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I. Introduction

In what follows we will use the vector model for forces within the framework of Classical mechanics. The mathematics will be done in a fixed reference frame and spherical polar coordinates (r, θ, f) referred to origin of frame as the pole. We represent a position vector as a matrix, in order to use a Cartan type of approach to geometry:

(1)

(For details of this approach, history, notation and outstanding achievements, the reader is encouraged to visit Professor Kiehn’s web page CARTAN’S CORNER. It is, indeed, a spiritual adventure!) Then the differential displacement will be given by

(2)

with the following notations:

(3)

The reference frame satisfies the equations of motion

(4)

and the equations of structure are identically satisfied. From equations (2) and (4) we have the second differential (not exterior differential!) of the position vector in the form

(5)

with the components are given by:

(6)

with t a continuity parameter that can be taken as time. These equations allow us to describe, within the framework of the Newtonian Dynamics, the motion of a unit mass particle. Specifically, this motion is described by equations

(7)

The radial motion is here described by the first of these equations, so its nature is dictated by the coefficient of r from that equation. The variation of that coefficient is obtained directly by differentiating it and using the last two equations (7). The result is

(8)

The classical case of central forces is characterized by f^θ = f^f = 0 or

(9)

where A is a real constant. As a consequence, in this case the first of equations (1) reduces to

(10)

We will describe first the motion of a free particle and then will focus on the forces banished from the realm of ideal gases, as shown somewhere else on this web page (see The Case of Banished Forces). The equation (10) for a free particle reads

(11)

Let’s see now what the other two components of the motion are. It is better to work directly on the equations (7) rather than starting from the fact that the motion is that of a free particle and work in Cartesian coordinates. Although this last approach is really interesting and helps enlightening the issue, it will be used under other circumstances. The last equations (7) reduce then to

(12)

Taking here the angle f as independent variable, and noticing that

(13)

along the trajectory, we get the following equation for h_θ:

(14)

Here h₀ and f₀ are two integration constants. Consequently, the other area rate h_f is given by

(15)

Using these results together with equation (9), we end up with the following solution of the equations of motion:

(16)

which represents geodesics of the unit sphere, as one might expect. This little exercise has no point but to provide the general solution in the form (16) in order to be compared with an interesting case where the forces are present. Let us describe that case.

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