applications of third order o.d.e.'s and cubic complex variables
DESCRIPTION
The search for a deeper understanding of the nature of physical interactions at an elementary level will continue to be severely stymied if we continue to use mostly second order governing equations. Nevertheless, Newtonian mechanics (and gravitational theory) as the original corner stone of modern physics, has played a major role in the mechanization of the modern world and even in the exploration of the Solar System.Quantum Electrodynamics can be characterized as a theoretical framework in which theoretical predictions agree with experimental results with a very high level of accuracy. QED spawned Quantum Filed Theory which is the basis of the Standard Model of particle physics. The Standard Model has had great success in describing elementary particle interactions.General Relativity has had good success in describing Solar System dynamics but when applied to much larger regions of the Cosmos, there arises such hypothetical (and so for totally mysterious) entities such as dark matter and dark energy. These are needed to hold GR together. It is estimated that nearly 70% of the Universe is composed of dark energy (which so far has not been directly observed)String Theory, which is supposed to unify all of the four fundamental interactions, is based upon a second order wave equation for strings that has no provisions for damping. The implicit assumption is that these tiny strings have the potential for vibrating forever.It has been nearly a century since Einstein initiated attempts to unify the then known fundamental forces: electromagnetism as gravity. Since then the weak and strong interactions were discovered. But all attempts to unify all 4 forces have fallen flat. Nevertheless, the Standard Model (which describes electromagnetism along with the weak and strong nuclear forces) has great success in matching theory with experimental results.But as a consequence of the many years of futility in the efforts to unify General Relativity and the Standard Model, perhaps serious consideration should be given to the possibility that other verifiable generalizations of Newtonian Gravity might exist: which also are compatible with the Standard Model. The objective of this paper is to present physicists with the first of many new theoretical and conceptual tools that very well may help science to achieve the scientific grand slam. That Majestic Slam would be the combining of gravity, electromagnetism, the weak force and the strong force all under one beautiful and symmetric theoretical umbrella.But you can be assured that if and when the mysteries leading to this quest are uncovered, the fifth and other exotic forces will be waiting to be discovered.Share these ideas with your friends and peers and watch the expansion of your careers!RHBTRANSCRIPT
Applications of Third Order O.D.E.'s and Cubic Complex Variables to Damped Harmonic Motion and RLC Electrical Circuits By Ronald H. Brady Table of Contents
Topic Page No.
Abstract 2
Second and Third Order O.D.E.’s 5
Cubic Complex Numbers 8
The Cubic Complex Exponential Function 13
The Three Fundamental Solutions of y’’’ + y = 0 16
Third Order Governing Equation for Damped Harmonic Motion 24
Hooke’s Law 28
Straight Forward Derivation of Newton’s Laws 32
Third and Higher Order Newtonian Kinematics 35
Third Order (Postulate of) Hooke’s Law 38
Experimental Procedure 40
Derivation of a Fundamental CCV Theory Identity 45
Table of Contents: cont’d
Topic Page No.
Norms and Pseudo Norms.................................. 49
Kinematics of Simple Harmonic Motion and Damped
Harmonic Motion .........................................50
Harmonic Oscillator Electrical Circuits .................64
Tetrahedronomy ..........................................69
Some Introductory Notes on Alternative Approaches to String Theory ............................ 73
Abraham Lorentz Force ...................................75
Philosophical Remarks ..................................76
A Note from the Author ..................................77
Abstract
This paper will present additional potential applications,
to those presented in a previous paper, of third order O.D.E.’s
and Cubic Complex Variables: a commutative generalization of
ordinary complex variables. The focus of the new applications
will be upon a special and important class of unforced damped
simple harmonic motion and RLC series electrical circuits. The
concepts introduced herein are also potentially applicable to
non-relativistic string theory. These concepts will be explored
in the last section of this paper.
Introductory concepts in the (non relativistic) generalization
of Newtonian Mechanics will also be presented. These proposed
higher order laws of motion will be invariant under accelerated
coordinate transformations. This paper is an expanded version of
the previous paper, 3rd Order O.D.E./Cubic Complex
Variables/Applications by the present author, and it illustrates
specific details of applications.
This is a preliminary draft. This gives the reader the
opportunity to discover some new and exciting theoretical
developments without further delay. In the following
discussions, a change of independent variables (for
convenience) may have been made one or more times than was
necessary. But if for a given function f = f(x), the reader can
see the truth of the following:
f(x) = f(t) and df(x)/dx = df(t)/dt whenever x = t, then he can
rest assured that the discussions presented herein will flow in
a logically consistent manner.
It is well known that a second order O.D.E. of the form
y’’ + y = 0, where the primes indicate differentiation with
respect to the independent variable (which may be chosen to be
time), may be used, with an appropriate choice of units, to
describe simple harmonic motion in one spatial dimension. If we
consider y to be a function of x, y = y(x), then the solutions of
that equation involves the periodic functions sin(x) and cos(x).
