Newsgroups: sci.physics
Subject: Re: Davis Mechanics
From: jim@netlink.cts.com (Jim Bowery)
Date: Fri, 04 Sep 92 08:35:44 PDT
Organization: NetLink Online Communications, San Diego CA

THE FOURTH LAW OF MOTION
 
BY DR. WILLIAM O. DAVIS
(Analog magazine May 1962)
 
DR. WILLIAM O. DAVIS
 
Dr. William O. Davis is Director of Research of the Huyck 
Corporation of New York City.  He was born in Buffalo, New York 
on November 11, 1919.  He received his A.B. degree from New 
York University in 1939, with study at Cambridge University, 
England, on a Carnegie Scholarship from 1937 to 1938.  He 
received his Ph.D. in physics from New York University in 1950.
In 1940 Dr. Davis joined what was then that Army Air 
Corps as a flying Cadet.  At the close of World War II he was 
commanding the 824th Bombardment Squadron (B-24's) of the 
Fifteenth Air Force.
 
After completion of his graduate training in 1950 he became 
a member of the staff of the Los Alamos Scientific Laboratory as 
an Air Force Officer, where he worked on nuclear weapons 
development.  From 1953 to 1955 he was Chief of Scientific 
Research, Headquarters, Air Research and Development 
Command and from 1955 to 1957 Deputy Commander of the Air 
Force Office of Scientific Research.  It was while in this post that 
Col. Davis initiated the Project Fireside program discussed in the 
article "Pie In The Sky" in the August, 1961 ANALOG, the first 
U.S. Spaceprobe program to get instruments out into the van 
Allen radiation belt.  And the one U.S. Space program to get there 
ahead of the Russians.
 
His last assignment in the Air Force, from 1957 to 1958, as 
Assistant to the Director of Laboratories, Wright Air 
Development Center.  He resigned as a Colonel from the Air 
Force in 1958.
 
In 1958-59 he was Vice President for Research, Executive 
Vice President and President, in succession, of the Turbo 
Dynamics Corporation.  His research activities have covered the 
fields of cosmic ray neutrons, special weapons development and 
the management of government and industrial research program
 
 
TEXT OF ARTICLE
 
Fundamentally, science must be a series of successive 
approximations to reality.  It simply is not possible to arrive at 
absolute truth with a finite number of investigations.  Physics at 
the Freshman level is a very straight-forward subject.  Facts are 
well known, relationships are stated in forthright terms without 
equivocation, and there is little room for doubt.  It takes three 
years or more of graduate school before it finally dawns on a 
budding scientist that the whole structure of science, so 
monumental when viewed from a distance, is a cracked and 
sagging edifice held together with masking tape and resting on the 
shifting sands of constantly changing theory.  Nothing is known 
with any real certainty.  Some things are merely more probable 
than others.  Well-known theories and even laws turn out to be 
only partially confirmed hypotheses, waiting to be replaced with 
somewhat better partially confirmed hypotheses.  If there is one 
thing we know about every theory in modern physics, it is that it is 
wrong, or at least incomplete.  Sooner or later somebody will come 
along with a more general theory of which the old theory is seen to 
be  a special case.
 
This is not a criticism of science, but merely a description of the 
scientific method.  Like democratic processes in general * for 
science is necessarily a democratic process * this method may 
seem a bit sloppy, and even bog down from time to time with 
authoritarian red tape, but it's the only method we know at the 
moment, and its results have been spectacular.
 
Progress tends to take place by a more or less random series of 
spurts in different fields at different times.  The Laws in a given 
field of science will appear adequate for many years and little 
attention will be paid.  The glamour is all in another field.  Then 
little anomalies start showing up in engineering or scientific 
applications of the Laws which are accounted for by engineers 
through "Finagle Factors" and explained by scientists as "second-
order effects,"  Usually, it takes a really major anomaly to generate 
the necessary speculation to derive a new and improved theory.
The field of Mechanics is a case in point.  Euler, Lagrange, 
Hamilton, and Newton did such a beautiful job of synthesizing a 
workable theory that no further attention was paid to Mechanics 
for many years.  Certain anomalies in astronomical data led 
Einstein to the Theory of Relativity, and the study of atomic 
physics led to the development of the Quantum Theory, but the 
mechanics of simple everyday systems is still founded squarely on 
the cornerstone of the three Laws of Motion of Newton.
 
These Laws are usually stated in the elementary textbooks as:
 
First Law:  Every body tends to remain at rest or in uniform 
motion in a straight line, unless acted upon by an outside force.
 
Second Law:  An unbalanced force acting on a body causes the 
body to accelerate in the direction of the force, and the 
acceleration is directly proportional to the unbalanced force and 
inversely proportional to the mass of the body.
 
Third Law:  For every action, there is an equal and opposite 
reaction.
 
