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

• tom.stoer
In summary, the conversation discusses the e.o.m. and their solution for a particle moving inside a sphere of homogeneous dust with density ρ. The solution is given by a harmonic oscillator with frequency ω. The conversation then moves on to discussing the introduction of a slowly varying, homogeneous density ρ(t) and the search for an ansatz that respects conservation of angular momentum.
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.

**********

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?

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

## 1. What is the equation of motion for a particle in a sphere of dust?

The equation of motion for a particle in a sphere of dust is given by Newton's Second Law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. In this case, the net force is due to the gravitational attraction between the particle and the dust particles in the sphere.

## 2. How is the equation of motion affected by the density of the dust sphere?

The equation of motion is affected by the density of the dust sphere because the gravitational force between the particle and the dust particles is directly proportional to the mass of the dust sphere. Therefore, a higher density would result in a stronger gravitational force and potentially a greater acceleration of the particle.

## 3. Can the equation of motion be used to calculate the trajectory of the particle?

Yes, the equation of motion can be used to calculate the trajectory of the particle in the dust sphere. By solving for the acceleration and using appropriate initial conditions, the position and velocity of the particle can be determined at any point in time.

## 4. How does the radius of the dust sphere affect the equation of motion?

The radius of the dust sphere affects the equation of motion because it determines the distance between the particle and the dust particles, which directly affects the strength of the gravitational force. A larger radius would result in a weaker force and potentially a smaller acceleration of the particle.

## 5. Is the equation of motion affected by the size or mass of the particle?

Yes, the equation of motion is affected by the size and mass of the particle. A larger or more massive particle would have a stronger gravitational force, resulting in a higher acceleration. Additionally, the size and mass of the particle would also affect its drag and buoyancy forces, which would need to be taken into account in the equation of motion.

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