These functions are related, via Euler’ formula,
eix = cos(x) + i*sin(x) .
to the Euler complex exponential function eix . The pure imaginary
unit i, of course, also plays a prominent role in the formula.
The present paper will present a derivation of a 3rd order
O.D.E. that is applicable to damped simple harmonic motion. The
solutions to that 3rd order O.D.E. will be shown to be related to
a generalization of Euler’s formula. This generalization, in
turn, will be based upon a natural generalization of complex
numbers from the plane to a Commutative Algebra in three
dimensions.
The presentation of potentially new identities and
symmetries that have the potential for physical application,
rather than formal mathematical rigor, is the
primary purpose of this paper. Therefore the style of
presentation will be informal. All of the indicated derivatives,
of all of the functions under discussion, are assumed to exist.
A brief word on the numbering of equations is in order. The
abbreviation (eq.15), for example, will be used denote
equation (15).
Second and Third Order O.D.E.’s
Now let us turn our attention to the general homogeneous
second-order ordinary differential equation
y’’ + By’ + Cy = 0 (eq.1)
where the coefficients B and C are constant. In particular
if B = 0 and C = 1, then (eq.1) reduces to
y’’ + y = 0 (eq.2)
As is well known, two (linearly) independent solutions of
(eq.2) are cos(x) and sin(x). These functions are as-
sociated with many identities such as
cos2(x) + sin2(x) = 1. (eq.3)
They also appear in the fundamental identity
eix = cos(x) + i*sin(x) (eq.4)
(known as Eulers’ Formula). A more general form of (eq.4) is
Given by
eikx = cos(kx) + i*sin(kx) (eq.4a)
where k is constant.
It is well known and can be easily shown that cos(kx) and sin(kx)
are linearly independent solutions of
y’’ + k2y = 0 (eq.4b)
which is a more general form of (eq.2), and that
cos2(kx) + sin2(kx) = 1. (eq.4c)
The reader will recall that since eikx is a linear combination of
cos(kx)and sin(kx), it is also a solution of the ordinary
differential equation (eq.4b). The equations (eq.4a),(eq.4b)
and (eq.4c), play indispensable roles in math, science, physics,
and engineering.
Let us pose the question: are there 3rd order O.D.E.’s with
constant coefficients such that similar identities are associated
with them? Well why don’t we start our search for the answer to
that question by creating a 3rd order O.D.E by differentiating
(eq.1)? Then we will see what we can find out?
Please note that later, when we discuss the kinematics of
simple and damped harmonic motion, we will replace the
independent variable x by t (which will denote time).
It is convenient to write (eq.1) in the following
form
y’’ = - By’ - Cy (eq.5)
Differentiating (eq.5), recalling that B and C are
constants, with respect to x results in
y’’’ = - By’’ – Cy’ (eq.6)
Now substitute the expression for y’’, from (eq.5)
into (eq.6) and simplify the results. We obtain
y’’’ + (C – B2)y’ – BCy = 0. (eq.6a)
If we now consider the special case for which C = B2,
then the equation above reduces to
y’’’ – B3y = 0. (eq.7)
Later on we will show that this special case corresponds
to the important case in which the quality factor q, asso-
ciated with damped simple harmonic motion, is equal to
unity. The concept of quality factor will be defined later. Before continuing, let’s recall that (eq.4b), re-written below,
y’’ + k2y = 0 [(eq.4b)]
where k is a constant, may be referred to as the one (spatial)
dimensional equation for un-damped simple harmonic motion. And
it is not difficult to show that cos(kx) and sin(kx) are
linearly independent solutions of (eq.4b). Therefore the general
solution of (eq.4b) may be written as
y = c1(cos(kx)) + c2(sin(kx)) (eq.8)
where the c1 and c2 are constants. The functions cos(kx) and sin(kx) as
indicated before are generated by the Euler Formula
eikx = cos(kx) + i*sin(kx)
where k is constant.
When B ≠ 0 then (eq.1) may be thought of as the
one spatial dimensional equation for damped harmonic motion.
Equation (eq.7) may be thought of as a 3rd order counter-
part of (eq.4b).
If we write y(n) to denote the nth derivative of y with
respect to x, then both (eq.7) and (eq.4b) may be derived
from
y(n) + Lny = 0 (eq.9)
where L is a constant. By letting n = 3 and L = -B we obtain
(eq.7). By setting n = 2 and L = k (or –k) we obtain (eq.4b).
The implications of (eq.9), when n is a non-negative
integer different from 2 or 3, will be discussed
in later papers.
Cubic Complex Numbers
In what follows we are motivated by the possibility of
finding identities, associated with (eq.7), that correspond to
the identities that are derived from (eq.2). We propose to indeed
do just that: but first we will need the help of a new1 algebraic
structure to do so. The name “Cubic Complex Numbers” (denoted by
C3) has been given to this “new” algebraic structure.
