Global Journal of Researches in Engineering, A: Mechanical & Mechanics, Volume 22 Issue 1

where a = acceleration of body B relative to body A, v = the relative velocity, r = the separation and = the unit vector from body A to body B. Also c = speed of light, K = G (m A + m B ) and v r is the radial component of velocity. Note that G is a constant which could be incorporated into the definition of the quantity of matter. These equations reduce to equation (1) when v << c . 2 2 2 φ 2 r V N e e a a a       + −= + = c v r K r K (4) where φ is the angle between the velocity and the radius. It should be noted that the velocities of the individual bodies do not appear in these equations, only the relative velocity. For two isolated bodies the relative motion is the only measureable value. A convenient definition of force is ( ) 2 1 22 2 2 2 r r ev v e a P × × +       − −= = cr mGm c v r mGm B A B A µ (5) where ) /( B A B A m mmm + = µ , the reduced mass. By definition of the centre of mass (or the centre of momentum) the total momentum is zero with reference to the centre of mass. It is now proposed that the motion of the centre of mass of two bodies is not affected by collision. From this it follows that for a group of particles the motion of the centre of mass is unaffected by internal impacts. The relative acceleration is only radial when the relative velocity is either radial or tangential. In general the moment of momentum can be shown to be a function of the relative position. So, for an elliptic orbit it remains within bounds. General inferences from equation (2). Reverts to Newtonian form when v << c . The second term of (2) is normal to the velocity. Moment of velocity (or moment of momentum per total quantity of matter) is shown to be a function of r . The equivalence of inertial mass to gravitational mass does not arise. c) Modified Equations An extra term is added in the direction of the relative velocity. This will affect the speed of light but not its deflection. As the term is a function of c 4 it only has a very small effect on the motion of large bodies in Solar orbits. The new equation is 2 3 2 2 2 2 t e e a a aa 2 r V N       +       + −= + + = c v c v r K c v r K r K r T φ (4a) Where v is the relative velocity and v r is the radial component. Also t = v/| v| is the vector in the direction of the velocity. The angle between the velocity and the radius is φ. For the third term the sign of the acceleration depends only on the sign of the radial velocity. K = G(m A + m B ) and c is the speed of light in a gravity free vacuum. This equation will be considered to be the basic for Post Newtonian Gravity. Justification will come from agreement with verified experimental data. © 2022 Global Journals Global Journal of Researches in Engineering (A ) Volume XxXII Issue I Version I 2 Year 2022 Gravitomagnetics a Simpler Approach Applied to Dynamics within the Solar System The equation can also be written in terms of the Newtonian part plus the gravitomagnetic part If v = c the first term of (2) vanishes so that there is no change of speed.