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Mathematical Theory

Connective Physics adds a third differential term (with respect to time) to the equations of classical mechanics and electrodynamics, by adding a time delay factor to the mass or inductance terms.  For example, in the case of Classical Mechanics:

F  =  D  m d3x/dt3  +  m d2x/dt2  +  u dx/dt  +  k x

                                                                                                                        (Equation 1)

Here F represents the net external force acting on a body of mass m, D is a constant, as is the viscosity u (resistance to the movement of the body through a medium) and the restoring force constant k (such as in a spring or pendulum -- Hooke’s Law). 

Meanwhile, dnx/dtn is the nth derivative of x (the displacement) with respect to the time, t.(For the sake of simplicity, this analysis assumes non-relativistic velocities.  However, for the more venturesome individuals, see Relativistic Variations on a Theme.)

Justification for adding this additional term is based on: 1) Experiential observations (i.e. those described in The Fifth Element), 2) the mathematical logic of an Infinite Series, 3) the work and writings of Davis and Stine [1,2,3,4,5], 4) the relationship with the Zero-Point Field, Zero-Point Energy, and Hyperdimensional Physics and 5) examples of potentially confirming evidence via Sonoluminesence, Heisenberg’s Uncertainty Principle, EPR Experiment, and the concept of Mass, inertia and Mach’s Principle.

The differential equation describing an electrical circuit -- which in general includes an electromotive force (i.e. Emf, E), inductance (L), capacitance (C) and resistance (R) -- is similar in mathematical form to mechanical systems, i.e.:

E   =   W L d3q/dt3 + L d2q/dt2 + R dq/dt + (1/C)q

 (Equation 2)

In Equation 2, a third order differential term (W L d3q/dt3) has been added in the same fashion as in the Newtonian Mechanics case (i.e. the addition of D m d3x/dt3).Equation 1 (and 2) can be solved by assuming an oscillatory force related to the cosine of the frequency and time and a constant amplitude of force, i.e. F  =  Fo cos (w t).  Here, Fo is the amplitude of the force (a constant), t is the time, and w is the frequency at which the force is applied.  The solution to Equation 1 can be shown to yield in this case:

x  =  Fo cos (w t - f) / Z


Z = [ (m w2 - k)2 + (D m w3 - u w)2]1/2


tan f = (D m w3 - u w) / (m w2 - k)

 (Equations 3)

Equation 2 can be solved in an identical manner, to yield:  

q  =  Eo cos (w t - f) / Zem


Zem = [(L w2 - 1/C)2 + (W L w3 - R w)2]1/2


tan f = (W L w3 - R w) / (L w2 - 1/C)

                                                                                                                        (Equations 4)

Power Equations   For the simple emf input given by: E = Eo cos wt, Equation 4(a) can be differentiated to obtain expressions for the current, i, as well as the rate of change of current, di/dt, and the “acceleration of current”, d2i/dt2.  These expressions are then substituted into Equation 2 (where each term is multiplied by the current) to obtain the power (energy per unit time) equation for the WLRC Circuit, i.e.

i E  =   i W L d3q/dt3 + i L d2q/dt2 +  i R dq/dt + i (1/C)q

or equivalently:

i E  =   i W L d2i/dt2 + i L di/dt +  i2 R + i q /C

                                                                                                                        (Equations 5)

Each of the terms in Equation 5(b) can be identified by noting the following:   (1)  i E  is the rate at which the seat of Emf delivers energy to the circuit, i.e.  

i E  =  - Eo2 w cos (wt) sin (w t - f) / Zem

                                                                                                                        (Equation 6)

 (2)  i q /C is the rate at which energy is either stored or removed from the Capacitor, i.e.:  

i q / C  = (1/C) q dq/dt  =  - (Eo2/Zem2)(w/C)(1/2) sin 2(w t - f)

                                                                                                                        (Equation 7)

(3)  i L di/dt is the rate which energy is stored or removed from the magnetic field of the inductor, i.e.:

i L di/dt  =  + (Eo2/Zem2)(L w3)(1/2) sin 2(w t - f)

                                                                                                                        (Equation 8)

(4) i2 R is the rate at which energy appears as load and/or heat in the resistor -- delivering heat from the circuit to the external world -- and equals:  

i2 R  =  + (Eo2/Zem2)(R w2) sin2(w t - f)

                                                                                                                        (Equation 9)

(5) i W L d2i/dt2 is the rate of energy associated with the time delay, i.e.:  

i W L d2i/dt2  =  - (Eo2/Zem2)(W L w4) sin2(w t - f)

                                                                                                                        (Equation 10)

Equation 6 (item 1)is the Emf input energy, and is precisely what would be expected from classical theory. Zem involves a square root, and thus can introduce a minus sign into the equation (as can both the sine and cosine terms).  Furthermore, it should be noted that as Zem ® 0, i.e. a super resonance occurs, and the Power Input represented by i E could approach infinity, as well -- effectively, a possibly infinite drain on the power source. 

