V.5 No 1

65

Three-body problem in collision theory

Proceeding from the said, to fulfil our plan of solving the many-body problem in collision theory, it is sufficient to obtain the solutions for the case of interaction of some mass with the local mass centre of other bodies of the system.

Suppose for simplicity the first body of the three-body system interacting with the local mass centre described by the system (49). With it, as it follows from the statement of problem shown in Fig. 8, this body interacts with the local mass centre in the plane passing through the points of initial location of the body and mass centre and the interaction point A. So we can immediately write the solution based on the general solution (36) of the two-body problem:

where m_m1 = m₂ + m₃. With it we are not interesting in the solution describing the parameters of motion, as for each body involved into the conservative system the equality (53) will be true; it will fully determine the after-interaction parameters of the body's motion. The same, for the system of n bodies, solution for the ith body (i = 1, 2, ..., n) will be the following:

where, according to (49),

Just as in case of two-body problem, these solutions have to be joined with the conditions complementing the initial parameters of bodies involved into the system. It is easy to do, given the conditions (37)- (38) and (41) have been yielded on the basis of additionally given location of the centre of interaction and time interval between the initial location of bodies and the instant of interaction. So these additional conditions can be determined through any pair of bodies involved into the system, under condition that the parameters of each body will be present at least in one condition. So from six parameters necessary for full determinacy of the initial location and speed of each body, we can determine three, - accordingly, for n bodies it will be 3n.

Fig. 9. Dynamic diagram of five scattering point masses. Parameters of diagram:

m₁ = 3 kg, m₂ = 1 kg, m₃ = 5 kg, m₄ = 0,3 kg, m₅ = 2 kg,

₁ = 200^o , ₂ = 160^o , ₃ = - 25^o , ₄ = 130^o , ₁ = 230^o ,

 ₁ = 35^o , ₂ = 60^o , ₃ = 40^o , ₄ = 115^o , ₅ = 140^o ,

v₁ = 4 m/s , v₂ = 2 m/s , v₃ = 3 m/s , v₄ = 6 m/s , v₅ = 1 m/s ,

x_A = 10 m,  y_A = 19 m , z_A = 5 m ,  t_A = 10 s  

To visualise the yielded solutions, we show in Fig. 9 the trajectory of scattering five bodies with arbitrarily chosen masses and other parameters, noting the complementing conditions.

Conclusions

Having studied sequentially the two-body and three-body problems and increasing the quantity of interacting bodies up to an arbitrary number, we established that general form of solution of two centrally interacting bodies (1 D case) fully remains in its vector form in transition to 2 D case, and the problem of three and more bodies is divided into a set of two-body problems describing the interaction of each body with the local centre of masses of the basic modelling system of equations formed on the basis of conservation laws of the energy and momentum of the system; so the problem gets its exact analytical solution.

13- 18 May, 2005

References:

1. Physical encyclopaedia, vol. 4. Moscow, Sovetskaya encyclopedia, 1965 (Russian)

2. Olkhovsky, I.I. The course of theoretical mechanics for physicists. Moscow, Nauka, 1970 (Russian)

3. Restricted Three-Body Problem

http://scienceworld.wolfram.com/physics/Three-BodyProblem.html

4. James P. Sethna The Restricted Three Body Problem

www.physics.cornell.edu/sethna/teaching/sss/jupiter/Web/Rest3Bdy.htm

5. Targ, S.M. The brief course of theoretical mechanics. Moscow, Nauka, 1970 (Russian)

6. Yavorsky, B.M., Detlaf, A.A., Milkovskaya, L.B. and Sergeev, G.P. The course of physics, vol. 1. Moscow, Viccaya schola, 1963 (Russian)