Continuous physics YouTube

Phase space

Figure 1.3 - Phase space

Figure 1.3 - Phase space

In continuous systems, the phase space is made up of pairs of real numbers. The number of pairs depends on the number of particles in the system, and in general, we can think of each pair as being made up of a position and a velocity, $(x, v)$.

Figure 1.3 shows a single pair of variables, where, for example, the point $(x, v) = (2, 3)$ means a particle is at position $x = 2$ and has velocity $v = 3$. The vector at that point shows the direction of the motion and the magnitude.

Nb. Prof. Susskind often uses $(x, v)$ to represent any number of pairs of positions and momenta.

Precision & prediction

Since we are dealing with real numbers which, in general, have an infinite number of digits, precision in classical mechanics is always within certain bounds.

Given a certain precision in our knowledge of the initial conditions, we would like to make a prediction of the state of the system after a certain length of time. If we had exact precision, then we could know the state for all time with infinite precision. In practise, better precision in the initial conditions means a better precision in the prediction.

Aristotle's Law of Motion

Aristotle thought that a force was required to keep an object moving, that the velocity of an object was proportional to the magnitude of the force and that the object moved in the direction of the force.

That is, $$F = m v$$

Since $F = F(x)$ is a known function, then we can easily calculate $v$.

Also, we can differentiate this equation with respect to time in two ways, $$\frac{\mathrm{d}F}{\mathrm{d}t} = m \frac{\mathrm{d} v}{\mathrm{d}t} = m a$$ and $$\frac{\mathrm{d}F}{\mathrm{d}t} = m \frac{\mathrm{d}F}{\mathrm{d}x} \frac{\mathrm{d}x}{\mathrm{d}t} = m \frac{\mathrm{d}F}{\mathrm{d}x} v$$

Hence, $$a = \frac{\mathrm{d}F}{\mathrm{d}x} v$$

Thus, given only the position $x$ at a certain time, Aristotle's equation allows us to know everything about the system. Needless to say, he was wrong.

Newton's Law

Of course, the correct law is Newton's Second Law of Motion, $$F = m a = m \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}$$

Here, even if we know $x$, and hence $F = F(x)$, we still cannot calculate $v$. $v$ has to be added as information into the system, as a given.

Second-order equations require knowledge of both position and momentum. If we are dealing with a single particle, then there will be, in general, three pairs of positions and momenta in the phase space, one for each of the three dimensions of space.