But this also raises the possibility of another source of power input.  There is always the implied assumption that the system is isolated, in which i E can not approach infinity!  But if the system is not isolated...Equations 7 and 8 (items 2 and 3) represent the fluctuating power between the capacitive and inductive elements in the circuit.  The capacitive and inductive terms are identical, except for the w/C and L w3 terms and fluctuate back and forth with the changing signs arising from the sin 2(w t - f) term. 

This is also as expected from classical theory.  At the same time, the inductive and capacitive terms are expected to be limited in terms of delivering an infinite power for the case of Zem ® 0, just as is the case of the i E term.Equation 9 (item 4) is noteworthy in that -- because of each of the terms being squared (and the assumption that one cannot have a negative electrical resistance) -- we encounter the situation where i2 R heating is one-directional.  In other words, the circuit will always act as a load and/or generate heat and radiate (conduct or convect) this energy and/or heat to the external world. 

Critically important, we see that the direction of the energy and/or heat is FROM the circuit TO the outside world.  This does not change.  Obviously then, the resistance or load cannot provide a power source (infinite or otherwise).In all respects, the power input, inductive, capacitive, and resistance terms in the equations appear to have all the characteristics normally associated in classical physics.Equation 10 (item 5), on the other hand, is something quite phenomenal!  The equation is identical in form to Equation 9 -- except for the change in sign and the term, W L w4 in lieu of  R w2. 

If we assume that both the time delay factor, W, and the inductance, L, are positive quantities, then the “inductive delay” term implies that there is a rate of energy being supplied TO the circuit FROM the external world.Furthermore, because of the potential for Zem ® 0, there is apparently no limit in the rate or amount of energy which might be supplied by the external world to the circuit!  In effect, the W term shown in Equation 10 might very well be an infinitely sized, universal source of energy and power!

The addition of the time delay term, W (and/or D) implies that the electrical circuit (or mechanical system) is NOT isolated from the rest of the universe (in accordance with Mach’s Principle), and that furthermore, the conservation of energy in the system is inapplicable in that the only “isolated system” now constitutes the entire universe!

Fundamentally, we can not arbitrarily limit the conservation principles to a system which is connected energetically and otherwise to the whole of the universe.  Such things as the Laws of Thermodynamics are ineffective here as conservation principles, just as E=mc2  eliminated the older Conservation of Mass principle.This condition is similar to that encountered in quantum physics, in which systems cannot always be isolated from one another, e.g. in the EPR Experiment, and the very special case of Alice in Barium-Titanate Land.  In addition, this analysis applies equally well to mechanical systems, where for example, the W L i d2i/dt2 term is replaced with an equivalent D mv d2v/dt2 term.


In the classical case,

q  = Eo cos (w t - f) / Zem


Zem = [(L w2 - 1/C)2 + (Rw)2]1/2


tan f =  - Rw / (Lw2 - 1/C),

(Equations 11)

a significant event occurs whenever:  

L w2 - 1/C  =  0

(Equation 12)

Equation 12 indicates that whenever the phase angle, f, equals (2n+1) p/2, the charge, q reaches a maximum value (assuming all other conditions being equivalent).  In the traditional interpretation, we have a resonant point where for a decreasing resistance R, we have a strongly increasing charge for a particular resonant frequency.  Importantly, Zem can never go to zero (and thus have the charge value reach infinity), because in a real physical circuit, R ¹ 0. This well-behaved, first resonant frequency involving the give and take of the inductance and capacitance is as expected from classical theory.In Equations 4, however, it is possible to encounter the situation where  

W L w3 - R w  =  0

(Equation 13)

and where the phase angle, f, has the value of  n p.

This second resonant frequency involving the resistance and the coefficient of the third order differential (and when f = n p) is considerably more interesting.A exceptional event occurs whenever both of the terms in the denominator Zem go to zero simultaneously.  On the one hand, the phase angle cannot equal both n p and (2n+1) p/2 simultaneously (except for n=0, or for very large n). However, finding just the right combination of w, W, L, R, and C (i.e. a particular frequency and system configuration, it is possible to obtain the conditions where:

w2   =  1 / L C  =  R / W L


W  =  R C

(Equation 14)

and the phase angle is now indeterminate. Under the conditions implied by Equation 14, Zem can go to zero, and thus the charge, q, can become infinite!.  This is an extremely important result, and suggests that the second resonant frequency (Equation 13), can, if combined with the first resonance condition (Equation 12) result in a super-resonance!  The fact that this super-resonance occurs when W = RC is an intriguing result, and one that is independent of the frequency!  The difficulty, obviously, is that in creating a system sufficiently large to make W long enough, the product of the resistance and capacitance might also exceed W.It is noteworthy that equation 14 could have just as easily been written:  

w2   =  1 / m k  =  m / D m D  =  m k

(Equation 15)

as the classical mechanical equivalent.  In both cases, we have a situation where a super resonance can occur for systems with just the right combination of physical characteristics.Equation 15 may be the reason why mechanical systems can, upon reaching resonant frequencies, not only grow unwieldy but physically come apart.

