The Web Site of Science-Philosopher Viv Pope

April 2003

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Pope's Unifying Angular Momentum Equation

The straight-line, so-called 'inertial' motion, or momentum mv, envisaged by Galileo and Newton was seen, by Pope, as counter-empirical. As he saw it, straight-line motion is purely theoretical, being a special, ideal case of angular momentum of magnitude mvr with r at theoretical infinity (a straight line and a circumference of infinite radius being one and the same). But with r at infinity, the magnitude of the angular momentum L = mvr is also infinite, which is unreal. It follows, then, that for all real (i.e., finite) angular momentum r is finite, hence the motion naturally curved or orbital, without having to postulate invisible in vacuo 'forces' being responsible for these curved trajectories.

Moreover, angular momentum is a naturally paired and balanced (i.e., non-local) relation between masses according to the equation

(3)        L = mvmrm + MvMrM

By this relation, the motions of the masses m and M are automatically linked, so that any forced change in the motion of either of the masses immediately affects that of the other, that is, directly, in accordance with Newton's third law of instantaneous and reciprocal action-reaction. This, of course, is without any need to postulate mediation by the fictitious and invisible 'forces', of classical conception, such as those of 'gravitation', 'electrostatics' and 'magnetostatics'.

Pope's unifying angular momentum formula (stated simply, for orbital motion assumed circular and in the same plane *) is therefore:

(4)        L = mvr = 2(KO + KX) r/v = G mM/v

where L is the orbital angular momentum, KO is the orbital kinetic energy and KX is some extra-orbital kinetic energy such as, for example, spin or some other convoluted form of angular momentum. G (call it 'curly gee') is a variable which, when KX is negligible, is equivalent to the usual 'gravitational' constant G (= 6.67259 10-11 N m2 kg-2).

* This simple, ideal formula, for magnitudes only and ideally circular motion, was later developed, by A.D. Osborne, to include the more real, elliptical orbits and angular momentum vectors. This development became the basis of the Pope-Osborne Angular Momentum Synthesis, or POAMS (see website ).

When KX is made equal, in mechanical units of joules, to the so-called 'electron charge' in coulombs (q = 1.602 10-19C) multiplied by the ionisation potential for elementary hydrogen (13.595 volts), then KX is 2.179907344 10-18 joule. The orbit defined in these purely mechanical terms, with G = 1,5141474 1029 N m2 kg -2, then becomes that of the electron mass m = 9.1094 10-31 kg around the proton mass (M = 1.6725 10-27 kg) in Bohr's 'electrodynamical' model of the hydrogen atom.

In this way, with suitable step-changes in the value of G, the free orbits of 'electric' and 'magnetic' particles are included with 'gravitation' in the unified angular momentum equation. (For details see Publications Nos. 18, 28, 30, 31)

In this way, all forces become real, i.e., measurable forces. For instance, the only 'gravitational' force it makes sense to speak of is that which a body exerts on the earth's surface (e.g., on a weighing-scale) when that surface prevents the body from orbiting where it should according to the angular momentum equation for the free-motion case of G = G. The real force pressing on the earth's surface is therefore:

(5)        F = (mM/r2) (G - G )

where m is, say, a standard kilogram mass, M is the mass of the earth and r and the radius of the earth's surface at that point. In this case, G has the value, from (4):

(6)        G = Lv/mM

where L is the angular momentum of m at that point due to the earth's rotation and v is the rotational speed at that same point. (For details of this see, again, Publications Nos. 18, 28, 30, 31).