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.

The job of classical mechanics is to predict the future.

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.

A collection of objects—particles, fields, waves, or whatever—is called a system.

Imagine an abstract object that has only one state. We could think of it as a coin glued to the table—forever showing heads. In physics jargon, the collection of all states occupied by a system is its space of states, or, more simply, its state-space.

The state-space is not ordinary space; it’s a mathematical set whose elements label the possible states of the system.

In classical mechanics we assume that systems evolve smoothly, without any jumps or interruptions. Such behavior is said to be continuous.

So let’s assume that time comes in discrete steps labeled by integers. A world whose evolution is discrete could be called stroboscopic.

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.

One very simple dynamical law is that whatever the state at some instant, the next state is the same. In the case of our example, it has two possible histories: H H H H H H . . . and T T T T T T .

A diagram for the second possible law is shown in Figure 3, where the arrows lead from H to T and from T to H. You can still predict the future. For example, if you start with H the history will be H T H T H T H T H T . . . . If you start with T the history is T H T H T H T H

We can even write these dynamical laws in equation form. The variables describing a system are called its degrees of freedom.

When we are considering a continuous evolution in time, we can symbolize it with t. Here we have a discrete evolution and will use n. The state at time n is described by the symbol σ(n), which stands for σ at n. The value of n is a sequence of natural numbers beginning with 1.

Let’s write equations of evolution for the two laws. The first law says that no change takes place. In equation form, In other words, whatever the value of σ at the nth step, it will have the same value at the next step.

All the basic laws of classical mechanics are deterministic.

The simplest way is to stick to diagrams such as Figure 5. Figure 5 says that given the numerical state of the die at time n, we increase the state one unit at the next instant n + 1. That works fine until we get to 6, at which point the diagram tells you to go back to 1 and repeat the pattern.

Not all laws are logically the same.

According to the rules of classical physics, not all laws are legal. It’s not enough for a dynamical law to be deterministic; it must also be reversible.

The meaning of reversible—in the context of physics—can be described a few different ways. The most concise description is to say that if you reverse all the arrows, the resulting law is still deterministic.

The law of Figure 9 does tell you, wherever you are, where to go next. If you are at 1, go to 2. If at 2, go to 3. If at 3, go to 2. There is no ambiguity about the future. But the past is a different matter. Suppose you are at 2. Where were you just before that? You could have come from 3 or from 1. The diagram just does not tell you.

There is a very simple rule to tell when a diagram represents a deterministic reversible law. If every state has a single unique arrow leading into it, and a single arrow leading out of it, then it is a legal deterministic reversible law.

The rule that dynamical laws must be deterministic and reversible is so central to classical physics that we sometimes forget to mention it when teaching the subject.

We could call it the first law, but unfortunately there are already two first laws—Newton’s and the first law of thermodynamics. There is even a zeroth law of thermodynamics.

what is undoubtedly the most fundamental of all physical laws—the conservation of information. The conservation of information is simply the rule that every state has one arrow in and one arrow out. It ...

There is no reason why you can’t have a dynamical system with an infinite number of states. For example, imagine a line with an infinite number of discrete points along it—like a train track with an infinite sequence of stations in both directions.

When the state-space is separated into several cycles, the system remains in whatever cycle it started in.

Whenever a dynamical law divides the state-space into such separate cycles, there is a memory of which cycle they started in. Such a memory is called a conservation law; it tells us that something is kept intact for all time.

All of the things that we discussed for simple discrete systems have their analogs for the more realistic systems but it will take several chapters before we see how they all play out.

Laplace may have been overly optimistic about how predictable the world is, even in classical physics. He certainly would have agreed that predicting the future would require a perfect knowledge of the dynamical laws governing the world, as well as tremendous computing power—what he called an “intellect vast enough to submit these data to analysis.”

there is another element that he may have underestimated: the ability to know the initial conditions with almost perfect precision.

In the real world, it’s even worse; the space of states is not only huge in its number of points—it is continuously infinite. In other words, it is labeled by a collection of real numbers such as the coordinates of the particles.

Real numbers are so dense that every one of them is arbitrarily close in value to an infinite number of neighbors.

In principle we cannot know the initial conditions with infinite precision. In most cases the tiniest differences in the initial conditions—the starting state—leads to large eventual differences in outcomes. This phenomenon is called chaos.

Perfect predictability is not achievable, simply because we are limited in our resolving power.

To describe points quantitatively, we need to have a coordinate system. Constructing a coordinate system begins with choosing a point of space to be the origin.

For example, the theory of the solar system would look more complicated if we put the origin anywhere but at the Sun.

the location of the origin is arbitrary—put it anywhere—but once it is chosen...

We want to describe a certain point in space; call it P. It can be located by giving the x, y, z coordinates of the point. In other words, we identify the point P with the ordered triple of numbers (x, y, z) (see Figure 2

The x coordinate represents the perpendicular distance of P from the plane defined by setting x = 0 (see Figure 3). The same is true for the y and z coordinates.

When we study motion, we also need to keep track of time. Again we start with an origin—that is, the zero of time.

The usual convention is that positive times are to the future of the origin and negative times are to the past. We could do it the other way, but we won’t.

Seconds are the physicist’s customary units, but hours, nanoseconds, or years are also possible. Once having picked the units and the origin, we can label any time by a number t.

There are two implicit assumptions about time in classical mechanics. The first is that time runs uniformly—an interval of 1 second has exactly the same meaning at one time as at another.

The other assumption is that times can be compared at different locations. This means that clocks located in different places can be synchronized.

Given the function f(t) = t2, we can plot the points on a coordinate system. We will use one axis for time, t, and another for the function, f(t)

We use trigonometry in physics all the time; it is everywhere.

To begin with, in physics we do not generally use the degree as a measure of angle. Instead we use the radian; we say that there are 2π radians in 360°, or 1 radian = 180° / π,

Thus a radian is about 57° (se...

The greek letter theta, θ, is defined to be the angle opposite the altitude, and the greek letter phi, φ, is defined to be the angle opposite the base.