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

by Leonard Susskind

the particle is left where it was (if it was at x = 1, then the new labeling puts it at x' = 2); only the label has changed.

In the second way of thinking about coordinate transformations, which is called active, you don’t relabel the points at all. The transformation x' = x + 1 is interpreted as an instruction: Wherever the particle is, move it one unit to the right.

A symmetry is an active coordinate transformation that does not change the value of the Lagrangian. In fact, no matter where the system is located in the configuration space, such a transformation does not change the Lagrangian.

In Eq. (5) the quantity δ can be any number. Later, when we consider transformations by infinitesimal steps, the symbol δ will be used to represent infinitesimal quantities, but for now it doesn’t matter.

Unless the potential is a constant independent of q, then the Lagrangian will change as q is shifted. In that case there is no symmetry.

The symmetry of moving a system in space by adding a constant to the coordinates is called translation symmetry, and we will spend a lot of time discussing it.

Again this is a case of translation symmetry, but in this case, to have a symmetry we must translate both particles so that the distance between them is unchanged.

If the potential is a function of a more complicated combination, it is not always clear that there will be a symmetry.

Since the potential is a function only of the distance from the origin, it doesn’t change if the system is rotated through an angle. Moreover, the kinetic energy is also unchanged by a rotation.

Now we come to an essential point about the transformations of translation and rotation. You can do them in small steps—infinitesimal steps.

Transformations like this are called continuous: They depend on a continuous parameter (the angle of rotation), and, moreover, you can make the parameter infinitesimal.

Since finite transformations can be compounded out of infinitesimal ones, in studying symmetries it’s enough to consider transformations with very small changes in the coordinates, the so-called infinitesimal transformations.

each coordinate shifts by an amount proportional to δ, but the proportionality factor depends on where you are in configuration space.

A continuous symmetry is an infinitesimal transformation of the coordinates for which the change in the Lagrangian is zero.

It is particularly easy to check whether the Lagrangian is invariant under a continuous symmetry: All you have to do is to check whether the first order variation of the Lagrangian is zero.

First of all, by definition, symmetry means that the variation of the Lagrangian is zero.

What Eq. (17) states is that a certain quantity, (18) does not change with time. In other words, it is conserved. The argument is both abstract and powerful. It did not depend on the details of the system, but only on the general idea of a symmetry.

But now we can say a far more general thing: For any system of particles, if the Lagrangian is invariant under simultaneous translation of the positions of all particles, then momentum is conserved.

Nothing in the laws of physics changes if everything is simultaneously shifted in space.

For any system of particles, if the Lagrangian is invariant under simultaneous rotation of the position of all particles, about the origin, then angular momentum is conserved.

Our goal will not be to solve the equations of motion. That we can always do, even if we have to put them on a computer and do it numerically. The goal is to find those equations.

Should I choose the second angle (the angle of the second rod) also to be measured from the vertical, or should I measure it relative to the angle of the first rod? The answer is that it does not matter. One choice may make the equations a little simpler, but either will get you to the answer.

Finding symmetries is not always a mechanical procedure; you may have to do some pattern recognition.

For instance, a translation is a symmetry that simultaneously shifts the Cartesian coordinates of all the particles in a system by the same amount.

Time-translation invariance does not always apply. For example, we live in an expanding universe. The effect of the expansion on ordinary laboratory experiments is usually negligible, but it’s the principle that counts.

Here is a more down-to-earth example. Suppose the system of interest is a charged particle moving in a magnetic field. If the magnetic field is constant then the motion of the particle will be time-translation invariant. But if the current that generates the field is being slowly increased, then the same initial condition for the particle—but starting at different times—will result in a different outcome.

How is time-translation symmetry, or the lack of it, reflected in the Lagrangian formulation of mechanics? The answer is simple. In those cases where there is such symmetry, the Lagrangian has no explicit dependence on time.

The value of the Lagrangian may vary with time, but only because the coordinates and velocities vary.

Explicit time dependence means that the form of the Lagrangi...

For example, if the experiment took place in a changing magnetic field, this could have a subtle effect on the atoms of the spring, which in turn could cause k to vary.

A system is time-translation invariant if there is no explicit time dependence in its Lagrangian.

There are three sources of time dependence of L. The first and second are due to the time dependence of the coordinates q and the velocities

The upshot is that there is no such thing as conservation of the Lagrangian.

An even more interesting way to say it is if a system is time-translation invariant, then the quantity H is conserved.

Notice what happens: Two terms proportional to combine to give the usual kinetic energy, and the potential term becomes + V(x). In other words, H just becomes the usual total energy, kinetic plus potential.

As long as the field is constant, the energy is conserved. But suppose the capacitor is being charged up so that ∊ is also ramping up. Then the Lagrangian has an explicit time dependence:

Where did that energy come from? The answer is that it came from the battery that was charging the capacitor. I won’t go into details, but the point is that when we defined the system to consist of just the particle, we narrowed our focus to just a part of a bigger system that includes the capacitor and the battery.

Consider the entire experiment, including the battery, capacitor, and particle. The experiment begins with an uncharged capacitor and a particle at rest, somewhere between the plates. At some moment we close a circuit, and current flows into the capacitor. The particle experiences a time-dependent field, and, at the end of the experiment, the capacitor is charged and the particle is moving. What if we did the entire experiment an hour later? The outcome, of course, would be the same. In other words, the entire closed system is time-translation invariant, so the entire energy of all items is conserved.

Nevertheless, it is often useful to divide a system into parts and to focus on one part. In that case, the energy of part of the system will not be conserved if the other parts are varying with time.

The Hamiltonian is important because (among other reasons) it is the energy. But its significance is far deeper: It is the basis for a complete revamping of classical mechanics, and it is even more important in quantum mechanics.

What do we gain by doubling the number of dimensions? The answer is that the equations of motion become first-order differential equations.

In less technical terms, this means that the future is laid out if we know only the initial point in phase space.

The first step in constructing the Hamiltonian formulation is to get rid of the and replace them with the p's. The goal is to express the Ha...

For particles in ordinary Cartesian coordinates, the momenta and velocities are almost the same thing, differi...

If at any time you know the exact values of all the coordinates and momenta, and you know the form of the Hamiltonian, Hamilton’s equations will tell you the corresponding quantities an infinitesimal time later. By a process of successive updating, you can determine a trajectory through phase space.

The harmonic oscillator is by far the most important simple system in physics. It describes all sorts of oscillations in which some degree of freedom is displaced and then oscillates about an equilibrium position.

let’s suppose a degree of freedom q has a potential energy V(q) that has a minimum. The minimum describes a stable equilibrium, and when the degree of freedom is displaced, it will tend to return to the equilibrium position. Without any real loss of generality, we can locate the minimum at q = 0.

One of the hallmarks of the Hamiltonian formulation is how symmetric it is between the q’s and p’s. In the case of the harmonic oscillator, it is almost completely symmetric.

this equation is second-order, meaning that it involves second time derivatives. By contrast, the Hamiltonian equations are each first-order. This somehow means that two first-order equations are equivalent to one second-order equation.

We usually “think” in configuration space. The harmonic oscillator is a system that moves back and forth along a single axis. But it is also an excellent starting point for getting used to “thinking” in phase space.