By Rodney Hill (Auth.)
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10 alone suffices to prove the orbit bounded when E < 0, it merely indicates that the orbit may be unbounded when E ^ 0 [in that (6) then formally admits real v as r -> oo]. That the orbit in fact extends to infinity when E ^ 0 follows only from full use of the equations of motion, either explicitly after complete integration or implicitly through just such an argument as Jacobi's. 12 Some Inequalities for an n-Body System I n the work of Jacobi, just discussed, no use is made of the conservation principle for the moment of momentum.
These are written here in a form exhibiting the (£', Y]') components of unit vectors attached to the (f, rj) axes. By differentiating these equations and grouping terms to exhibit the same unit vectors: together with / ' = r. Consequently, the rate of change relative to 0 has the components (p — qip q + pip, r) on the axes in a. This simple b u t fundamental result can be stated in the following way. , q, r) has a rate of change (p, q, f) relative to a, but a rate of change (p -qtt,q + (i) 9 relative to any other frame 0, where (0, 0, Q) is the spin of a relative to /?.
E, say, expression (7) also reduces to e. That is, G moves as if the whole mass were concentrated there, or as if the separate masses were concentrated at P , Q, . . Moreover, the accelerations of P , g , . . relative to G are then just a , a , . . given by (4) with (6) in terms of inverse-square contributions from the system itself. 6 under the mass-point idealization. On the other hand, the collective motions of centres P , Q, . . 10). , as just proved when e = e , etc. In general the vector products d o not vanish since the mutually induced accelerations in bodies of a pair are not usually in the line of mass-centres.