The fundamentals of Cubic Complex Numbers, which can be
shown to be a commutative algebraic ring over the set of ordered
triples of complex numbers, will now be discussed. But first we
will review some basic concepts from ordinary complex numbers.
Recall that the complex “imaginary” unit i may
be defined by
i = (-1)(1/2)
But the multiplicative identity 1 and i may be put into a
one to one correspondence with unit vectors in the plane
as follows:
1 ↔ (1,0)
i ↔ (0,1)
These one to one correspondences may be replaced by the defining equalities of
1 = (1,0)
i = (0,1)
The present author was motivated to write, in the early 1980’s, a more abstract definition of i.
i = (-1, 0)(1/2)
so that
i2 = (-1, 0) = -1
The reader will recall from elementary abstract algebra
that i is the generator of a cyclical group of order 4. We
may write
i1 = i
i2 = -1
i3 = -i
i4 = 1
Making use of the fact that i2 = -1, as the reader knows
well,we may form the product of two arbitrary complex
variables u = x + iy and v = a + ib, as follows
uv = vu = ax –by + i(ay + bx).
Using the ordered pair notations (corresponding to vectors
in the plane), we may also write
(x, y)(a, b) = (ax – by, ay + bx)
Now we will turn our attention to Cubic Complex Numbers.
We will now provide a preliminary description of the algebra
of Cubic Complex Numbers (to be denoted by C3). The reader will
recall that every field [such as the complex numbers] is also an
algebraic ring but not necessarily vice versa. It will be
convenient to denote the ring of complex numbers by C2. The
algebra C3 may be described as the set of ordered triples of
ordinary complex numbers (C2) endowed with the operations of
vector addition and a commutative rule for multiplication (which
will be given later).
Clearly C3 contains the set of 3D vectors (with real
components) as a subset. We will now make the following
identification.
1 <=> (1,0,0).
And now we will define the symbol j by the following:
j = (-1,0,0)^(1/3) (equa.10a)
so that
j3 = (-1,0,0) = -1, j4 = -j, j5 = - j2 and j6 =1. Therefore j is
the generator of a cyclic group of order 6: a fact that will be
exploited later.
For brevity we write2
j = (-1)^(1/3) (eq.10b)
and
j^3 = -1 (eq.11)
The reader will note that j as defined above is different from
any of the three cube roots of -1 <=> (-1,0), because the three
ordinary complex cube roots of -1 are -1 and a pair of (ordinary)
complex conjugate numbers.
The cyclical group (of order six) properties of the powers
of j are readily seen in the following (in which the symbol ^
denotes exponentiation.
j^3 = -1 (eq.12a)
j^4 = -j (eq.12b)
j^5 = -j^2 (eq.12c)
j^6 = j^0 = 1 (eq.12d)
j^1 of course is just j.
An arbitrary cubic complex variable u may be written as
u = a + jb + (j^2)c (eq.12e)
where, in general, a, b and c are (ordinary) complex numbers. If
v = x + jy + (j^2)z (eq.12f)
is another arbitrary cubic complex variable
then we may write the sum and product of u and v as
u + v = a + x + j(b + y) + (j^2)(c + z) (eq.13)
uv = ax - bz- cy + j(ay + bx - cz) + j^2(az + cx + by) (eq.14)
The operation of addition is obviously commutative and
it is not difficult to show that the product is also
commutative.
The sum and product operations may also be expressed in
terms of ordered triples as follows:
(a, b, c) + (x, y, z) = (a + x, b + y, c + z)
(a,b,c)(x,y,z) = (ax – bz - cy, ay + bx - cz, az + cx + by)
By definition, if u = a + jb + (j^2)c is a cubic complex
number and if v = x + jy + (j^2)z is also a cubic complex
number, then
u = v if and only if
a = x
b = y
c = z
The additive identity of C3 is (0,0,0) <=> 0. Also for a
given ordered triple of complex numbers (a,b,c),
the additive inverse is (-a, -b, -c). And it can be shown that
the multiplicative inverse of (a,b,c) is given by
(a, b, c)^(-1) = (L/S, -N/S, -M/S) (eq.14a)
where S ≠ 0 and where L, M, N and S are defined by
L = a^2 + bc (eq.14b)
M = ac – b^2 (eq.14c)
N = c^2 + ab (eq.14d)
S = a^3 – b^3 + c^3 + 3abc (eq.14e)
So then every cubic complex number
(a, b, c) ≡ a + jb + (j^2)c
for which S, as defined above does not vanish, has a unique
multiplicative inverse. This can easily be verified as follows:
(a, b, c)*(a, b, c)^(-1) = (a, b, c)*( L/S, -N/S, -M/S)
By using the rules of multiplication, the definitions for L, M, N
and S, and simplifying, it is seen that the right side of the
above equation reduces to (1, 0, 0). Therefore
(a, b, c)*(a, b, c)^(-1) = (1, 0, 0) <=> 1
when S = a^3 – b^3 + c^3 + 3abc ≠ 0, as required.
