- #1
Oliver321
- 59
- 5
I was recently working on the two body problem and what I can say about solutions without solving the differential equation. There I came across a problem:
Lets consider the Kepler problem (the two body problem with potential ~1/r^2). If I use lagrangian mechanics, I get two differential equations. Now if I plot the phase portrait it looks like a vectorfield in the x-y-plane. Trajectories on this phase plane describe (under certain conditions) ellipses and circles. But they never cross and are closed!
Now if I consider the two body problem with an other potential, say ~1/r^5. Its told that if the potential is not like ~1/r^2 or 1/r there are not necessarily closed orbits but possibly rosetta orbits (https://en.wikipedia.org/wiki/Rosetta_orbit). But here the trajectories cross if I plot it in 2D phase space like in the kepler problem. But trajectory can never cross in phase space! How is this possible?
My solution: If I have a n-body problem the phase space is not a 2D vector field anymore. So the trajectories don’t cross at all, it’s only because of the projection on the plane. But I can’t find a solution to the problem if I use a potential like ~1/r^5. The motion should be planar and only two body’s are considered. So I should get no crossing of trajectories in 2D phase space.
Thanks for every awnser!
Lets consider the Kepler problem (the two body problem with potential ~1/r^2). If I use lagrangian mechanics, I get two differential equations. Now if I plot the phase portrait it looks like a vectorfield in the x-y-plane. Trajectories on this phase plane describe (under certain conditions) ellipses and circles. But they never cross and are closed!
Now if I consider the two body problem with an other potential, say ~1/r^5. Its told that if the potential is not like ~1/r^2 or 1/r there are not necessarily closed orbits but possibly rosetta orbits (https://en.wikipedia.org/wiki/Rosetta_orbit). But here the trajectories cross if I plot it in 2D phase space like in the kepler problem. But trajectory can never cross in phase space! How is this possible?
My solution: If I have a n-body problem the phase space is not a 2D vector field anymore. So the trajectories don’t cross at all, it’s only because of the projection on the plane. But I can’t find a solution to the problem if I use a potential like ~1/r^5. The motion should be planar and only two body’s are considered. So I should get no crossing of trajectories in 2D phase space.
Thanks for every awnser!