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

by Leonard Susskind

Chemical energy is also a special case: The energy stored in chemical bonds is a combination of the potential energy and kinetic energy of the constit...

Electrostatic energy is just another word for the potential energy associated with the forces of attraction and repulsion between electrically charged particles.

Magnetic energy is tricky, but the force between the poles of magnets is a form of potential energy. The tricky part comes when we think about the forces between magnets and charged particles.

Finally, there is the energy stored in electromagnetic radiation. It can take the form of heat from the sun, or the energy stored in radio waves, laser light, or other forms of radiation.

In some very general sense, it is a combination of kinetic and potential energy, but it is not the energy of particles (not until we get to quantum field theory, anyway) but, of fields.

All you have to remember is that the action is always stationary.”

The principle of least action—really the principle of stationary action—is the most compact form of the classical laws of physics.

Let’s begin with a general observation about the basic problem of classical mechanics, namely this problem is to determine the trajectories (or orbits) of systems from their equations of motion.

we can formulate the problem of classical mechanics in another way that also involves specifying 2N items of information. Instead of providing the initial positions and velocities, we provide the initial and final positions.

The two ways of posing the problem of motion are analogous to two ways of formulating the problem of fixing a straight line in space.

the problem is similar to asking how we have to aim a line from some initial point so that it passes through another point. The answer: Find the shortest path between the points.

Formulating the action principle involves exactly the same parameters as formulating Newton’s equations. You have to know the masses of the particles, and you have to know the potential energy.

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.

The principle of stationary action is really very remarkable. It almost seems that the particle must have supernatural powers to feel out all the possible trajectories and pick the one that makes the action stationary.

The process of minimizing the action is a generalization of minimizing a function.

Time is not really stroboscopic, and a real trajectory is a function of a continuously infinite number of variables.

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.

despite the differences from ordinary functions, the condition for a stationary action strongly resembles the condition for a stationary point of a function.

Later in this lecture we will work out the equations for minimizing the action. They are called the Euler-Lagrange equations.

the particle does not have to have supernatural powers to test out all future trajectories—at least no more so than it needs to follow Newton’s equations of motion.

The first replacement is just the usual approximation of replacing the integral by a discrete sum of terms, each weighted with the small time interval Δ t. The second is also familiar.

What this derivation shows is that there is no magic involved in the ability of the particle to feel out the entire path before deciding which way to go. At each stage along the trajectory, the particle has only to minimize the action between a point in time and a neighboring point in time.

The principle of least action for more degrees of freedom is essentially no different than the case with only a single degree of freedom.

It’s like minimizing a function of many variables: There is an equation for each variable. The same is true for the Euler-Lagrange equations: There is one for each variable xi.

There are two primary reasons for using the principle of least action. First, it packages everything about a system in a very concise way.

Once you know the Lagrangian, the only thing left to specify is the initial conditions.

In future volumes, we will find that whole theories—Maxwell’s theory of electrodynamics, Einstein’s theory of gravity, the Standard Model of elementary particles—are each described by a Lagrangian.

The second reason for using the principle of least action is the practical advantage of the Lagrangian formulation of mechanics.

Take the case of a particle in one dimension that, from the point of view of someone standing at rest, satisfies Newton’s laws. The physicist at rest—call him Lenny—uses the coordinate x to locate the object.

A second physicist—George—is moving, without rotation relative to Lenny, and he wants to know how to describe the object relative to his own coordinates. First of all, what does it mean to talk about George’s coordinates? Because George moves relative to Lenny, the origin of his coordinate frame moves relative to Lenny’s origin.

When Lenny sees a particle moving on the trajectory x(t), George sees the same particle moving on the trajectory X = x(t) − f(t). If George does not want to keep asking Lenny what the trajectory is, then he wants his own laws of motion to describe the object from his coordinates. The easiest way to transform the equations of motion from one coordinate system to another is to use the principle of least action, or the Euler-Lagrange equations.

The Coriolis force is a velocity-dependent force.

The main point of this exercise was not so much to derive the centrifugal and Coriolis forces as to show you how to transform a mechanics problem from one coordinate system to another by simply rewriting the Lagrangian in the new coordinates.

There is really nothing very general about Cartesian coordinates. There are many coordinate systems that we can choose to represent any mechanical system.

Even more general would be an object rolling on a general curved surface like a hilly terrain. In such a case, there may not be any special set of coordinates. That’s why it is important to set up the equations of classical mechanics in a general way that applies to any coordinate system.

In a general coordinate system, the equations of motion may be complicated, but the action principle always applies.

Sometimes the Lagrangian is guessed on the basis of some theoretical prejudices or principles, and sometimes we deduce it from experiments. But however we get it, the Lagrangian neatly summarizes all the equations of motion in a simple package.

The concept of conjugate momentum transcends the simple example in which momentum comes out to be mass times velocity.

This quantity should be familiar. It is the angular momentum of the particle. Angular momentum and pθ are exactly the same thing.

In other words, angular momentum is conserved.

We can see that is a constant. That’s why angular velocity increases as a particle gets closer to the origin.

As we’ve just seen, it sometimes happens that some coordinate does not appear in the Lagrangian though its velocity does. Such coordinates are called cyclic (I don’t know why.)

Whenever a coordinate is cyclic, its conjugate momentum is conserved.

there is nothing particularly cyclic about them—it’s just a word. Therefore, all of the components of momentum are conserved.

The relationship between symmetries and conservation laws is one of the big main themes of modern physics.

At first, the fact that certain quantities are conserved will seem somewhat accidental—hardly things of deep principle.

the conservation law has not been lost; it just changed a little bit. By multiplying the first equation by b and the second by a and then adding them, we can see that b p1 + a p2 is conserved.

What determines whether there are conservation laws and what they are? The answer has been known for almost 100 years from the work of the German mathematician Emmy Noether.

There are two ways to think about a change of coordinates. The first way is called passive. You don’t do anything to the system— just relabel the points of the configuration space.

According to the passive way of thinking, the transformation consists of erasing all the labels and replacing them with new ones.