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alt="images"/> due to the gravitational field created by the mass
is therefore given by
(2.44)
and is independent of the test mass, .
The concept of gravitational potential between a pair of isolated masses, , can be extended to a system of point masses, where the net gravitational acceleration caused by point masses, , on the particle of mass is the vector sum of all the individual gravitational accelerations given by
(2.45)
or
(2.46)
where
(2.47)
is the net gravitational potential experienced by the particle due to the gravity of all the other particles.
The potential energy, , of the ‐particle system can be defined by
(2.48)
to be the net work done by the gravitational forces to assemble all the particles, beginning from an infinite separation, , where . Thus a finite separation of the particles results in a negative potential energy (a potential well), escaping from which requires a positive energy expenditure.
The gradient of with respect to gives the negative of the gravitational force, , on the particle as follows:
(2.49)
Hence the right‐hand side of Eq. (2.37) is expressed as follows:
(2.50)
A substitution of Eqs. (2.38) and (2.50) into Eq. (2.37) yields the important result that the total energy of the system is conserved:
(2.51)
or This is true for any system solely governed by gravity.
To demonstrate another constant of the ‐particle system, consider the vector product of Eq. (2.33) with , followed by summing over all particles:
(2.52)
Because , all the terms on the right‐hand side of Eq. (2.52) vanish, resulting in the following:
(2.53)
or
(2.54)
This implies that the ‐particle motion takes place in a constant (or invariant) plane containing the centre of mass. The constant vector is normal to the invariant plane, and is termed the net angular momentum of the system about the origin . This is the law of conservation of angular momentum in the absence of a net external torque about .
The conservation of linear and angular momentum, as well as the total energy of the ‐particle system, is valid for any system ruled only by gravitational forces. The conservation principles are also valid for ‐bodies of arbitrary shapes, as no restrictions have been applied in deriving those principles for the ‐particle system. A body is defined to be a collection of a large number of particles. Thus the particles can be grouped into several bodies, each translating and rotating with respect to a common reference frame. However, solving for the motion variables (linear and angular positions and velocities) of a system of bodies (referred to as the ‐body problem) requires a numerical determination of the individual gravity fields of the bodies, as well as an integration of the first‐order, ordinary differential equations governing their motion. The next section discusses how such differential equations are derived for a body. The solar system is an example of the
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