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 dimensional phase space:
The coordinates will evolve in time according to:
It then follows from the substitution rule for multivariable integral that:
to get the last equality, observe the fact that only the diagonal does not pick up any
dependence and the
terms vanish by antisymmetry. Therefore:
and the phase-space volume remains unchanged in time.
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 as considered in the previous post:
Then the equations of motion read:
Now consider the continuum limit at which:
for some equilibrium separation
, which yields:
a wave equation with a derivative correction to the propagation speed. This is not exactly the phi fourth alluded to earlier.
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 and two springs attached to it opposite one another, each having equilibrium length . Now displace the mass in a direction to the springs by . 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
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