V.5 No 1

59

Three-body problem in collision theory

The passing to the mass centre frame is based on the Galilee principle that means conservation of the state of motion of material bodies conservative system. "Because of absence of external forces, the mass centre of the system moves in relation of S (stationary reference frame - Authors) uniformly. So the speed of mass centre and its radius-vector are equal to

where

and ₁ , ₂ , ₁ , ₂ are the initial locations and speeds of corresponding points.

Fig. 1. The motion of point masses relatively the uniformly moving system of mass centre S_m

Consider further the motion of points relatively the uniformly moving system of mass centre S_m (Fig. 1). We call so the reference frame whose origin is located in the mass centre of mechanical system and axes do not change their orientation as to the system S (i.e., the angles between the axes of systems S_m and S are constant). In this case the system S_m is inertial, as the mass centre moves uniformly in relation to the system S.

Consequently, the locations, speeds and accelerations of points in relation to S_m and S (noting (10) - Authors) are interrelated as follows:

where 'undotted' vectors relate to the system S and 'dotted' vectors relate to the system S_m… However, locations of points 1 and 2 in the system S_m are not independent. Actually, from the definition of mass centre

(where m_i and r_i are the mass and radius-vector of the ith point of the system, m is the mass of the total system, N is the number of material points of the system) and from the definition of the system S_m we have

So the radius-vector

characterising the relative location of points, is expressed through '₁ and  '₂:

and the radius-vectors  '₁ and '₂ are linked with through the relationship

Having differentiated (13)- (16) over the time, we yield

where

[2, p. 104- 105].

We see from this derivation that the passing to the mass centre frame allowed us to determine the relationship of speeds of bodies with reference to the mutual speed of material points. This gives us an additional relationship for the modelling system of equations (8)- (9) in collision theory.

At the same time we encounter new difficulties. If we look again at Fig. 1, we will see that relationships (19) are written for bodies moving as to each other spatially, irrespectively of, whether they interact. (11)- (13) to be true, it is sufficient to remember that the system is conservative, as this concept includes both interaction and independent motion.

In the scattering theory this difficulty is surmounted so. The calculation results in the mass centre frame are substituted into the differential equations of motion (1). Forces of interaction involved to these equations in fact complement (11)- (20), introducing necessary conditions of interaction.

In the collision case such completion of model is absent, since both the equation of energy conservation (8) and equation of momentum conservation (9) are equally true both for pulse interaction and for non-interaction. So (19) that interrelates the speeds of bodies in the mass centre frame with the relative speed of these bodies is unable to complement the equation of momentum conservation (9). In particular, (19) says nothing as to, which direction will have the speeds u₁ and u₂ after collision.

Another difficulty of collision theory is, even if having complemented the system (8)- (9), it appears indefinite with growing number of simultaneously interacting bodies. Just because of it the three-body problem remained unsolved several centuries. To it there added the problem of scattering that, above the immediate importance of its approaches for celestial mechanics, particle physics etc. allows to estimate the solution of two-body problem in collision theory, considering the initial and final locations of interacting bodies at considerable distances from the point of interaction [3, chapter 2, item 13, p. 109]. But this approach is helpless in solving the problem of three and more bodies in the collision theory, as this problem has not been solved in frames of the very scattering theory. On the contrary, if this problem were solved in collision theory, it would be of help to solve the problem in scattering theory, showing the trajectories of scattered bodies in the asymptotic approximation at large distances from the point of interaction.

Since the three-body problem is important both in collision theory and to develop the solving technique in scattering theory, we in our study will refine the two-body problem in collision theory and develop this solution for three, four etc. bodies.