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:(adsbygoogle = window.adsbygoogle || []).push({});

[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)?

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# 1D three-body problem (with spherical shell)

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