Book Review: The Theoretical Minimum

It's hard to imagine a book that seems more tailor-made for me than The Theoretical Minimum. I feel like I have a decent understand of physics at a pop science level but not at a mathematical level (especially when it comes to quantum mechanics), and I'm often frustrated by the lack of depth in general science books. From a math perspective, I know some calculus but have forgotten a lot of the details. The idea of The Theoretical Minimum is to teach the basics of theoretical physics—specifically classical mechanics—to people in that exact boat.

Does it work? Kind of! I don't necessarily feel like I can start "doing physics" now (which is the promise of the title), but I do have a better grasp of what classical mechanics is all about, at least from a mathematical perspective. (Other books in the series cover quantum mechanics and relativity, and I'm looking forward to reading them.)

The emphasis here is very much on the theoretical: there's a lot of math and few real-world examples. The most complex physical system described is a double pendulum, and there's barely a mention of cumbersome real-world phenomena like friction. Instead, it's basically a whirlwind tour of the Newtonian, Lagrangian, and Hamiltonian formulations of classical mechanics.

Newtonian Mechanics

After introducing the basic ideas of states in physics providing some quick calculus refreshers, the book dives into variations of the familiar Newton equations of motion (in a nutshell: "How many ways can you say $F=ma$?") along with related ideas like phase space and energy conservation. What I found most challenging about this section (which was a continuing theme throughout the book) was the mathematical notation—physicists seem to have a penchant for extreme notational brevity and a lot of implicit convention, leading to elegant-but-cryptic formulas like:

$$ \displaylines{\dot{p_i}=F_i\{(x)\}. \\ \dot{r_i}=\frac{p_i}{m}.} $$

That's kind of hard to interpret if you're a physics newbie! (Let me see if I can remember well enough to try: it's something like "the time-derivative of the momentum of a particle equals the force of a particle given the state of all particles in the system; the time-derivative of the position of a particle equals its momentum divided by its mass.") I'd have appreciated a more gradual introduction to the notational conventions, but I guess it's all part of the learning curve for understanding how physicists do their thing.

Despite the notational struggles, I found the basic concepts through Lecture 5 to be mostly intuitive (thanks to vague recollections of high-school-level physics).

Lagrangian Mechanics

Things got more interesting (for me) with the introduction of the Principal of Least Action and Lagrangian Mechanics. The fundamental formulas involved are the definition of Action:

$$ \mathcal{A} = \int_{t_0}^{t_1} L(q,\dot{q})dt $$

And the Principal of Least Action:

$$ \delta \mathcal{A} = \delta \int_{t_0}^{t_1} L(q,\dot{q})dt = 0 $$

Which leads to the Euler-Lagrange Equation:

$$ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} = 0 $$

(Where $q$ means "a coordinate of any kind".)

There's a lot to unpack with those equations, starting with a basic question: what is $L$? It's called a Lagrangian, it's a function related to a physical system, and it's often equal to "kinetic energy minus potential energy" (although not always). But how would one find a Lagrangian in the real world? Unfortunately I'm not really sure—the examples in the book kind of assume you have one lying around.

But once you have a Lagrangian and initial conditions, life is good. You can determine the state of the system at any point of time through Euler-Lagrange equations, and you can translate the system specification into different coordinate systems without too much trouble (which, if I understand right, is the biggest advantage over Newtonian formulations).

What does the Principal of Least Action actually mean? The book's discussion was a little bit hard for me to parse, but if I'm understanding right it's something like "changes in configuration follow a path that always minimizes changes", which is a pretty mind blowing property of the universe. (Side note: Wikipedia informs me that the rate of change is not always minimal, just stationary.) I'm hoping there's more discussion in sequels to The Theoretical Minimum because I'm not sure I fully understand the implications yet.

Hamiltonian Mechanics

After a few chapters in Lagrange-land, the discussion switches over to Hamiltonian Mechanics, which is based on Hamilton's Equations:

$$ \displaylines{\frac{\partial H}{\partial p_i} = \dot{q_i}. \\ \frac{\partial H}{\partial q_i} = -\dot{p_i}.} $$

Where $H$ is another function (the Hamiltonian) defined in terms of the Lagrangian:

$$ H = \sum_{i}(p_i \dot{q_i}) - L $$

The big change is that instead of dealing with configuration space (basically "positions of things"), Hamiltonian Mechanics deals with phase space (where "positions of things" and "momenta of things" are all variables). (In typical Physics Notation fashion, you just have to know that $q$ generally means a coordinate and $p$ generally means a momentum to understand the equations.) The upshot is that if you know the positions and momenta of everything in a system at a point in time, plus the Hamiltonian, you can predict the future of the system forever (using only first-order differential equations).

The book doesn't have much discussion of the big-picture differences between Lagrangian and Hamiltonian Mechanics (like when you would want to use one or the other), although it does allude to the fact that Hamiltonian Mechanics becomes very important in quantum mechanics.

The Rest

After introducing Hamiltonian Mechanics, the book moves onto a grab bag of related topics. Lecture 9 uses the Hamiltonian to discuss Liouville's Theorem, which states "the phase space fluid is incompressible". This was the one section that lost me entirely—I just don't understand what any of that means from a real-world perspective, or what the mathematical implications are.

Then the book introduces the Poisson bracket notational system, which I actually really like (since it comes with a nice set of axioms for algebraic manipulation). After that, there's some (very fast) discussion of angular momentum and electricity and magnetism, but I'll definitely need more practice to feel comfortable with either of those topics.

A Note on the Math

The book includes a fair number of exercises, which is nice because there's a lot of math that's easy to almost-understand but hard to internalize. The exercises ranged in difficulty from "trivial" to "way beyond my (limited) math abilities", but your mileage will vary based on how faded your memory of calculus is. Thankfully there are quite a few sets of solutions online (I found these ones to be particularly helpful).

In terms of the math background you need to get through The Theoretical Minimum, if you've never taken Calculus III you're gonna have a bad time—there are partial derivatives on basically every page. Some background knowledge on differential equations is also useful, although thankfully none of the exercises require solving non-trivial differential equations.

(It's quite funny to me that this book was a bestseller—I can't help but wonder how many of the people who bought it actually made it all the way through.)

The Upshot

The Theoretical Minimum takes on an extremely ambitious—some might say impossible—challenge: introduce technically-minded readers with minimal physics background to classical mechanics in about 200 pages. Despite a few gripes, I'd say it rises to the challenge pretty well. (The companion lectures by Leonard Susskind were helpful for me when I got stuck, since they cover the concepts more slowly.)

If I have one overarching complaint, it's that I wish the authors spent a bit more time discussing "big-picture" ideas before diving into the math. I now understand how one would use a Lagrangian or a Hamiltonian, but I'm not quite sure I have a clear picture of what either of them really means in relation to the physical universe or how one would find them through experiment. (Oh and a side-complaint: a few more practical examples would have helped drill in the concepts.)

Overall, The Theoretical Minimum is a great jumping-off point for a longer amateur physics journey. I fully intend to read the other books in the series (I've already started Quantum Mechanics), and if I have time in the future I hope to dive deeper into classical mechanics (The Structure and Interpretation of Classical Mechanics looks promising).