1D pendulum problem in phase space
Posted by peeterjoot on February 15, 2013
Problem 2.6 in  asks for some analysis of the (presumably small angle) pendulum problem in phase space, including an integration of the phase space volume energy and period of the system to the area included within a phase space trajectory. With coordinates as in fig. 1.1, our Lagrangian is
As a sign check we find for small from the Euler-Lagrange equations as expected. For the Hamiltonian, we need the canonical momentum
Observe that this canonical momentum does not have dimensions of momentum, but that of angular momentum ().
Our Hamiltonian is
Hamilton’s equations for this system, in matrix form are
With , it is convient to non-dimensionalize this
Now we can make the small angle approximation. Writing
Our pendulum equation is reduced to
With a solution that we can read off by inspection
Let’s put the initial phase space point into polar form
This doesn’t appear to be an exact match for eq. 1.0.3, but we can write for small
This shows that we can rewrite our initial conditions as
Our time evolution in phase space is given by
This is plotted in fig. 1.2.
The area of this ellipse is
With for the period of the trajectory, this is
As a final note, observe that the oriented integral from problem 2.5 of the text , is also this area. This is a general property, which can be seen geometrically in fig. 1.3, where we see that the counterclockwise oriented integral of would give the negative area. The integrals along the paths give the area under the blob, whereas the integrals along the other paths where the sense is opposite, give the complete area under the top boundary. Since they are oppositely sensed, adding them gives just the area of the blob.
Let’s do this integral for the pendulum phase trajectories. With
 RK Pathria. Statistical mechanics. Butterworth Heinemann, Oxford, UK, 1996.