# Phase space and phase trajectories.

The phase space and phase trajectories are the space of ‘s and ‘s of a mechanical system (always even dimensional, with as many ‘s as ‘s for N particles in 3d: 6N dimensional space).

The state of a mechanical system the point in phase space.

Time evolution a curve in phase space.

Example: 1 dim system, say a harmonic oscillator.

Our phase space can be illustrated as an ellipse as in figure (\ref{fig:phaseSpaceAndTrajectories:phaseSpaceAndTrajectoriesFig1})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{phaseSpaceAndTrajectoriesFig1}

\caption{Harmonic oscillator phase space trajectory.}

\end{figure}

where the phase space trajectories of the SHO. The equation describing the ellipse is

which we can put into standard elliptical form as

## Applications of .

\begin{itemize}

\item Classical stat mech.

\item transition into QM via Poisson brackets.

\item mathematical theorems about phase space “flow”.

\item perturbation theory.

\end{itemize}

## Poisson brackets.

Poisson brackets arises very naturally if one asks about the time evolution of a function on phase space.

Define the commutator of and as

This is the Poisson bracket of with , defined for arbitrary functions on phase space.

Note that other conventions for sign exist (apparently in Landau and Lifshitz uses the opposite).

So we have

Corollaries:

If has no explicit time dependence and if , then is an integral of motion.

In QM conserved quantities are the ones that commute with the Hamiltonian operator.

To see the analogy better, recall def of Poisson bracket

Properties of Poisson bracket

\begin{itemize}

\item antisymmetric

\item linear

\end{itemize}

### Example. Compute , commutators.

So

Similarly .

How about

So

This provides a systematic (axiomatic) way to “quantize” a classical mechanics system, where we make replacements

and

So

Our quantization of time evolution is therefore

These are the Heisenberg equations of motion in QM.

### Conserved quantities.

For conserved quantities , functions of ‘s ‘s, we have

Considering the components , where

We can show (3.27) that our Poisson brackets obey

(Prof Poppitz wasn’t sure of the sign of this and the particular bracket sign convention he happened to be using, but it appears he had it right).

These are the analogue of the momentum commutator relationships from QM right here in classical mechanics.

Considering the symmetries that lead to this conservation relationship, it is actually possible to show that rotations in 4D space lead to these symmetries and the conservation of the Runge-Lenz vector.

# Adiabatic changes in phase space and conserved quantities.

In figure (\ref{fig:phaseSpaceAndTrajectories:phaseSpaceAndTrajectoriesFig2}) where we have

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{phaseSpaceAndTrajectoriesFig2}

\caption{Variable length pendulum.}

\end{figure}

Imagine that we change the length very *slowly* so that

where is the period of oscillation. This is what’s called an adiabatic change, where the change of is small over a period.

It turns out that if this rate of change is slow, then there is actually an invariant, and

is the so-called “adiabatic invariant”. There’s an important application to this (and some relations to QM). Imagine that we have a particle bounded by two walls, where the walls are moved very slowly as in figure (\ref{fig:phaseSpaceAndTrajectories:phaseSpaceAndTrajectoriesFig3})

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{phaseSpaceAndTrajectoriesFig3}

\caption{Particle constrained by slowly moving walls.}

\end{figure}

This can be used to derive the adiabatic equation for an ideal gas (also using the equipartition theorem).

# Appendix I. Poisson brackets of angular momentum.

Let’s verify the angular momentum relations of 1.21 above (summation over implied):

So, as claimed, if we have

# Appendix II. EOM for the variable length pendulum.

Since we’ve referred to a variable length pendulum above, let’s recall what form the EOM for this system take. With cylindrical coordinates as in figure (\ref{fig:phaseSpaceAndTrajectories:phaseSpaceAndTrajectoriesFig4}), and a spring constant our Lagrangian is

\begin{figure}[htp]

\centering

\includegraphics[totalheight=0.2\textheight]{phaseSpaceAndTrajectoriesFig4}

\caption{phaseSpaceAndTrajectoriesFig4}

\end{figure}

The EOM follows immediately

Or

Even in the small angle limit this isn’t a terribly friendly looking system

However, in the first equation of this system

we do see the dependence mentioned in class, and see how this being small will still result in something that approximately has the form of a SHO.