The two-body problem consists of a system of two particles, one of mass m 1 = M and the other of mass m 2 = m. The sole forces on the system are the result of a Newtonian gravitational force field...

The two-body problem consists of a system of two particles, one of mass m¹
= M and the other of mass m²
= m. The sole forces on the system are the result of a Newtonian gravitational force field that has a potential energy function

(a) For this system, show that the linear momentum of the system is conserved and that the center of mass C moves at a constant speed in a straight line. In addition, show that

Here, r¯ is the position vector of the center of mass.

(b) Show that the angular momentum of the system of particles relative to C is conserved.

(c) Show that the total energy of the system of particles is conserved.

(d) Show that the differential equations governing the motions of m₁
and m₂
can be written in the following forms:²⁶

(e) Argue that the results of Section 2.8 can be applied to (4.44) to determine the orbital motions of the particles about their center of mass C.

Chapter 5