Phase space YouTube

Incompressibility

Figure 1.2 - States of a die

Figure 1.2 - States of a die

Recall the discrete system we set up in the first lecture. We had a space ot two states, $\left\{H, T\right\}$. There were two possible evolutions of this system, $$\left\{ \begin{matrix} H \to H \\ T \to T \end{matrix} \right.$$ and $$\left\{ \begin{matrix} H \to T \\ T \to H \end{matrix} \right.$$

Both have the feature that, if you know the state of the system at a given instant, then you will know the state at the next instant. The rule for updating is precise.

A phase space, for both discrete and continuous systems, is a set of all known possible states of that system, along with a rule for updating all of the states at each instant. For finite discrete systems, the rule for updating is just a permutation of the states.

We saw, for systems with more than two elements (see figure 1.2), there were many possible rules, some of which broke the phase space into closed loops of connected states - leading to conservation laws.

The other feature of a phase space is state is that, after updating, there is no loss of possible states in the system. We could imagine a marker at every point in the system and that the updating rule moves each of these markers to new points. No loss of states means that, after every update, no two markers will be in the same position and all states will always contain exactly one marker.

This feature of phase space is called imcompressibility, and generalises to the continuous case.

Hamiltonian phase space

The phase space described by Hamilton's form of the equations of motion is incompressible - we shall see the proof of this in Liouville's Theorem in the next lecture.

For continuous systems, it describes the positions and the momenta of all of the states of the system, for all time. This can be compared with Newton's formulation, where the space we normally consider would be just the positions of the system, and the rule for updating must be derived from the second-order equations.

We can always convert one second-order equation into two first-order equations. Given $$m_i \frac{\mathrm{d}^2 x_i}{\mathrm{d} t^2} = F_i$$ we can define $$m \frac{\mathrm{d} x_i}{\mathrm{d} t} = p_i$$ which implies that $$\frac{\mathrm{d} p_i}{\mathrm{d} t} = F_i$$

This is a pair of first-order equations $$\left\{\begin{matrix}\frac{\mathrm{d} x_i}{\mathrm{d} t} = \frac{p_i}{m}\\\frac{\mathrm{d} p_i}{\mathrm{d} t} = F_i\end{matrix}\right.$$

Phase space is made up of the pairs, $\left\{x_i, p_i\right\} \quad \forall i = 1, \dots, N$, and we can think of trajectories as flows in this space. That is, we can consider all states of a system as points flowing on trajectories through phase space.