Singular trajectories in the restricted problem of four bodies
Annali di Matematica Pura ed Applicata
The restricted problem of three bodies has occupied a prominent place in recent researches in celestial mechanics. It maybe formulated as follows: Two bodies, S and J, revolve round their common centre of gravity in circular orbits under the influence of thoir mutual attraction (that is, constitute an exact solution of the problem of two bodies). A third body P without mass (that is~ such that it is attracted by S and J, but does not disturb their motion) moves in the same plane as S and J. The
... restricted problem of three bodies is to determine the motion of P. K corresponding particular problem of four bodies has received some attention (*). It has to do with the motion of an infinitesimal body in the plane of three finite bodies which are in motion according to one or the other of the Lagrangian exact solutions of the problem of three bodies, and attracted by each of them according to Newton's law, but itself incapable of affecting their motion. The problem of the motion of the infinitesimal body might be called a restricted problem of four bodies, distinction between the Lagrangian solutions being made where necessary. Pxxsi, Ev¢, has shown that in the general problem of three bodies the motion is regular as long as there are no collisions. The theorem applies a fortiori to the restricted problem of three bodies. _A corresponding theorem dots not hold when the number of bodies is greater than three, as PAI~L~-~ (~) MOULTON, Particular solutions of the problem of four bodies.