuum
- Spontaneous symmetry breaking

Each idea and topic he covers are supported by numerous motivations, logically or historically. Learning the history of a subject or concept is a unique experience in this lecture series.

## My new understanding about fluid dynamics

To end this post, I want to talk about fluid dynamics. I have failed to find motivations learning fluid dynamics for a long time. In my previous Engineering training, fluid mechanics simply means applying the Bernoulli’s principle (conservation of energy, basically) to flow in a pipe. It seems very trivial. However, after finishing watching the first 14 lectures of Polyakov, I realized the power of fluid dynamics and its position in modern physics.

Whenever the first-principle microscopic description of a system involves too many particles and is too complicated, you have to find a way to simplify the description: a low-energy effective theory. Fluid dynamics is such an effective theory. When the microscopic details are not important, many seemingly difficult physical system can be equally-well described by fluid dynamics. This means Fluid dynamics has a very wide range of applications. For example, recently I realize all the following systems can be analyzed using fluid mechanics:

- For very large scale, the formation of galaxy and expanding universe.

When you add gravity in your fluid mechanics equation of motion, you will discover instability in the linearized equation, called Jeans instability. See Polyakov’s Lecture 11 and 12 for an excellent discussion. What’s more, you can also see the expanding universe solution in this equation of motion called Friedmann equations**. **Although Friedmann equations are derived historically after Einstein’s general relativity, it is possible to derive it just using Newtonian gravity. I was astonished to know this when watching Professor Susskind’s lecture on cosmology. Polyakov discussed this in fluid dynamics context in his Lecture 12.

- For very small scale, quark-gluon plasma.

This is the physics inside a nucleus. I learned this from two lectures in the Quantum Connects summer school, one by Barbara Jacak and the other by Krishna Rajagopal.

- For ordinary scale physics: why plane can fly, jets of plane, explosion of a bomb, tornado, or the motion of superfluidity of helium under low temperature (a very quantum behavior).

Sadly, it is very hard to find an excellent modern textbook to learn this stuff. Two possible choice might be Landau and Lifshitz or *Modern Classical Physics* by Thorne and Blandford.

Thank You for the notes