Here is a problem that has been on my mind for some time. I have made some progress, but there is yet much to be done!
Suppose there is a system of uncountably many particles under Newtonian gravitation. Assume these particles may pass through one another; i.e., that they never collide. This is essentially an extension of the N-body problem; however, the presence of uncountably many bodies makes possible a phase-space density formulation. Let ⍴ be a density function over a 6-dimensional phase-space (consisting of 3 position and 3 velocity dimensions). Using Newton’s law of gravitation the acceleration at any position is found to be
Then the change in the phase-space density at any position x and velocity y may be derived:
where the gradients are three-dimensional and taken with respect to the position or velocity components as indicated.
I am curious about stationary states of this equation. A stationary state is a particular phase-space density which undergoes no change under the influence of gravitation. So far I have found:
- all mass focused at a point with null velocity
- a uniform distribution of matter throughout all space with the same arbitrary velocity distribution at each point
- a uniform ring of matter orbiting the center of gravity at the orbital speed
- the superposition of two such rings but with opposite orbital directions
- a uniform spherical shell of matter with orbiting the center of gravity at the orbital speed with velocities uniformly distributed in all tangential directions
Furthermore, I suspect that solutions along the following lines may be found:
- spherical shells of different sizes may be superposed but with decreased speed for inner shells and increased speed for outer shells to account for change in the net force of gravity
- a gaussian distribution of matter about a center with velocities pointing exactly inwards and outwards with some gaussian-like dependence of speed on distance from center
- a gaussian distribution of matter with velocities distributed gaussian-like among all directions and speeds
The questions I would like to answer include
- Can all stationary states of a certain dimention be simply charicterized?
- Which stationary states are stable? I.e., which deviate slightly when slightly perturbed?
- What is the Lagrangian formulation of the UMBP?
- The quantum mechanical version?