Using the definitions provided it can be shown in a straight
forward fashion that C3 is a commutative ring with identity.
The Cubic Complex Exponential Function
We will now make use of the cyclicality (of order six) of
j^n, (where n is an integer) to define the Cubic Complex
Exponential e^(jx). In a later paper we will examine the more
General form of the Cubic Complex Exponential: e^{jx + (j^2)y}.
It will be shown that the Cubic Complex Exponential e^(jx) will
generate solutions of the third order O.D.E.
y’’’ – y = 0,
just as the complex exponential e^ix generates solutions of the
second order O.D.E.
y’’ + y = 0
We have
e^jx = 1 + jx + (1/2)(jx)^2 + (1/6)(jx)^3 + ... + (1/n!)(jx)^n
+ ...
e^jx = 1 + jx + (1/2)(j^2)(x^2) + (1/6)(j^3)(x^3) + ... +
(1/n!)(j^n)(x^n) + ... (eq.15)
Now since j^n will always equal (+/-)1, (+/-)j or
(+/-)j^2, where the symbol +/- denotes “plus or minus”,
the above equation may be written as
e^jx = 1 – (1/3!)(x^3) + (1/6!)(x^6) + ...+ j[x-(1/4!)(x^4)
+(1/7!)(x^7)+...]+j^2[(1/2!)(x^2)-(1/5!)(x^5)
+(1/8!)(x^8)+ ...] (eq.16)
It will be noted that three separate infinite series
are indicated on the right of (eq.16). We may define
them, using the notation F1, F2 and F3, as follows:
F1 = 1 – (1/3!)(x^3) + (1/6!)(x^6) + ... (eq.17a)
F2 = x - (1/4!)(x^4) + (1/7!)(x^7) +... (eq.17b)
F3 = (1/2!)(x^2)-(1/5!)(x^5) +(1/8!)(x^8) + ... (eq.17c)
The reader is reminded that even though the (conventional)
notation + ... appears after the third term, on the right sides
of each of the three equations directly above, in actuality the
terms will alternate in sign.
It is easily seen that the general or the “nth” term of the
expressions for F1, F2 and F3 are x3(n-1)/(3(n-1))!,
x(3n-2)/(3n-2)! and x(3n-1)/(3n-1)! respectively.
It can also be shown in a straight forward fashion, using
the ratio test for example, that each of these infinite series
converges (absolutely) for all real values of x.
So then with the aid of equations (17a, 17b and 17c), we may
re-write (eq.16) simply as
e^(jx) = F1 + jF2 + (j^2)F3 (eq.18a)
or equivalently as
ejx = F1 + jF2 + j2F3 (eq.18b)
We will refer to (eq.18a or 18b) as the Cubic Complex
(version of the) Euler Formula. These are actually
identities because in each equation, by the definitions
set forth, each side is merely an equivalent
representation of the other side.
Using summation notation we may also write (eq.18b) as
ejx = ∑ j(k-1)Fk
where k is an integer and ranges from 1 to 3.
The three functions F1, F2 and F3 will be called the three
fundamental functions (of a single independent variable) of Cubic
Complex Variable (CCV) theory. It should be noted that similar
functions have been defined in connection with the more
general cubic complex exponential ejx + (j^2)y (or exp(jx + j2y)).
But they are functions of the two independent variables x and y.
The first function F1 (as defined in (eq.17a)) will be
called the principal function of CCV and it will be seen to play
a role similar to the cosine function in ordinary complex
variable theory.
The Three Fundamental Solutions of y’’’ + y = 0
We will now show that the three CCV fundamental functions (of a
single variable), derived in the preceding section, are
independent solutions of (eq.19).
y’’’ + y = 0, (eq.19)
which may be obtained from the more general form of (eq.19a)
y’’’ + k3y = 0. (eq.19a)
by letting the constant k = 1. We will first derive the solutions of (eq.19). The solutions of (eq.19a) will then follow in a straight forward fashion.
Recall from (eq.17a) that
F1 = 1 – (1/3!)(x^3) + (1/6!)(x^6) + ...
differentiating (term by term) with respect to x gives us
F1’ = -[(1/2!)(x^2)-(1/5!)(x^5) +(1/8!)(x^8) + ...]
where the symbol ( ’ ) denotes differentiation (with respect
to x). But the expression inside of the brackets, in the
above equation, is just F3 as defined by (eq.17c). Therefore
we may re-write the equation directly above as
F1’ = - F3 (eq.20).
Differentiation of both sides of (eq.20) results in
F1’’ = - F3´ (eq.21)
To determine F3´ we differentiate the right side of
(eq.17c). We obtain
F3´ = x - (1/4!)(x^4) + (1/7!)(x^7) +...
It is seen that, according to (eq.17b), the right of the
the above equation is F2. So we have
F3´= F2 (eq.22)
From equas.(21) and (22) it immediately follows that
F1’’ = - F2 (eq.23)
Differentiation of both sides of (eq.23) with
respect to x results in
F1’’’ = - F2
’ (eq.24)
Differentiation, with respect to x, of the
defining equation for F2, (eq.17b), we obtain
F2´ = 1 – (1/3!)(x^3) + (1/6!)(x^6) + ...
