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:

[itex]h_1(0)=H(0)=h_0[/itex]

[itex]h_2(0)=0[/itex]

[itex]h_1'(0)=h_2'(0)=H'(0)=0[/itex] (time derivative)

(the mass m1 is in the center of the spherical shell at time t=0, but I'm trying to prove that for every t)[itex]\frac{d^2h_1(t)}{dt^2}=-G\frac{m_2}{(h_1-h_2)^2}[/itex]

[itex]\frac{d^2h_2(t)}{dt^2}=G\frac{m_1}{(h_1-h_2)^2}+G\frac{M}{(H-h_2)^2}[/itex]

[itex]\frac{d^2H(t)}{dt^2}=-G\frac{m_2}{(H-h_2)^2}[/itex]

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

 

[maxverywell]

Αnyone? At least tell me if this problem is solvable or not.
It seems quiet easy but the system off differential equantions is hard to solve. Is there any other way to solve it?

Actually what i am trying to prove is [itex]\frac{d^2 h_1(t)}{dt^2}=\frac{d^2 H(t)}{dt^2}[/itex] and that the mass m1 will remain in the center of spherical shell.