are the speeds of the particle at the points A and B, respectively. Thus the net work done on a particle equals the net change in its kinetic energy, .
2.5 The n‐Body Problem
Gravity, being the predominant force in space flight, must be understood before constructing any model for space flight dynamics. Consider two particles of masses, and , whose instantaneous positions in an inertial frame, OXYZ, are denoted by the vectors, and , respectively. The relative position of mass, , with respect to the mass, , is given by the vector . By Newton's law of gravitation, the two particles apply an equal and opposite attractive force on each other, which is directly proportional to the product of the two masses, and inversely proportional to the distance, , between them. The equations of motion of the two particles are expressed as follows by Newton's second law of motion:
(2.29)
(2.30)
being the universal gravitational constant. Adding the two equations of motion yields the important result that the centre of mass of the two particles is non‐accelerating:
is the position of the centre of mass. This approach can be extended to a system of particles, where the particle has the following equation of motion:
where locates the particle from the ith particle (Fig. 2.2). A summation of Eq. (2.33) for all the particles produces the following result:
(2.34)
because . Thus Eq. (2.31) is seen to be valid for the ‐particles problem, where the centre of mass is located by
(2.35)
The non‐accelerating centre of mass is the result of the law of conservation of linear momentum in the absence of a net external force on the system of particles. Integrated twice with time, Eq. (2.31) shows that the centre of mass moves in a straight line at a constant velocity:
(2.36)
where are constants (the initial position and the constant velocity, respectively, of the centre of mass).
Figure 2.2 A system of particles in an inertial reference frame OXYZ.
For the convenience of notation, consider the overdot to denote the time derivative relative to the inertial reference frame. Taking a scalar product of Eq. (2.33) with , and summing over all particles, we have
Before proceeding further, it is necessary to consider that gravity is a conservative force, because, as will be seen later, it has no influence on the net energy, , of a system. A force which depends only upon the relative position of the masses (as gravity does) is a conservative force, and can be expressed as the gradient of a scalar function, called the gravitational potential. The gradient of a scalar, U, with respect to a position vector, , is defined to be the following derivative of the scalar with respect to the given vector,
(2.39)
Thus, the gradient of a scalar with respect to a column vector is a row vector.
Consider, for example, an isolated pair of masses,