But the infinite series on the right side of the
equation above is by definition F1 . So we have
F2´ = F1 (eq.25)
Substitution of this result into (eq.24) results in
F1’’’ = - F1 (eq.25a)
or equivalently
F1’’’ + F1 = 0 (eq.26)
Now let y = F1 and obtain
y’’’ + y = 0. (eq.27)
Therefore y = F1, one of the three fundamental
functions, is a particular solution of (eq.19), which
for convenience has also been labeled as (eq.27),
as asserted. In a similar fashion, it can also be
shown that F2 and F3 are also independent
solutions of (eq.19). It will also be noted that
(eq.19) may be obtained from (eq.7) by setting
B = -1.
So then we have shown that the components of the
Cubic Complex exponential
ejx = F1 + jF2 + j2F3
are independent solutions of the third order O.D.E.
y’’’ + y = 0
just as the ordinary complex exponential
eix = cos(x) + i*sin(x)
has components that are solutions of the second order O.D.E.
y’’ + y = 0.
It is hereby conjectured that these results may be
generalized to the nth order case involving
y(n) + y = 0 (eq.27a)
where y(n)indicates the nth derivative of y with respect
to x and where n independent solutions of (eq.27a) are
conjectured to be generated by the nth order Euler complex
exponential exp((jn)*x) where jn is
defined by
jn = (-1)^(1/n) (eq.27a.1)
where
-1 = (-1,0,0,...) (eq.27b)
so that
1 = (1,0,0,...) (eq.27b.1)
is a unit vector in an nth dimensional commutative
algebraic ring which is a generalization of the
commutative algebra C3. It therefore follows that
(jn)n = (-1,0,0,...) = -1 (eq.27c)
Now we will show that the three fundamental functions,
F1,F2 and F3 of Cubic Complex Theory can be obtained, using Taylor
Series, as solutions for initial value problems. First we will
find a particular solution
of
y’’’ + y = 0 (eq.28a)
such that the following conditions are satisfied:
y(0) = 1, (eq.28b)
y’(0)= 0 (eq.28c)
y’’(0) = 0 (eq.28d)
Now let x = 0 in (eq.28a) and obtain
y’’’(0) + y(0)= 0
but y(0)= 1, from (eq.28b) so we have
y’’’(0) + 1 = 0 or
y’’’(0) = -1 (eq.29)
If we now differentiate (eq.28a) with respect to x we
will obtain
y(4) + y’ = 0 (eq.30)
where the notation y(4) is employed to denoted the 4th
derivative of y with respect to x. In general, we will
denote the nth derivative of y with respect to x by
the symbolism y(n) for (positive) integer n ( > 3 ).
By setting x = 0 in (eq.30) we arrive at the following
result
y(4)(0) + y’(0)= 0
but from (eq.28c) we have
y’(0)= 0, therefore we may write y(4)(0) + 0 = 0 or
y(4)(0) = 0 (eq.31)
Successive differentiation of (eq.30) will give us
y(5) + y’’ = 0 (eq.32)
and
y(6) + y’’’ = 0 (eq.33)
Setting x = 0 in (eq.32), we have
y(5)(0) + y’’(0) = 0
now substituting
y’’(0) = 0, from (eq.28d), will lead us to
y(5)(0) = 0 (eq.34)
And finally, in this sequence of calculations,
if we set x = 0 in (eq.33) and then substitute
y’’’(0) = -1 from (eq.29), we arrive at the
following
y(6)(0) = 1 (eq.35)
Now, thanks to the information provided by
equas.(28b,28c,28d,29,31,34 and 35), we have the
values of y and its first six derivatives evaluated
at x = 0. Therefore we can write the first seven
term of a Taylor Series3 expansion of the function
y = y(x) near x = 0. We obtain
y = y(0) + y’(0)x + (1/2)y’’(0)x2 + (1/6)y’’’(0)x3 +(1/24)y(4)(0)x4 + (1/5!)y(5)(0)x5
+ (1/6!)y(6)(0)x6 + ...
Now making the appropriate substitutions and dropping the
zero terms (of course) we arrive at
y = 1 – (1/3!)x3 + (1/6!)x6 + ... (eq.36)
Except for the differences in the notation used for
exponents, it is seen that the rights of (eq.36) and
of (eq.17a) are identical. Accordingly, they must
represent the same (convergent) infinite series and so
we can state that y = F1, as defined by (eq.17a), satisfies
(eq.28a). It can be easily verified that y, y’ and y’’ satisfies
the auxiliary conditions of equas.(28b, c and d).
Following a procedure, similar to that used above, it
can also be shown that y = F2 satisfies the equation
y’’’ + y = 0
such that the following conditions are satisfied:
y(0) = 0, y’(0)= 1 and y’’(0) = 0. Finally it can also be demonstrated that y = F3 satisfies the
same equation subject to the conditions
y(0) = 0, y’(0) = 0 and y’’(0) = 1.
