Classical Mechanics: The Theoretical Minimum (Theoretical Minimum 1)

by Leonard Susskind

Classical physics includes Newton’s equations for the motion of particles, the Maxwell-Faraday theory of electromagnetic fields, and Einstein’s general theory of relativity.

In classical physics, if you know everything about a system at some instant of time, and you also know the equations that govern how the system changes, then you can predict the future. That’s what we mean when we say that the classical laws of physics are deterministic.

collection of objects—particles, fields, waves, or whatever—is called a system.

In physics jargon, the collection of all states occupied by a system is its space of states, or, more simply, its state-space.

A world whose evolution is discrete could be called stroboscopic.

A system that changes with time is called a dynamical system. A dynamical system consists of more than a space of states. It also entails a law of motion, or dynamical law. The dynamical law is a rule that tells us the next state given the current state.

The variables describing a system are called its degrees of freedom. Our coin has one degree of freedom, which we can denote by the greek letter sigma, . Sigma has only two possible values; = 1 and = −1, respectively, for H and T.

in each case the future behavior is completely determined by the initial state, such laws are deterministic. All the basic laws of classical mechanics are deterministic.

There must be one arrow to tell you where you’re going and one to tell you where you came from.

the conservation of information. The conservation of information is simply the rule that every state has one arrow in and one arrow out.

Whenever a dynamical law divides the state-space into such separate cycles, there is a memory of which cycle they started in. Such a memory is called a conservation law;

it tells us that something is kept intact for all time.

To make the conservation law quantitative, we give each cycle a numerical value called Q. In the example in Figure 15 the three cycles are labeled Q = +1, Q = −1, and Q = 0. Whatever the value of Q, it remains the same for all time because the dynamical law does not ...

the ability to know the initial conditions with almost perfect precision.

The ability to distinguish the neighboring values of these numbers is the “resolving power” of any experiment, and for any real observer it is limited.

In most cases the tiniest differences in the initial conditions—the starting state—leads to large eventual differences in outcomes. This phenomenon is called chaos.

If a system is chaotic (most are), then it implies that however good the resolving power may be, the time over which the system is predictable is limited. Perfect predictability is not achievable, ...