  ## Davis Mechanics [A non-mathematical discussion of Davis Mechanics is presented in Davis and Stine.]

The equations of Classical Mechanics rest, in part, upon the foundation laid by Newton’s equations of motion.  In their simpler form, Newton’s Three Laws are:

I.  A body at rest or in uniform motion will continue at rest or in a uniform motion unless acted upon by a net external force.

II.  A net external force upon a body will result in an acceleration of the body which is proportional to the applied force.

III.  For every action, there is an equal and opposite reaction.

Newton’s First Law -- also known as Galileo’s Law -- is essentially a special case of the Second Law.  The Third Law, meanwhile, may be thought of as a conservation law -- in the sense that a net external force on a body will result in an equal and opposite, reactive force by the same body.  This leads ultimately to other fundamental laws such as the laws of conservation of energy and/or conservation of momentum.

Newton’s Second Law, and for that matter much of Classical Mechanics, can be derived from the following:

F  =  m Sn=0  Dn dnX/dtn

(Equation 1)

where F represents the net external force acting on a body of mass m, Dn are constants, and dnX/dtn is the nth derivative of X (the displacement) with respect to the time t.

Classically, the upper limit of the summation corresponds to n = 2, such that:

F  =  m D2 d2X/dt2  +  m D1 dX/dt  +  m Do X

(Equation 2)

where mass is defined such that D2 º 1, D1 is associated with the viscosity or resistance (m or R) to the movement of the body through a medium and equal to m/m, Do is associated with the restoring force constant k (such as in a spring or pendulum) and equal to k/m.

Davis, et al  extended the summation in Equation 1 to a case of n = 3 and thus assumed a modification of equation 2 such that:

F  =  D3  m d3X/dt3 +  m d2X/dt2 +  m dX/dt +  k X

(Equation 3)

where D3 -- like m, m, and k -- is assumed to be a constant.  (Inasmuch as mass is not constant in relativistic situations, the below analysis continues to assume non-relativistic velocities.)  The exact nature of D3 may not be immediately obvious, but with respect to what we have termed Davis Mechanics, The Fifth Element, or Connective Physics, we know that D3 has units of time.

[Relativity is briefly considered in Relativistic Variations on a Theme.]

Davis, Stine, Victory, and Korff also considered the case of a long projectile impacting armor plate at high velocity such that the general solution of Equation 3 is given by :

X  =  C1 exp A1 t  +  C2 exp A2 t  +  C3 exp A3 t  +  Xo(t)

(Equation 4)

where the C’s are constants to be determined by the boundary conditions, the A’s are functions of the constant coefficients D3, m, m, and k, the expression Xo(t) depends upon the nature of the applied force F(t), and exp is the exponential function (natural log).

According to Davis, et al,  “an examination of the homogeneous solution reveals the differences in behavior predicted by the introduction of third derivative force into the analysis.  The solutions of the classical Equations of Motion involving forces proportional to displacement, velocity, and acceleration predict behavior of several sorts.  Depending on the coefficients, the motion may be oscillatory, critically damped, or overdamped.  If the motion is oscillatory, resonance is predicted at a basic frequency and its harmonics, with the sharpness and amplitude of the resonance a function of the first derivative term.

“The addition of the third derivative has two significant consequences.  First, a new set of resonant frequencies is determined by the relationship between the first and third derivative, with damping determined by the second derivative coefficient, to wit, the mass.  In addition, a number of additional resonant frequencies are possible due to the linear superposition of solutions.  Second, certain combinations of the coefficients can lead to hyperbolic solutions which would indicate that systems are possible which are unstable in the presence of certain boundary conditions regardless of the form or frequency of the driving force.  No consideration has yet been given to the form such systems might take other than to note that they would in general be characterized by very large delay times and a strong restoring force.  These two factors are probably incompatible in the type of mechanical systems to be generally found in practice.”

Furthermore, “the significance of this approach to analysis is not limited to high speed impacts or isolated starting transients.  It can also be applied to systems subjected to oscillatory force where the period of the driving force is substantially less than the Critical Action Time of the system to which the force is applied.”

This latter point is critical in that most dynamic systems in physics and engineering involve oscillatory forces and/or are cyclical in nature (or can be represented mathematically as such), and can thus be written as a combination of sines and cosines.  A “sawtooth” or “square” wave pattern, for example, can, despite the abrupt discontinuities in the wave shapes, be closely approximated by an Infinite Series of trigonometric sines (or cosines).

Accordingly, it is reasonable to make assumptions with regards to the external force acting upon a body, and the resulting displacement of that body, both of which reflect this cyclical nature.

For example, the oscillatory force given by:  F  =  Fo cos (w t) yields a solution of the form:

X  =  a cos wt  +  b sin wt  =  [(a2 + b2)]1/2 cos (wt - f)

(Equation 5)

where

tan f  =  b / a

(Equation 6)

In this case a and b are constants, and f is the phase angle.

