The theoretical minimum of physics

The theoretical minimum.

What you need to know to start doing physics

by Susskind and Hrabovsky, 2013

This crash course of classical mechanics is targeted at those who “regretted not taking physics at university” or who perhaps did but have forgotten most of it, or anyone who is just curious and wants to learn how to think like a physicist. Since first year university physics courses usually have rather high drop out frequencies, there must be some genuine difficulties to come over. Instead of dwelling on the mind-boggling paradoxes of quantum mechanics and relativity as most popular physics books do, and wrapping it up in fluffy metaphores and allusions to eastern philosophy, the theoretical minimum offers a glimpse of the actual calculations and their theoretical underpinnings in classical mechanics.

This two hundred page book grew out of a series of lectures given by Susskind, but adds a series of mathematical interludes that serve as refreshers on calculus. Although covering almost exactly the same material as the book, the lectures are a good complement. Some explanations may be more clear in the classroom, often prompted by questions from the audience. Although Susskind is accompanied by Hrabovsky as a second author, the text mysteriously addresses the reader in the first person singular.

A typical first semester physics text book may cover less theory than the theoretical minimum in a thousand pages volume, although it would probably cover relativity theory which is not discussed in this book. There are a few well chosen exercises in the theoretical minimum, some quite easy and a few that take some time to solve. “You can be dumb as hell and still solve the problem”, as Susskind puts it in one of the lectures while discussing the Lagrangian formulation of mechanics versus Newton's equations. That quote fits as a description of the exercises too, as many of them can be solved without really gaining a solid understanding of how it all works.

The book begins by introducing the concept of conservation of information and how it applies to deterministic, nonreversible systems (all systems considered in classical mechanics are deterministic and nonreversible). Halfways through the book the first more advanced ideas come into play: the Lagrangian and the principle of least action. In general, one gets an idea of what kinds of questions physicists care about, such as symmetries and conservation laws. Examples of symmetries that are discussed include spatial translation invariance and time shift invariance, and the conservation of energy is a recurrent theme. The trick is simple: take a time derivative of the Lagrangian or the Hamiltonian, and show it to be zero. The principle of least action requires more sophisticated mathematics (functional analysis), although the authors try to explain it in very simple terms. Nonetheless, that part is not very easy to follow.

The writing is concise, yet almost colloquial, with only a few typos. Mathematical rigour is thrown out whenever it would clutter the exposition. Susskind does not care for limits in the formulation of derivaties, but uses a delta or an epsilon that is supposedly infinitesimal in a loosely nonstandard analysis kind of way. Most derivations are easy to follow, using elementary calculus and patiently laid out step by step. Some background in one variable and vector calculus will be necessary to follow the text, although all math that is needed (which is not very much) is summarized in the mathematical interludes.

Why should we need to know about Lagrangians, Hamiltonians and Poisson brackets, a student may ask. Susskind's answer might be that Lagrangians make the solution of certain problems much easier than trying to apply Newton's equations, and that Hamiltonians play an important role in quantum mechanics.

The theoretical minimum is probably the most concise introduction to advanced physics out there, highly suitable for self-study. It provides much of the essential background needed for books reviewed here in previous posts, such as Steeb's Nonlinear Workbook or Haken's Synergetics.