1D three-body problem (with spherical shell)

[maxverywell]

Let's suppose that we have two point particles with masses m1,m2 and the spherical shell with mass M, placed in a line, at distances h1,h2 and H from 0 in that line (0 is the center of some inertial frame of reference). The initial conditions and the equations of motion are the following:

$h_1(0)=H(0)=h_0$

$h_2(0)=0$

$h_1'(0)=h_2'(0)=H'(0)=0$ (time derivative)

(the mass m1 is in the center of the spherical shell)$\frac{dh_1^2}{dt^2}=-G\frac{m_2}{(h_1-h_2)^2}$

$\frac{dh_2^2}{dt^2}=G\frac{m_1}{(h_1-h_2)^2}+G\frac{M}{(H-h_2)^2}$

$\frac{dH^2}{dt^2}=-G\frac{m_2}{(H-h_2)^2}$

Is there any way to solve this problem (to find the positions of the masses as functions ot the time)?

 

[Pythagorean]

integrate twice with respect to t?

Or are H and h actually arbitrary functions of t?

 

[maxverywell]

integrate twice with respect to t?

Or are H and h actually arbitrary functions of t?

Yes they are functions of t, i think, so it's more complicated.

Edit: I'm going to create the same thread in Classical Physics section so the moderators can close this one. Thnx!