## 1D pendulum problem in phase space

Posted by peeterjoot on February 15, 2013

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Problem 2.6 in [1] 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

where

Our time evolution in phase space is given by

or

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

We have

# References

[1] RK Pathria. *Statistical mechanics*. Butterworth Heinemann, Oxford, UK, 1996.

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