E.o.m. for a particle in a sphere of dust

[tom.stoer]

I am looking for the e.o.m. of a particle moving inside a sphere of homogeneous dust with density ρ. I start with the Lagrangian (in cartesian coordinates with i=1,2,3)

$$L = \frac{m}{2}\dot{x}_i^2 - \frac{4\pi}{3}Gm\rho x_i^2$$

The e.o.m. and their solution are given by the harmonic oscillator with frequency

$$\omega^2 = \frac{8\pi}{3}G\rho$$

$$x_i(t) = a_i\cos\omega t + b_i\sin\omega t$$

with

$$a_i = (a_x,a_y,0);\;\;b_i = (b_x,b_y,0)$$

for motion in the xy-plane.

Of course angular momentum is conserved. This follows either from the solution or directly from the Lagrangian in spherical coordinates

$$J_z = \frac{\partial L}{\partial \dot{\phi}} = mr^2\dot{\phi} \to mr^2\omega$$

This a well-known warm-up.

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In the next step I want to study a slowly varying, homogeneous density ρ(t). What I want to do now is to introduce a general ansatz which can be studied in perturbation theory or in a kind of adiabatic approximation. I am interested in perturbations to exactly circular motion. Of course I can use something like time-dependent parameters a(t), b(t) and ω(t). Angular momentum is still conserved b/c the time-dependent density does not affect the kinetic term. And here comes my problem:

I can't find an ansatz respecting conservation of angular momentum!

Any ideas?

 

[tom.stoer]

OK; I found the Ansatz; rather simple ;-)