5 Planar Orbital Motion Kinematics In D0.In this Section, the equation of an orbital motion is first derived and then solved as a second illustration of the manner in which the concept of Existence Velocity, may be applied to relativistic problems of this nature in Pseudo - Euclidean Space - Time.
5.1 Derivation of the General Curvi - Linear Equation of Motion in a Plane in D0.Repeating (2.7) for convenience:
where for mathematical convenience spatial polar axes have been chosen and where, with reference to some stationary origin, r is the radial distance of the orbit.
Differentiating (5.1) with respect to time yields the force equation thus:
If F is purely spatial, then the temporal part of (5.2 ) is zero and m can be determined by simple integration to be:
where mo is the rest mass. Thus the mass rate is:
Substitution of (5.3) and (5.4) into (5.2) then yields after reduction, (F
is now purely spatial),
This is the most general form of the force equation for spatially accelerated curvi - linear motion in a plane in D0 and which clearly possesses a distinct symmetry.
5.2 The Case of a Purely Radial ForceIf F is purely radial (constant angular momentum ) , then in (5.2), in addition to the temporal component, the radial normal component will also be zero. Thus
which from (5.3) and (5.4) becomes
which can also be obtained from the radial normal component of (5.5).
Substitution of (5.7) into (5.5) gives
This is the equation of motion of a point mass in a plane in
D0 subjected to an arbitrary spatial radial force. Note that
substitution of (5.7) into (5.4) yields after reduction
which when substituted into (5.8) gives
This is clearly seen to be identical to the rectilinear case and provides further confirmation that the mass rate effect only exists along coincident elements of the force and velocity vectors.
5.3 Conversion of the Equation of Motion to Proper TimeTo determine the equation of the orbit it is first necessary to convert the equation of motion to the proper time of the point mass.
Conversion of (5.8) to the proper time of the point mass, is achieved as
follows. With
then
consequently with
Substitution from (5.11) and (5.12) gives
But from (5.7)
Which, when substituted into (5.14) gives after reduction
Also from (5.8) after taking the magnitude
and substitution of this into (5.16) then yields
but from (5.11)
so that this gives in (5.18)
and, from (5.19), with
and rearrangement of this then gives
so that (5.20) finally becomes
and for a purely spatial radial force, is the equation of planar motion in D0 expressed as a function of the proper time.
5.4 Derivation of the Equation of the OrbitTo obtain the equation of the orbit from (5.25), it is initially necessary to evaluate the first integral of (5.6). Rearrangement of that equation yields
Integrating (5.26) gives
and in line with convention this constant is designated, h. The equation of
the orbit may now be obtained in the usual way thus.
Putting
then
and
Insertion of (5.28), (5.29) and (5.30) and (5.31) into (5.25) then gives the
desired result for the equation of the orbit.
5.5 Solution of the Equation of the Orbit for Two Oppositely Charged Particles In a Vacuum.Assuming conditions are such that the only effect between the two particles is an electrostatic one and that their relative size is such that the smaller has negligible effect upon the larger, and ignoring any spin effects, then the force of attraction between them may be expressed as
The equation of the orbit of the smaller particle, from (5.32) and (5.33)
then becomes
to solve this equation put
this being the inverse of the perpendicular distance from a focal point of
the orbit to a tangent at any point on the spatial trajectory.
Differentiating (5.35) with respect to j gives
Substitution of (5.35) and (5.36) into (5.34) then yields
This equation can now be solved using standard methods to yield
where the constant of integration has, together with (5.35), been inserted. Rearranging (5.38) for dm/dj gives
This equation is also a standard type that can be solved using conventional
methods to yield, after some reduction
where
and where initial conditions have been chosen such that the constant of integration is zero. Equation (5.40) describes the spatial trajectory of the smaller particle about the larger and clearly, as in the literature [5], is seen to be a rotating conic section. From the second part of (5.40), this rotation is seen to be a function of the finite Spatial Terminal Velocity, c, within D0, and also, that the precession angle is a constant one being, unlike the gravitational case, independent of the term m.
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