Periodic Elliptic Motion in the Problem of Trhee Bodies

Abstract

We consider the Newtonian three-body problem in the plane for three mass points P _k with arbitrarily given masses m _k >0, (k=1, 2, 3). Identifying the plane of motion with the ordinary complex plane we introduce the relative (complex) vectors

$$ u = {{P}_{2}}{{P}_{3}} = u(t),\;v = {{P}_{1}}{{P}_{4}} = v(t), $$

where P ₄ is the center of mass of P ₂ and P ₃ . Assuming the center of mass of the three bodies to be at rest at the origin and defining the mass parameters µ and v by

$$ {{m}_{2}} + {{m}_{3}} = \mu ({{m}_{1}} + {{m}_{2}} + {{m}_{3}}),\;{{m}_{3}} = v({{m}_{2}} + {{m}_{3}}) $$

the equations of motion take the form (where • =d/dt)

$$ \ddot{u} = \mu F(u) = (1 - \mu )[F(v - vu) - F(v + u - vu)]\ddot{\nu } + F(\nu ) = F(\nu ) - (1 - \nu )F(\nu - \nu u) - F(\nu + u - \nu u) $$

(1)

with$F(u) = u{{\left| u \right|}^{{ - 3}}}.$ For small $\left| {u/v} \right|$ this can be approximated by

$$ \ddot{u} + \mu F(u) = 0,\;\ddot{v} + F(v) = 0; $$

(2)

and these two uncoupled Kepler-problems have periodic solutions of the form u = u* = u*(t;ε, k, m), v = v* = v*(t), in particular, where

$$ {{u}^{*}} = {{u}^{{1/3}}}{{c}^{2}}{{(1 - \varepsilon \cos s)}^{{ - 1}}}{{e}^{{is}}},\;{{v}^{*}} = {{e}^{{it}}},\;0 < \varepsilon < 1,\;t = {{c}^{3}}\int\limits_{0}^{s} {{{{(1 - \varepsilon \cos s)}}^{{ - 2}}}ds,\;c = {{{(m/k)}}^{{1/3}}}{{{(1 - {{\varepsilon }^{2}})}}^{{1/2}}},} $$

(3)

describing elliptic motion of eccentricity ɛ for u. Here m>0 and k are relatively prime integers, and the motion is periodic in the sense that u*(t + 2πm) = u* (t) and v*(t +2πm) = v* (t) identically in t.