We now summarize these results as follows: The equation
Fn’’’ + Fn = 0
for n = 1, 2 and 3 is satisfied respectively by the functions F1, F2 and F3, which are defined as in the equations (eq.17a, b and c). Therefore y = Fn(x), for n = 1, 2 or 3 is a solution of
y’’’ + y = 0
We will now outline the proof that z = F1(u) is a solution of
z’’’ + k3z = 0 (eq.36a)
where the primes indicate differentiation with respect to x and
where F1 = F1(x) is defined by (eq.17a), where u = kx and where
k is constant.
Outline of Proof of Solution for z’’’ + k3z = 0
Let y = F1, where F1 = F1(x), be one of the three independent solutions of
y’’’ + y = 0
as shown above, where F1 and y are thrice differentiable
functions and where the primes indicate differentiation with
respect to x.
Therefore, after changing the name of the independent variable
(and then taking the third derivative with respect to that new
variable) we obtain
d3F1(u)/du3 + F1(u) = 0 (eq.36a.1)
for an arbitrary function u.
Now let
z = F1(u)
where u = kx (as above) and k is a constant. We use the
following notation:F1* = dF1(u)/du, F1** = d2F1(u)/du2 and F1*** = d3F1(u)/du3,
z’= dz/dx, z’’ = d2z/dx2, Z’’’ = d3z/dx3 and u’ = du/dx = k
Starting with z = F1(u) we have by the chain rule,
z’ = (F1*)u’ = (F1*)k = kF1*
z’= kF1*
z’’ = k2F1**
z’’’ = k3F1***
so that
z’’’ + k3z = k3F1*** + k3F1(u)
z’’’ + k3z = k3(F1*** + F1(u))
z’’’ + k3z = k3(d3F1/du3 + F1(u))
But d3F1(u)/du3 + F1(u) = 0 per (eq.36a.1), therefore,
z’’’ + k3z = k3(0)
z’’’ + k3z = 0
as required.
More generally it can be shown that z = Fn(u) is a solution of
z’’’ + k3z = 0
where Fn = Fn(x), n =1, 2 and 3, is defined by (eq.17a),(eq.17b)
and (eq.17c) respectively. And where u = kx and where k is a
constant. We can therefore write
Fn’’’ + k3Fn = 0 (eq.37a)
for n = 1, 2 and 3.
Third Order Governing Equation For Damped Harmonic Motion
We will now recall (eq.6a)
y’’’ + (C – B2)y’ – BCy = 0.
which was derived from the equation
y’’ + By’ + Cy = 0
which is the second order governing equation for unforced damped
harmonic motion when the primes indicates differentiation with
respect to the independent variable t and B and C are constants.
Now let us consider the special case for which C = B2.
Therefore the third order equation above reduces to
y’’’ – B3y = 0 (eq.37b)
By a simple change of notation we denote the dependent variable in the last equation by z and this gives us
z’’’ - B3z = 0,
and by setting the constant B equal to (-k), B = -k, or
k = -B, we obtain
z’’’ + k3z = 0
which is (eq.36a), an equation we know how to solve.
We will now explain why the special case, when C = B2, is
important. Let’s turn our attention again to, (eq.1),
which is the governing equation for damped (unforced) simple
harmonic motion, which we re-write and re-label, for
convenience, below
y’’ + By’ + Cy = 0 (eq.37b.d.1)
and we will let B = 2ζω0 and C = (ω0)2 and so obtain
y’’ + (2ζω0)y’ + (ω0)2y = 0 (eq.37b.d.2)
where ζ is the damping ratio and ω0 is the angular frequency.
The equation above is the conventional form of the governing equation for damped (unforced) harmonic motion (See Wikipedia, Harmonic Oscillator). The quality factor q, of damped harmonically oscillating
systems, is defined by
q = 1/(2ζ). As noted above we are interested in the (important) special case when C = B2. We write
C = B2 (eq.37b.d.3)
(ω0)2 = (2ζω0)2 (eq.37b.d.4) Therefore ζ = 0.5 and substituting that value of ζ into the
formula for q, q = 1/(2ζ), gives us q = 1.
So when (eq.37b.d.3) or it’s equivalent (eq.37b.d.4) holds true then the quality factor q is equal to unity.
