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

by Leonard Susskind

If the determinant and the trace of the Hessian are positive then the point is a local minimum. If the determinant is positive and the trace negative the point is a local maximum. If the determinant is negative, then irrespective of the trace, the point is a saddle point.

One caveat, these rules specifically apply to functions of two variables. Beyond that, the rules are more complicated.

If—as Laplace believed—natural systems are composed of particles, then the laws of nature must be the dynamical laws of motion that determine the motion of those systems of particles.

What is it that determines the force on a given particle? It is the positions of all the other particles.

There are many types of forces—such as friction, the drag force exerted by the wind, and the force exerted by the floor that keeps you from falling to the basement—that are not fundamental. They originate from the microscopic interactions between atoms and molecules.

The fundamental forces are those that act between particles, like gravity and electric forces. These depend on a number of things: Gravitational forces between particles are proportional to the product of their masses, and electric fo...

Charges and masses are considered to be intrinsic properties of a particle, and specifying them is part o...

Apart from the intrinsic properties, the forces depend on the location of the particles. For example, the distance between objects determines the electric and gravitatio...

What this equation means is that the force on the ith particle is a function of the positions of all the particles. The symbol stands for the collective location of every particle in the system. Another way of saying this is that the symbol represents the set of all position vectors.

This last set of equations makes it clear that there is an equation for each coordinate of every particle, which would tell Laplace’s vast intellect how every particle moves if the initial conditions were known.

The answer is three for each particle, so if there are N particles the grand total is 3N equations.

The formal meaning of the state of a system is, “Everything you need to know (with perfect accuracy) to predict its future, given the dynamical law.”

In Aristotelian mechanics, assuming that the forces on an object are known, the state is specified by simply knowing the location of the object. In fact, from Aristotle’s law, the force determines the velocity, and the velocity tells you where the particle will be at the next instant.

Newton’s law is different from Aristotle’s: It tells you the acceleration, not the velocity. This means that to get started, you need to know not only where the particles are but also their velocities. Knowing the velocity tells you where the particle will be at the next instant, and knowing the acceleration tells you what the velocity will be.

All of this means that the state of a system of particles consists of more than just their current locations; it also ...

if the system is a single particle, its state consists of six pieces of data: the three components of its position and th...

vz. Now let’s consider the motion of the particle. At each instant of time, the state is specified by the values of the six variables x(t), y(t), z(t), vx(t), vy(t), vz(t). The history of the particle can be pictured as a trajectory through the six-dimensional state-space.

If you are struck by a moving object, the result depends not only on the velocity of the object but also on its mass. Obviously, a Ping-Pong ball at 30 miles per hour (about 13 meters per second) will have much less of a mechanical effect than a locomotive moving at the same speed.

the effect is proportional to the momentum of the object, which for now we shall define as the product of the velocity and the mass.

Since velocity and momentum are so closely linked, we can use momentum and position instead of velocity and position to label the points of the state-space.

When the state-space is described this way, it has a special name—phase space.

You may wonder why we go to the trouble of replacing the intuitive concept of velocity with the more abstract concept of momentum in describing the state of a particle. The answer should become clear as we develop the basic framework of classical mechanics in later chapters.

This simple, elegant set of equations is exactly what Laplace imagined the laws of nature to be: For each coordinate of phase space we have a single equation to tell you how it changes over an infinitesimal interval of time.

The principle of the conservation of momentum is a profound consequence of abstract general principles of classical mechanics that we have yet to formulate.

The simplest way to think of the third law is to suppose first that particles interact in pairs. Each particle j exerts a force on each other particle i, and the total force on any particle is the sum of the forces on it exerted by all the other particles.

In other words, the rate of change of the momentum of any particle is the sum of the forces due to all the other particles.

The total momentum of an isolated system never changes.

One often learns that there are many forms of energy (kinetic, potential, heat, chemical, nuclear, . . . ) and that the sum total of all of them is conserved. But when reduced to the motion of particles, classical physics really has only two forms of energy: kinetic and potential.

The basic principle—call it the potential energy principle— asserts that all forces derive from a potential energy function denoted V({x}).

The force is always directed in a way that pushes the particle toward lower potential energy (note the minus sign).

Potential energy by itself is not conserved. As the particle moves, V(x) varies. What is conserved is the sum of potential energy and kinetic energy.

Roughly speaking, as the particle rolls down the hill (in other words, as it moves toward lower potential energy), it picks up speed. As it rolls up the hill, it loses speed. Something is conserved.

As the particle rolls along the x axis, the two types of energy individually vary, but always in such a way that the sum is conserved.

The key is to realize that V(x) changes with time because x changes.

We now have what we need to prove energy conservation: Newton’s law, F = m a, is exactly the condition that the factor in parentheses vanishes, which in turn tells us that the total energy is constant.

One point before we go on to many-dimensional motion. We have shown that energy is conserved, but why is it that momentum is not conserved in this case? After all, in the previous chapter we showed that for an isolated system of particles, Newton’s third law implies that total momentum does not change. The answer is that we have left something out of the system—namely, the object that exerts the force on the one-dimensional particle.

When the particle falls, its momentum changes, but that change is exactly compensated for by a tiny change in the motion of the Earth.

It is a fact that the components of force are derivatives of potential energy, but it is not a definition.

It is quite possible to imagine force laws that do not come from differentiating a potential energy function, but nature does not make use of such nonconservative forces.

For each coordinate, there is a mass mi and a component of force Fi. Each component of force can depend on all positions {x}.

But when there is more than one dimension, things get more complicated. It is generally not true that if you have a set of functions Fi({x}), that they can all be derived by differentiating a single function V({x}). It would be a brand-new principle if we asserted that the components of force can be described as (partial) derivatives of a single potential energy function.

For any system there exists a potential V({x}) such that

Picture a terrain with the function V({x}) representing the height or altitude at each point. First of all, the minus sign in Eq. (5) means that the force points in the downhill direction.

For example, on a contour map, there is no force pushing along the contour lines. The force vector points perpendicular to the contours.

Precisely as in the one-dimensional case, Eq. (8) says that the time derivative of the total energy is zero—energy is conserved.

imagine that the terrain has a ball rolling on it with no energy being lost to friction. Whenever the ball rolls toward a lower altitude it picks up speed, and whenever it rolls uphill it loses speed. The calculation tells us this happens in a special way that conserves the sum of the kinetic and potential energies.

then why the principle of least action? Ultimately, the answer can be traced to the laws of quantum mechanics and to the origin of forces in field theory—subjects that, for the moment, are still out of range for us.

At some point we have to give up and say that’s just the way it is. Or, not give up and push on.

Mechanical energy usually refers to the kinetic and potential energy of large visible objects such as planets or weights being hoisted by a crane. It often refers to gravitational potential energy.

The heat contained in a gas or other collection of molecules is also kinetic and potential energy. The only difference is that it involves the large and chaotic motion of so many particles that we don’t even try to follow it in detail.