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

by Leonard Susskind

In other words, the processes of integration and differentiation are reciprocal: The derivative of the integral is the original integrand.

Can we completely determine F(t) knowing that its derivative is f(t)? Almost, but not quite. The problem is that adding a constant to F(t) does not change its derivative.

The ambiguous constant is closely related to the ambiguity in choosing the other endpoint of integration that we earlier called a.

There are some tricks to doing integrals. One trick is to look them up in a table of integrals. Another is to learn to use Mathematica. But if you’re on your own and you don’t recognize the integral, the oldest trick in the book is integration by parts.

You might wonder how often this trick works. The answer is quite often, but certainly not always. Good luck.

Aristotle lived in a world dominated by friction. To make anything move—a heavy cart with wooden wheels, for example—you had to push it, you had to apply a force to it. The harder you pushed it, the faster it moved; but if you stopped pushing, the cart very quickly came to rest. Aristotle came to some wrong conclusions because he didn’t understand that friction is a force.

It would have been obvious that heavier things are harder to move than lighter things, so somehow the mass of the object has to be in the equation.

One suspects that Aristotle never went ice skating, or he would have known that it is just as hard to stop a body as to get it moving.

Consider one-dimensional motion of a particle along the x axis under the influence of a given force. What I mean by a given force is simply that we know what the force is at any time.

One obvious difference is that Aristotle’s equation is not stroboscopic—that is neither t nor x is discrete. They do not change in sudden stroboscopic steps; they change continuously.

In other words, wherever the particle happens to be at time t, at the next instant its position will have shifted by a definite amount.

Equations for unknown functions that involve derivatives are called differential equations.

The left side of the equation is the integral of a derivative. That’s where the fundamental theorem of calculus comes in handy. The left side is just x(t) + c.

The analogous procedure to reversing the arrows when time is continuous is very simple. Everywhere you see time in the equations, replace it with minus time. That will have the effect of interchanging the future and the past.

The implication is simple: The reversed equation of motion is exactly like the original, but with a different rule for the force as a function of time. The conclusion is clear: If Aristotle’s equations of motion are deterministic into the future, they are also deterministic into the past. The problem with Aristotle’s equations is not that they are inconsistent; they are just the wrong equations.

Frictional forces do exist, and in many cases they are so important that Aristotle’s intuition—things stop if you stop pushing—is almost correct.

Frictional forces are not fundamental. They are a consequence of a body interacting with a huge number of other tiny bodies—atoms and molecules—that are too small and too numerous to keep track of. So we average over all the hidden degrees of freedom.

Aristotle’s mistake was to think that a net “applied” force is needed to keep an object moving. The right idea is that one force—the applied force—is needed to overcome another force—the force of friction.

An isolated object moving in free space, with no forces acting on it, requires nothing to keep it moving. In fact, it needs a force to stop it. This is the law of inertia.

What forces do is change the state of motion of a body. If the body is initially at rest, it takes a force to start it moving. If it’s moving, it takes a force to stop it. If it is moving in a particular direc...

Newton’s law of motion involves three quantities: acceleration, mass, and force. Acceleration we studied in Lecture 2. By monitoring the position of an object as it moves, a clever observer—with a bit of mathematics—can determine its acceleration. Mass is a new concept that is actually defined in terms of force and acceleration. But so far we haven’t defined force.

To define two units of force, we could pull just hard enough to stretch the spring to two ticks. But this assumes that the spring behaves the same way between one tick and two ticks of stretching.

When we do this experiment in free space, we discover the interesting fact that object A accelerates along the direction in which we pull the hook. More exactly, the acceleration is proportional to the force—twice as big for two units of force, three times as big for three units, and so on.

What we find is that when we apply a single unit of force (by pulling the whole thing with a single spring stretched to one tick) the acceleration is only half what it was originally. The inertia (mass) is now twice as big as before.

We can do many more experiments in which we hook any number of springs to any number of A’s. The observations are summarized by a single formula, Newton’s second law of motion, which tells us that force equals mass times acceleration,

In other words, force equals mass times the rate of change of velocity: no force—no change in velocity.

Both force and acceleration are vectors because they not only have magnitude but also direction.

A mathematician might be content to say that the length of a line segment is 3. But a physicist or engineer—or even an ordinary person—would want to know, “Three what?”

[x] = [length] = meters. Despite its appearance, this is not an equation in the usual sense. The way to read it is x has units of length and is measured in meters.

Once we have units for length and time, we can construct units for velocity and acceleration.

To compute the velocity of an object, we divide a distance by a time. The result has units of length per time, or—in our units—meters per second.

Similarly, acceleration is the rate of change of velocity, and its units are velocity per unit time, or length per unit time per unit time:

The unit of mass that we will use is the kilogram; it is defined as the mass of a certain lump of platinum, that is also kept somewhere in France.

One might define it in terms of some particular spring made of a specific metal, stretched a distance of 0.01 meter, or something like that. But in fact, we have no need for a new unit of force. We already have one—namely the force that it takes to accelerate one kilogram by one meter per second per second.

One kilogram meter per second squared is called a Newton, abbreviated N. Newton, himself, being English, probably favored the British unit, namely the pound.

This, incidently, is often referred to as Newton’s first law of motion: Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

Equations (1) and (2) are called Newton’s second law of motion, The relationship between an object’s mass m, its acceleration a, and the applied force F is F = ma.

But, as we have seen, the first law is simply a special case of the second law...

The harmonic oscillator is an enormously important system that occurs in contexts ranging from the motion of a pendulum to the oscillations of the electric and magnetic fields in a light wave.

The calculus of multivariable functions is a straightforward generalization of single-variable calculus. Instead of a function of a single variable t, consider a function of several variables.

It’s an interesting and important fact that the mixed derivatives do not depend on the order in which the derivatives are carried out.

Notice that there are places on the curve where a shift in y in either direction produces only an upward shift in F. These points are called local minima. In Figure 2 we have added dots to indicate the local minima.

For each local minimum, when you go in either direction along y, you begin to rise above the dot in F(y). Each dot is at the bottom of a little depression. The global minimum is the lowest possible place on the curve.

One condition for a local minimum is that the derivative of the function with respect to the independent variable at that point is zero.

Local maxima, local minima, and other stationary points can happen for functions of more than one variable.

The very tops of hills are places where no matter which way you move, you soon go down. Valley bottoms are the opposite; all directions lead up. But both are places where the ground is level.

There are other places where the ground is level. Between two hills you can find places called saddles. Saddle points are level, but along one axis the altitude quickly increases in either direction. Along another perpendicular direction the altitude decreases. All of these are called stationary points.

To have a minimum, or for that matter to have any stationary point, both derivatives must vanish.

Suppose we found such a point. How do we tell whether it is a maximum, a minimum, or a saddle. The answer is a generalization of the criterion for a single variable.

Matrices, determinants, and traces may not mean much to you beyond these definitions, but they will if you follow these lectures to the next subject—quantum mechanics.