Also, for a value of the damping ratio ζ = 0.5, the quantity
B = 2ζω0 becomes
B = ω0 (eq.37b.d.5)
And the (eq.37b) becomes
y’’’ – ω03y = 0 (eq.37b.d.6)
The scientific literature contains a more than ample collection of discussions about under damped, critically damped and over damped types harmonic oscillators. But here we are mainly interested in the cases in which the damping ratios ranges from zero (pure sinusoidal motion) to 1 (critical damping)
The following table should be helpful
ζ
Q
Description
0 ∞
No damping: pure simple harmonic motion. The quality factor is “infinite”
0.5
1.0
ζ has a value precisely mid way between 0 and 1 and the quality factor is
q = 1
1.0
0.5
Critical Damping
Table 1
Let cos(x) and sin(x), which are two well known
linearly independent solutions of (eq.2), be denoted by
f1 = cos(x) and f2 = sin(x). We then have
f1’ = df1/dx = - f2 = - sin(x)
So then the famous identity
cos2(x) + sin2(x) = 1 may be written as
(f1)2 + (f1’)2 = 1 (eq.37b.d7)
It can also be easily shown that
(f2)2 + (f2’)2 = 1 (eq.37b.d8)
We will now show how fundamental trigonometric identities
of the form of (eq.4c), rewritten below,
cos2(kx) + sin2(kx) = 1.
may be used to prove that the total energy of a simple harmonic
system is constant. We will then derive, via cubic complex
variable theory, the counterpart of this relation for the damped
harmonic oscillator case. These new identities and/or equations
should have extensive physical applications: since no real world
oscillator is strictly sinusoidal, there is always some damping
present.
Speaking of sinusoidal variation, we will briefly discuss
the case when there is a sinusoidal driving force. The governing
equation is (see Wikipedia: Harmonic Oscillator)
(eq.37b.d9)
where the driving amplitude is F0 and the driving frequency is
denoted by . The other parameters are as previously defined.
The following quote verifies our assumption that the case when
Q = 1 (or when the damping ratio ζ= 0.5) is special.
“For a particular driving frequency called the resonance, or
resonant frequency , the amplitude (for a given
) is maximum. This resonance effect only occurs when ,
i.e. for significantly underdamped systems.”
Clearly ζ = 0.5 < .
More detailed descriptions of the applications of Cubic Complex Variable theory to
forced damped harmonic oscillators will be presented in a later paper. But for now it is
interesting to note that when ζ = 0.5, then the quantity (1- ζ2)^(0.5) is equal to the
quantity which is the imaginary part of one of the ordinary complex cube roots of
unity which is (-1/2 + i *( ) ) = (-ζ + i *( ) ).
Hooke’s Law
From numerous references in the literature it is seen that
Hooke’s Law, as applied to the simple harmonic oscillator, may be
written as
F = -kx
Where F is force, x is the displacement of the vibrating mass
from equilibrium and k is the spring constant. Since the force is
equal to mass times acceleration, we may write F = mx’’, where
x’’ indicates the second order derivative of displacement x with
respect to time. The governing equation associated with Hooke’s
Law may then be written as
mx’’ = - kx or
mx’’ + kx = 0
or
x’’ + (k/m)x = 0 (eq.37c)
From (Ref.1) it is seen that a solution of an equation
Of the form (eq.37c) may be written as follows:
x = A*Sin(ωt) (eq.37d)
where A is the amplitude and ω is the frequency.
It can easily be shown that
ω2 = k/m (eq.37e)
Successive differentiation of (eq.37d) with respect to t
Will result in the following:
x’ = ω*A*Cos(ωt) (eq.37f)
x’’ = -(ω2)*A*Sin(ωt) (eq.37g)
Substituting (eq.37g) and (eq.37d) into (eq.37c),
simplifying and then solving for ω2 will give us
ω2 =k/m which establishes (eq.37e).
From (eq.37d) we have
Sin(ωt) = x/A (eq.37h)
And from (eq.37f) we obtain
Cos(ωt) = x’/(ω*A) (eq.37i)
Now recall the trigonometric identity
cos2(ωt) + sin2(ωt) = 1. (eq.37j)
Substituting equations (eq.37h) and (eq.37i) into
(eq.37j) will give us
(x’)2/(ω2*A2) + x2/A2 = 1 (eq.37k)
Substituting the expression for ω2, from (eq.37e),
Into the equation above will give use
(x’)2/((k/m)*A2) + x2/A2 = 1 (eq.37L)
From which we obtain
(m/k)*(x’)2/A2 + x2/A2 = 1
Now multiply both sides by (k/2)*A2 to obtain
(1/2)*m*(x’)2 + (k/2)*x2 = (k/2)*A2 (eq.37m)
The first term on the left side of (eq.37m) denotes the kinetic
energy (KE)of the system and the second term denotes the
potential energy (PE). We therefore have the identity
KE + PE = (k/2)*A2 (eq.37m.1)
A review of some of the fundamental concepts of dimensional
analysis will be helpful in the interpretation of (eq.37m) and
in the introduction to higher order Newtonian mechanics which
will follow.
In all of the relevant discussions in this paper we will
adopt the usage of mass m, time t and length l as the fundamental
units to be used in dimensional analysis. The reader will recall
that the physical dimensions of angular frequency ω, the spring
constant k and the amplitude A are those as given in Table 2.