By and large, the Laws of Motion have stood the test of time 
remarkably well.  In spite of the newer concepts of Relativity and 
Quantum Mechanics, the basic meaning of the Laws has merely 
been interpreted, not challenged.  Even then, interpretation has 
only been required in the case of very small or very large systems 
and in everyday life Newton still reigns supreme, much to the 
distress of a host of would-be inventors.
 
There have been anomalies from time to time, of course, but it 
has normally been possible to attribute these to lack of sufficient 
information or understanding.
 
For a number of years, the author has been intrigued with a 
group of anomalies having to do essentially with rate of strain in 
materials and mechanical systems.  It is well known, for example, 
that the strength of many materials is a function of rate of strain.  
Some materials such as silicon putty, display this characteristic to 
a marked extent.  The field of terminal ballistics is replete with 
odd rate of strain effects such as the tendency of long projectiles to 
penetrate targets by means of a series of hammer blows.  There are 
many other examples which come to mind from almost every field 
of science and engineering.  Generally speaking, it is very difficult 
to explain such effects in terms of Newtonian physics, although it 
has been done.
 
Several years ago, the importance of rate-of-onset effects in 
general was highlighted for the author by studies of the dynamics 
of the high speed pressing of paper and textiles in the laboratories 
of Huyck Corporation based on a theoretical study by E.L. 
Victory.  The need to solve certain problems relating to very high 
speed paper machines led to the creation of a small research 
program involving both theory and experiment.
 
Interest in this field was further heightened by the discussions 
of the Dean Drive started by John Campbell in these pages in 
1960.  Our earlier investigations led the author to the hypotheses 
that the Drive, if it worked might be explained in some way by 
rate-of-onset effects.
 
This article is a preliminary report on findings of general 
interest which have resulted from this research program.  It is not 
primarily designed to explain "reactionless" drives, but certain 
conclusions will certainly have a bearing on the evaluation of these 
devices.  Because this is a preliminary report, there will be no 
attempt to offer proof of assertions as such, although it is believed 
that existing data will tend to confirm theoretical predictions in the 
areas that have been investigated after the analysis has been 
completed.
 
A speculative theory will be proposed which has been logically 
and mathematically explored to some extent and checked against 
known data in various areas of science and engineering.  No claim 
is made that this theory has been "proven" any more than any 
theory is "proven."  We do believe that this approach will shed 
some light on certain problems.
 
Let us take a moment to review the meaning of the terms 
describing a body in motion.  First of all, a body has a position, 
described in terms of its distance and direction from some 
arbitrary point or "origin."  To this distance we assign the letter, s.  
If the body is in motion it will move a very small distance, ds, in a 
very short time, dt, so that its instantaneous velocity will be given 
by 
 
    v=ds/dt.
 
Note that the body might move in such a way that its distance 
from the origin remained constant and only the  angle or direction 
changed; it might move directly away from the origin in a straight 
line so that only its distance changed; or it might move in some 
intermediate way so that both its distance and direction change at 
the same time.  Thus in general v must represent a rate of change 
of position in any of these ways.  It represents not only a speed, 
but a speed in a certain direction, and therefore, like the 
displacement, it is called a "vector."
 
If now, we make a small change in the velocity, dv, in a small 
period of time, dt, the body will have an instantaneous 
acceleration vector given by:
 
    a=dv/dt                     (1)
 
But, v=ds/dt, so that we can also say
 
a=d(ds)/(dt(dt))  which is normally written:
 
    a=(d^2)s/(dt^2)             (2)
 
An expression like ds/dt or dv/dt is called a "derivative," and 
therefore v is called the "first derivative" of the displacement with 
respect to time.  Since the derivative must be taken twice to get it , 
we say that a is the "second derivative" of the displacement.  Thus, 
Newton's Second Law could be written (and frequently is),
 
F=M(d^2)s/(dt^2)                        (3)
Let us now go one step further and assume that the acceleration 
is not constant.  In other words, if we make a small change in 
acceleration, da, in a short time, dt, the body will have an 
instantaneous "surge" given by:
 
    a'=da/dt=(d^2)v/(dt^2)=d^3s/(dt^3)          (4)
 
Surge is thus the "third derivative" of displacement.  This 
quantity has previously been referred to as "rate of onset," "jerk," 
"kick," and a number of other terms.  We use "surge" because it 
seems more appropriate to describe an action that may be 
repetitive or even continuous in some systems.  We use the symbol 
a' because it is conventionally used as an abbreviation for da/dt.
It is obviously possible to continue indefinitely to define fourth, 
fifth, and higher derivatives of displacement with respect to time.  
In some cases, such as very high speed impact, this may prove 
later to be worthwhile.  In most cases, higher derivatives will 
probably shed little light on the problem and in the case of cyclic 
or oscillating motion, it can be shown that a dynamic system can 
be completely described with not more than three derivatives.  
Higher orders may be present, but can always be expressed in 
terms of combinations of the first three.

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