In using Equations 5 and 6 to solve the differential form of Newton’s Second Law (Equation 2) -- and assuming m and k to be negligible -- we obtain the particular solution:

X  =  Fo cos (wt - f) / m w2

where

tan f  =  0

(Equations 7)

We can use the same logic and methodology in solving Equation 3 as we did in solving Equation 2.  The particular solution of Equation 3 for an applied oscillatory force given by F = Fo cos (w t) -- again assuming m and k to be negligible -- is:

X  =  Fo cos (wt - f) / [ m w2  ( 1 + D32  w2 )1/2 ]

tan f  =  D3 w

(Equations 8)

This is in sharp contrast to Equations 7, where D3 had been assumed to be zero.

Davis goes on to show many of the possible implications of this result -- including the fact that: 1) a phase angle (f) exists between the applied force and the resulting acceleration whose tangent increases with increasing frequency, and 2) D3 can be viewed as a time delay for a non-point mass to interact to an applied force.  Inasmuch as any physical body has dimensions, a force applied at any point can not simultaneously effect the entire body, the delay in its effects being the time required for the information on the force to make its way across the mass’ dimension.  This delay time D3 was defined by Davis as the Critical Action Time.  As such the CAT imposed a new restriction on Newton’s Third Law in terms of the physical body’s value of D3.

Typically, the value of the Critical Action Time is extremely small.  It takes only a short time for the information of the initiation of a force on a metal rod, for example, to reach all parts of the rod.  However, if an oscillatory force with an extremely high frequency is applied, then the product of D3 w may no longer be a neglectable term.  In the case of a force applied to a mass, but then removed such that the time of application is less than D3, portions of the mass would not receive the message in time to respond.

Davis goes on to consider the possibility of the nonconservation of angular momentum.  For example, a body orbiting a central force field has a resulting equation of motion of:

D3 m R3  +  m R2  =  m Ro q12    +  V / R

(Equation 9)

where Rn is the nth derivative of R with respect to t, qn the nth derivative with respect to t, and  V / R the partial derivative of the system potential energy with respect to R. The solution to Equation 9 is:

Po  º  m Ro2 q1    =   C1  +  C2  m exp( - t / D3)

(Equation 10)

where C1 and C2 are constants and exp is the exponential of e, the base of the natural logarithm.  Equation 10 suggests that the angular momentum Po is not conserved -- except in time periods large in comparison to the Critical Action Time, D3.

Equation 10 might also be applied to the mechanics of the hydrogen atom, if one, for example, were to assume that the increment of kinetic energy of the electron in its orbit at a given moment is equal to the time-rate-of-change of the angular momentum, i.e.:

E  =  d Po / d t  =  - ( C2  m / D3 ) exp( - t / D3)

(Equation 11)

At t = 0, Eo = - C2  m / D3.  At a later time, t2 (which is assumed to be significantly larger than D3), E2 = 0 and thus:

D E  =  E2  -  Eo  =  C2  m / D3

But if we also assume that D E = h n, then

C2  =  D3 h n / m

For the special case of D3 = 1/n, C2  =  h / m; thus an interesting hint to the possibilities inherent in Davis Mechanics.

Davis has also written the energy equation of the system being described as:

X1              X1                          t1                                           t1               t1

ò F dX  =    ò dV  +  [m v2 / 2]½  +  D3 [ d (m v2 /2) / dt ]½ - D3 m  ò a2 dt

Xo               Xo                              to                                           to             to

(Equation 12)

The first two terms are Newtonian. The third term represents the change of kinetic energy during the Critical Action Time that is not immediately available to the system. The fourth term very probably represents a radiation (but not necessarily of the electromagnetic variety).  If we set the variation of this expression to zero (i.e. Hamilton’s method), we find that for a free system such as the orbiting system of the hydrogen atom, the sum of the two right hand terms must vanish.  This represents an extension of the principle of the Conservation of Energy to a system possessing third derivative characteristics.  In effect, each quantum of kinetic energy net available to the mechanical system must be counterbalanced by an equal and opposite quantum of radiant energy.

The work on these theories of Davis, Stine, Victory, and Korff was done primarily in the 1960s.  It is assumed that experiments were also conducted, but for inexplicable reasons the theories barely saw the light of day.  This is despite the fact that Davis, et al were not adverse to publication.  Unfortunately, establishment physics was apparently not ready for the implications of what amounts to the early stages of Connective Physics.  Both Davis and Stine published in Analog Science Fiction/Science Fact Magazine, where the editors did not have to decide whether the theories were fiction or fact, and where both authors could spent considerable time considering the experiential basis for their theories, as well as the ramifications of what happens if its true.

Forward to:

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References:

  William O. Davis, “The Fourth Law of Motion”, Analog Science Fiction/Science Fact Magazine, May, 1962, and William O. Davis, G. Harry Stine, E. L. Victory, and S. A. Korff, “Some Aspects of Certain Transient Mechanical Systems”, Presentation to the American Physical Society, April 23, 1962, New York University.

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