Electromagnetic circuits may not be different -- other than the fact that their explosions might not be quite so dramatic.  And, of course, electrons are inherently less likely to come apart.

Power Ratios  

Power Ratios are typically defined as the useful power output divided by the power input.  Effectively, this is the ratio of the right-hand side of Equation 5 (less power losses within the system), divided by the left-hand side.  Because there are inevitably system losses, we consequently arrive at the more traditional view of a power ratio -- which in accordance with traditional expectations will be less than unity, i.e. the difference between unity and what is actually realized as useful power constitutes the system losses.I.e., traditionally, a Power Efficiency, P%, might be defined as:  

P%  º  1 - e  =  {L i di/dt  + i2 R + i (1/C) q } / { i E }  - e

                                                                                                                                                                (Equation 16)

where we have identified e as representing the ratio of the internal system power losses to the input power, i.e. e = system power losses / i E. [Obviously, if there are no system losses, the Power Efficiency is identically equal to unity; which in the traditional view, is the limiting factor of power output/input.]

However, assumptions associated with the conservation of energy require that the Power Efficiency term (which includes the system losses) is valid only if the system is isolated and not connected to a larger system (see discussion below).  This assumption that the system is indeed isolated from the rest of the universe is considered herein to be an assumption which -- under circumstances which include highly time-dependent systems -- can no longer be considered to be valid!

Accordingly, it will be necessary to account for the fact that in reality, and within the time constraints of non-simultaneity, the useful power output (as well as the power input) has additional terms related to i W L d2i/dt2, both of which effectively connect the system to the rest of the universe.

Another view is to think in terms of a power input which consists of the power usage of conventional sources plus a power input derived from universal sources (and which does not deplete any local, conventional sources).  In this case, the power factor or power efficiency could be greater than unity, but only because the system is not being charged for or being required to account for the universal free source of power input.It is also possible to assume that the power input term (i E) -- or more specifically, the power used to energize the system-- is theoretically reduced by the inability of the system to simultaneously respond to the imposition of the emf in an “equal and opposite” fashion.  The system is thus not utilizing all of the power input, and instead, using a smaller portion (i.e. that power input derived from what is traditionally viewed as the conventional energy or power sources).  Meanwhile, the amount of power not utilized in meeting the electromagnetic equivalent of Newton’s Third Law, becomes the power exemplified in Equation 10 which is coming from the universe to the system.

For these purposes, define:

W  º  [ i W L d2i/dt2]


Y  º  [L i di/dt + i2 R + i (1/C) q]

(Equations 17)

The Power Ratio, PR, is then defined by the following:  

PR  º  { W  + Y } / { i E -  W }  -  e


PR  º  { W  + Y } / { W  + Y  -  W }  -  e

which simplifies to:

PR  º  1 -  e  +  W / Y

                                                                                                                                                                (Equations 18)

The validity of Equation 18 must ultimately be based not wholly upon any logical analysis (as described above), but upon its agreement with experimental results.  Can a system be constructed whereby the power input usage is, in fact, not reduced in its entirety such that the effective power ratio is given by Equation 18?  It can if a system is arranged such that W  ¹  0, and is large enough to account for e!

By setting the time delay factor, W, equal to zero, we return to the traditional form of the conservation of energy and PR  = 1 - e (and the so-called Power Ratio term is nothing more than the Power Efficiency in Equation 16).  The same effect would occur simply by neglecting W in terms of a potentially larger e -- or inadvertently, unknowingly including the effects of W in the determination of e.

Conservation of Energy  

Another way of looking at the implications of Equation 10 is via a very simple circuit consisting of a power supply and a resistance coil in one circuit, and a load connected to the same resistance coil in a second circuit.  A power supply can be thought of as a device where the directionality of the current and voltage are opposite.  If this power supply is now suddenly switched on -- with only the resistance coil in the circuit with the power supply -- the current and voltage will have the same direction in the resistance coil (which can be said to be a means of identifying resistance).  If the power supply is then abruptly taken out of the circuit and replaced with the load, the momentum of the current through the resistance coil will initially have the charges continue to travel in the same direction as before.  Only now the voltage is switched in direction.  In effect, the resistance coil is for a very brief time -- based on the inertia of the current -- acting as a power supply.  In this connection, we are assuming that the source of its power is the whole of the universe.

