The Theoretical Minimum: What You Need to Know to Start Doing Physics

by Leonard Susskind

The term classical physics refers to physics before the advent of quantum mechanics. Classical physics includes Newton’s equations for the motion of particles, the Maxwell-Faraday theory of electromagnetic fields, and Einstein’s general theory of relativity. But it is more than just specific theories of specific phenomena; it is a set of principles and rules—an underlying logic—that governs all phenomena for which quantum uncertainty is not important. Those general rules are called classical mechanics.

In classical physics, if you know everything about a system at some instant of time, and you also know the equations that govern how the system changes, then you can predict the future. That’s what we mean when we say that the classical laws of physics are deterministic. If we can say the same thing, but with the past and future reversed, then the same equations tell you everything about the past. Such a system is called reversible.

A system that changes with time is called a dynamical system. A dynamical system consists of more than a space of states. It also entails a law of motion, or dynamical law. The dynamical law is a rule that tells us the next state given the current state.

The fundamental forces are those that act between particles, like gravity and electric forces. These depend on a number of things: Gravitational forces between particles are proportional to the product of their masses, and electric forces are proportional to the product of their electric charges.

The quantity T – V is called the Lagrangian of the system, and it’s denoted by the symbol L. The things you need to know to specify L are the mass of the particle (for the kinetic energy) and the potential V(x). It is, of course, no accident that these are the same things you need to know to write Newton’s equation of motion.

A function of a function—a quantity that depends on an entire function—is called a functional. Minimizing a functional is the subject of a branch of mathematics called the calculus of variations.

There are two primary reasons for using the principle of least action. First, it packages everything about a system in a very concise way. All the parameters (such as the masses and forces), and all the equations of motion are packaged in a single function—the Lagrangian. Once you know the Lagrangian, the only thing left to specify is the initial conditions. That’s really an advance: a single function summarizing the behavior of any number of degrees of freedom.

Is one formulation better than the other? Did Lagrange have the final word or did Hamilton? You can decide for yourself, but wait a while before you do. We still have a couple of courses on relativity and quantum mechanics before the real meanings of the Lagrangian and Hamiltonian become completely clear.