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

by Leonard Susskind

It is easy to see that the trajectories of the oscillator in phase space are concentric circles about the origin.

Especially interesting is the fact that the angular velocity in phase space is the same for all orbits, independent of the energy of the oscillator.

the fact that the phase point stays on a contour of constant energy is universal.

Focusing on a particular initial condition and following it along its specific trajectory through phase space are very natural things to do in classical mechanics.

Instead of putting your pencil down at a point in phase space and then following a single trajectory, try to do something more ambitious. Imagine you had an infinite number of pencils and used them to fill phase space uniformly with dots (by uniformly, I mean that the density of dots in the q, p space is everywhere the same). Think of the dots as particles that make up a fictitious phase-space-filling fluid.

In Lecture 8 we saw that each dot moves in a circular orbit with uniform angular velocity. (Remember, we are talking about phase space, not coordinate space. In coordinate space, the oscillator moves back and forth in one dimension.)

In other words, there is a surface for each value of E; as you scan over values of E, those surfaces fill up the phase space. You can think of the phase space, along with the surfaces defined in Eq. (2) as a contour map (see Figure 1), but, instead of representing altitude, the contours denote the value of the energy.

For a general mechanical system, the energy surfaces are far too complicated to visualize, but the principle is the same: The energy surfaces fill the phase space like layers and the flow moves so that the points stay on the surface that they begin on.

What is it that characterizes an allowable law? The answer is that every point must have exactly one incoming arrow and one outgoing arrow. If at any point the number of incoming arrows exceeds the number of outgoing arrows (such a situation is called a convergence), then the law is irreversible. The same is true if the number of outgoing arrows exceeds the number of incoming arrows (such a situation is called a divergence). Either a convergence or divergence of the arrows violates reversibility and is forbidden.

Forget about phase space for the moment, and just consider an ordinary fluid moving through regular three-dimensional space labeled by axes x, y, z. The flow can be described by a velocity field.

Or we may describe the velocity field to be the components of the velocity: vx(x, y, z), vy(x, y, z), vz(x, y, z). The velocity at a point might also depend on time, but let’s suppose that it doesn’t.

Now let’s suppose the fluid is incompressible. This means that a given amount of the fluid always occupies the same volume.

By the way, the term incompressible also means indecompressible. In other words, the fluid cannot be stretched out, or decompressed.

The divergence is aptly named; it represents a spreading out of the molecules, or an increase in the volume occupied by the molecules.

One way to think about incompressibility is to imagine that each of the molecules, or points, of the fluid occupies a volume that cannot be compromised. They cannot be squeezed into a smaller volume, nor can they disappear or appear from nowhere. With a little bit of thought, you can see how similar incompressibility is to reversibility.

The obvious question that this raises is whether the flow through phase space is incompressible. The answer is yes, if the system satisfies Hamilton’s equations. And the theorem that expresses the incompressibility is called Liouville’s theorem.

The concept of a divergence in Eq. (4) is easily generalized to any number of dimensions. In three dimensions it is the sum of the derivatives of the velocity components in the respective directions.