The basic assumption (which ultimately underlies the form of Equation 5) is the so-called “power supply” of the resistance coil (albeit a very temporary one) can be represented by the i W L d2i/dt2 term.  Furthermore, just as the original power supply may be connected to other parts of the universe (with, for example, an electrical extension cord), the resistance coil “power supply” may be connected with other parts of the universe via Mach’s Principle or Puthoff’s charge-connection [6].  More importantly, based on Davis and Stine’s, work, this inductive delay power term may exceed the value of the i2 R power input!  The inertial power source is now the expanded system, i.e. the universe.

There is no violation of the conservation of energy here, but merely the recognition of an as yet unidentified “extension cord” which links the i W L d2i/dt2 power term to other parts of the universe.  Very importantly, the suggested nature of this link is likely to be due in some manner with the inertial field -- and probably connected in some manner to the Heisenberg Uncertainty Principle, the Zero-Point Field, and Mach’s Principle. 

Puthoff [6] has already argued that a charged particle is continually exchanging energy with all other charges in the universe.  This critically important work provides clear evidence of what must be considered to be an extension of Mach’s Principle which connects the masses of the universe in order to explain inertia.

It is worth noting that there are two aspects of what constitutes The Law of the Conservation of Energy (aka Laws of Thermodynamics).  These are:

1)  Energy can be neither created or destroyed in a closed system.

2)  The total energy of a closed system is constant.

The first aspect thinks in terms of a transformation or transfer of energy from one form to another, while the second aspect is a more general condition of what has already been implied in Newton’s First Law (an isolated mass continues at rest or in uniform motion).

But if the only truly closed system is a universal one, then all other so-called closed systems do not exist. From a purely practical viewpoint, variations from a local system‘s adherence to energy conservation may be sufficiently minute that we can operate in a normal manner (where apparently energy conservation laws continue to function).  But such systems would operate only in the same context that sub-luminal velocities do not need to include relativistic terms.  A fundamental truth, however, is that many macro examples are occurring whereby the deviation for local energy conservation laws is no longer sufficiently minute so as to be neglectable.

One could also challenge that the three-dimensional spatial universe (with its one dimension of time) might not be a closed system either.  For example,theories of Zero-Point-Energy, Hyperdimensional Physics, and Superstrings assume additional dimensions as a crucial aspect.  Mach’s Principle of connectedness may very well utilize such additional dimensions as a means of connecting the universe -- see, for example, Puthoff [6].  Furthermore, if we were to consider cosmological theories originating from Hoyle and others of a “quasi-steady-state universe [7], we might enlarge upon this and view the universe as a “lumpy mattress” where entropy is increasing in one locale (our observable universe, for example) and decreasing in another locale.  Meanwhile, all locales could be connected via Mach’s Principle (or Puthoff’s [6,8] argument), the means of connection being extra-dimensional.  The fact that the connecting link might be dependent upon Heisenberg’s Uncertainty Principle is also relevant.

It is worth noting that prior to Einstein’s E = m c2, there existed two conservation laws: one of mass and one of energy.  By relating energy and mass, the concept of the law of conservation of energy was expanded.  Subsequently, in an attempt to account for the possibility of a Maxwell’s Demon, information was recognized as a form of energy and the law of conservation of energy again expanded.  Clearly, Mach’s Principle is just another expansion of the law (while zero-point and/or superstring theories may be yet another).And if the above is insufficient, there’s More Math, and, of course, The Sixth Element.Or for the ramifications:  

Zero-Sum Games          Inertial Propulsion         Relativistic Variations on a Theme

New Energy Ramifications         Sonoluminescence         Consciousness


Connective Physics         Causality         Objectivity         Lothar Schäfer



[1]  Davis, William O., private notebook, June 9th, 1961.

[2]  Davis, W. O., Stine, G.H., Victory, E.L., and Korff, S.A., “Some Aspects of Certain Transient Mechanical Systems”, Presentation to the American Physical Society, April 23, New York University, 1962.

[3]  Davis, William O., “The Fourth Law of Motion”, Analog Science Fiction/Science Fact Magazine, May, 1962.

[4]  Stine, G. Harry (Correy, L.), “Star Drive”, Del Rey, New York, 1980.

[5]  Stine, G. Harry, “Detesters, Phasers and Dean Drives”, Analog Science Fiction/ Science Fact Magazine, June, 1976.

[6]  Puthoff, H.E., “Source of Vacuum Electromagnetic Zero-Point Energy”, Physical Review A, Vol. 40, 1989, pg 4857-4862.

[7]  Burbidge, Geoffrey, Hoyle, Fred, and Narlikar, Jayant V., “A Different Approach to Cosmology”, Physics Today, April, 1999.

[8]  Puthoff, H.E., Haisch, Bernhard, and Rueda, Alfonso., “Advances in the Proposed Electromagnetic Zero-Point Field Theory of Inertia,” American Institute of Aeronautics and Astronautics, AIAA-98-3143, 1998.



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