V.5 No 1

63

Three-body problem in collision theory

3. The problem of three and more bodies in collision theory

To make clear the way to solve the problem of three and more bodies in collision theory, it is sufficient to consider the three-body problem, since, as we show below, the problem of more bodies is solved identically.

To solve the three-body problem, let us write its modelling equations. As in previous problem, we have them two:

the conservation equation for the system energy

and the conservation equation for the system momenta

Generally the three-body problem has to be solved in three dimensions; totally we will have four modelling equations. But we have to find 9 unknowns - three to determine each vector of bodies speeds after collision.

As we see, already the three-body problem is generally five times degenerate and the degree of indeterminacy grows with the number of interacting bodies.

On the other hand, besides the above equations (41), (42), there are absent any additional conservation laws for point masses which would facilitate to complement the basic system of equations. Namely this is the difficulty of solving and the reason of unsuccessful attempts to solve during previous generations of scientists.

Furthermore, the formulated difficulty indicates, we have to seek the solution in limits of known equations (41), (42).

To reveal, which possibilities we have to complement this system of equations in limits of this system itself, let us analyse in more details, just as in the two-body problem, the behaviour of mass centre of three interacting bodies. For it, suppose we have three bodies of mass m₁ , m₂ and m₃ which at the initial moment of time are located at the points Q₁(x₁, y₁, z₁), Q₂(x₂, y₂, z₂) and Q₃(x₃, y₃, z₃) of the stationary inertial frame S and move towards the point A with the speeds ₁ , ₂ and ₃. Taking into account the above study of two-body problem, stipulate that all bodies will reach A in some time t_A, as is shown in Fig. 7.

Fig. 7. Visualised statement of three-body problem in collision theory. Red line means the trajectory of mass centre of the system of bodies

Of course, by an analogy with the two-body problem we can determine the location and mass centre speed for the system of three bodies:

where m = m₁ + m₂₁ + m₃ . In Fig. 7 the mass centre trajectory is denoted by a red line.

The same as in case of two bodies, we can easily show that the mass centre will move inertially. Actually, we can write the kinetic energy of the system as follows:

In turn, proceeding from (44) and noting (43), we yield

Consequently,

and this proves our statement.