CM: Phase Space Volume Conservation

During the last CM session I failed to recall immediately how the volume conservation can be derived from Hamiltonian equations. There is a beautiful explanation in Tong's notes on dynamics and relativity - the credit goes him.

To spare you reading through the whole notes (if you have time, I really recommend it ;)), consider the evolution of a small volume in 2N dimensional phase space:

 V = \int \Pi_{k=1}^N d q_k d p_k

The coordinates will evolve in time according to:

 p_k(t+dt) = p_k' = p_k(t) + \dot{p}_k dt = p_k - \frac{\partial H} {\partial q_k} dt \\ q_k(t+dt) = q_k' = q_k(t) + \dot{q}_k dt = q_k + \frac{\partial H}{\partial p_k} dt

meaning that:

 dV(t+dt) = dV' = \Pi_{k=1}^N d q_k' d p_k'

It then follows from the substitution rule for multivariable integral that:

 \int \Pi_{k=1}^N d q_k' d p_k' = \int \Pi_{k=1}^N |det \, J | \, d q_k d p_k

where  |det\, J| is the 2N \times 2N Jacobian determinant:

 \left| \begin{array}{cc} \frac{\partial q_j'}{ \partial q_k}  & \frac{\partial q_j'}{ \partial p_k} \\ \frac{\partial p_j'}{\partial q_k} & \frac{\partial p_j'}{\partial p_k} \end{array} \right| = \left| \begin{array}{cc} \delta_{jk} + \frac{\partial^2 H}{\partial p_k \partial q_k} dt & \frac{\partial^2 H}{\partial q_j \partial p_k} dt \\ -\frac{\partial^2 H}{\partial p_k \partial p_j} dt & \delta_{jk} - \frac{\partial^2 H}{\partial p_k \partial q_k} dt \end{array} \right| = 1 + O(dt^2)

to get the last equality, observe the fact that only the diagonal does not pick up any dt dependence and the O(dt) terms vanish by antisymmetry. Therefore:

 \int \Pi_{k=1}^N d q_k' d p_k' = \int \Pi_{k=1}^N d q_k d p_k

and the phase-space volume remains unchanged in time.

CM: A Pure Quartic Chain

This extends on the previous idea on purely quartic oscillators by considering an infinite chain of them, to see if this leads to a certain "interacting" field theory. Consider the following Lagrangian for a long string of masses coupled by Hooke's springs subject to horizontal displacement \phi_k as considered in the previous post:

 \mathcal{L} = \sum_{k= - \infty}^\infty \frac{1}{2}m \dot{\phi_k}^2 - \frac{\kappa}{8}\frac{(\phi_k - \phi_{k+1})^4}{L^2}

Then the equations of motion read:

 m\ddot{\phi_k} = -\frac{1}{2} \frac{(\phi_k - \phi_{k+1})^3 + (\phi_k - \phi_{k-1})^3}{L^2}

Now consider the continuum limit at which:

 (\phi_k - \phi_{k-1}) \rightarrow a \frac{\partial \phi}{\partial x} (x_k)

and therefore:

 m\ddot{\phi} \approx \frac{ka^4}{2}\frac{(\frac{\partial \phi}{\partial x} (x_{k+1}))^3 - ( \frac{\partial \phi}{\partial x} (x_k))^3}{aL^2} \approx \frac{ka^4}{2L^2} \frac{\partial}{\partial x} \left( \frac{\partial \phi} {\partial x} \right)^3

for some equilibrium separation a, which yields:

 \ddot{\phi} = \frac{3}{2} \frac{k a^4}{L^2 m} \left( \frac{\partial \phi}{\partial x} \right)^2 \frac{\partial^2 \phi}{\partial x^2}

a wave equation with a derivative correction to the propagation speed. This is not exactly the phi fourth alluded to earlier.

CM: A Pure Quartic Oscillator

I am currently a teaching assistant for a course in Classical Mechanics here at ETH. As such, I have to design a problem once in a week (or two) to torture the Poly second years. Today I got quite surprised by a beautiful result I always overlooked. That is how easily can one get quartic oscillations from a simple spring system. The problem goes as following:

Consider a mass m and two springs k attached to it opposite one another, each having equilibrium length L. Now displace the mass in a direction \textit{perpendicular} to the springs by \delta x. The increase in potential energy will be proportional to extension of the springs:

 \delta l =  \frac{1}{2}\frac{\delta x^2}{L}

and the Lagrangian for the problem is therefore given by:

 \mathcal{L} = \frac{1}{2} m (\dot{\delta x})^2 - \frac{1}{4} k \frac{\delta x^4}{L^2}

Isn't that marvelous? A purely quartic potential directly from two springs and a single mass! The equations of motion lead to:

 \ddot{\delta x} = - \frac{k}{m L^2} \delta x^3

which means no Hooke's law in the horizontal direction, no force \propto displacement, nuthin!

I was surprised how horrendous the analytic solutions to this problem are (Jacobi elliptic functions anyone?). Also, one may speculate about considering a chain of such "horizontally plucked" masses and therefore possibly recovering the classical massless phi-fourth theory in the continuum limit.

I used to be told that all oscillators are all harmonic when slightly perturbed around equilibria, up to (possibly) some parasitic cases. Well, here you have (a perfectly non-parasitic) one :)