# 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:

The coordinates will evolve in time according to:

meaning that:

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

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

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:

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:

Then the equations of motion read:

Now consider the continuum limit at which:

and therefore:

for some equilibrium separation $a$, which yields:

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:

and the Lagrangian for the problem is therefore given by:

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

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