Energy of Motion
The inverse-square force, being spherically symmetric, conserves the total energy of the particle. Let us determine the value of total energy in terms of the constants of motion p and e.
In fact, the constant is related to angular momentum of the particle, p = 1/A, where
We shall, therefore, determine an expression for E in terms of constant L and e. To do that, we look at one of the extreme points of orbit, the perigee or apogee. At these points, we have said that velocity of the particle has only transverse component:
Hence, kinetic energy for the particles at perigee P is given by,
Hence, the total energy, which remains conserved, can be written in terms of its value at perigee:
Substituting these two, we get,
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