Physical Quantity PhysicalDimensions
Angular frequency (ω) t-1
Spring constant (k) m* t-2
Amplitude (A) lVelocity l* t-1
Acceleration l* t-2
Force m* l* t-2
Energy m* l2 t-2
Table 2
By referring to Table 2, it is easy to see that every term of
(eq.37m) has the physical dimensions of energy.
The first term on the left, which is proportional to the square
of the velocity, is the (varying) kinetic energy. The second term
is proportional to the square of displacement and is the
(varying) potential energy. The term on the right is the constant
total energy.
++++
A musical tone may be found in the vibration zone of a tuning
fork. And a vibrating string could provide the zing and the
energy of every quark.
++++
Straight Forward Derivation of Newton’s Laws
Before we consider third order Newtonian mechanics it will be
helpful to review the derivation of the fundamental concepts of
ordinary Newtonian mechanics. We will refer to ordinary Newtonian
mechanics as being of the 2nd order because the fundamental
concept of force is defined in terms of the 2nd derivative of
displacement with respect to time. It is to our advantage to take
note of the fact that Newton defined force in such a way that for
uniformly accelerated motion the force would be constant. The
reason he did not define force (or a higher order fundamental
quantity) in terms of jerk (or higher order rates of change of
displacement) could possibly be due to the fact that
instantaneous jerk and higher order rates of motion were very
difficult to measure during the time of Newton.
In the following we will define the third (and higher) order
generalizations of Newton’s force law.
We will now present a derivation of Newton’s Force law in terms
of the kinematic moment of a particle. This leads naturally to
the concepts of momentum and force.
Let x = x(t) be a function of t and let x denote the
displacement of a particle p along the x axis of a one spatial
dimensional coordinate system. Let m be the constant magnitude of
a physical quantity that is associated with the tendency of the
particle p to resist changes in its velocity. We will now define
the (variable) moment M = M(t) of the particle, with respect to
the given coordinate system, as follows:
M(t) = m*x(t) (eq.37.N.1a)
So that
x(t) = M(t)/m (eq.37.N.1b)
We will now consider the simplest type of motion for which the
velocity is not constant. That is the case of uniformly
accelerated motion. We may therefore write
x(t) = (1/2)a*t2 + (vo)*t + d (eq.37.N.2a)
where a, vo and d are the constant acceleration, initial velocity
and displacement respectively. Equating the right sides of
(eq.37.N.1b) and (eq.37.N.2a) will result in
M(t)/m = (1/2)a*t2 + (vo)*t + d
So that
M(t) = m*((1/2)a*t2 + (vo)*t + d) (eq.37.N.3)
Differentiation of the above equation with respect to t results
in the following:
dM(t)/dt = m*(a*t + vo) (eq37.N.4)
If we now define the momentum P = P(t), in general, to be the
time rate of change of the moment M(t), we may then write
P(t) = dM(t)/dt (eq.37.N.5)
And for the present example we have (referring to (eq.37.N.4))
P(t) = dM(t)/dt = m(at + vo)
P(t) = m(at + vo) (eq.37.N.6)
Setting t = 0 on both sides of the above equation will give us
the initial momentum P(0)
P(0) = mvo (eq.37.N.7)
If we now define the force F, in general, to be the time rate of
change of momentum we may write
F = dP(t)/dt (eq.37.N.8)
Making use of (eq.37.N.5) we obtain
F = d/dt(dM(t)/dt)
F = d2M(t)/dt2 (eq.37.N.9)
Where M = M(t) is the instantaneous moment as defined
in (eq.37.N.1a).
Equations (eq.37.N.8) and (eq.37.N.9) are equivalent forms of
Newton’s Second Law when the (kinematical) moment M(t) is defined
by (eq.37.N.1a).
For our present example we see that, after differentiating
(eq.37.N.3) twice,
d2M(t)/dt2 = ma
Therefore (eq.37.N.9) becomes
F = ma (eq.37.N.10)
Which is an expression of Newton’s Second Law when the force F is
constant.
Newton’s Third Law, as is noted in various accounts in the
literature, can be obtained by assuming that the total momentum
of two particles is constant. Let particles p1 and p2 have
momentums of P1 = P1(t)and P2 = P2(t) and let the total momentum
have the constant value k. We may then write
P1(t) + P2(t) = k (eq.37.N.11)
Differentiating both sides of the above equation with respect to
t will give us
dP1(t)/dt + dP2(t)/dt = 0
dP1(t)/dt = - dP2(t)/dt
Now since the force Fi is the time rate of change of the momentum
Pi, let Fi = dPi(t)/dt, for i = 1,2 denote the force acting upon
particle pi and we may therefore write
F1 = - F2 (eq.37.N.12)
The above equation is basically a statement of Newton’s Third
Law: for every action there is an equal and opposite reaction.
Recall that m was previously described as “a physical quantity
that is associated with the tendency of the particle p to resist
changes in its velocity”. From Wikipedia we have “(Inertial) mass
is a quantitative measure of an object's resistance to changes in
velocity”. So then if we give that quantity m the name “mass”,
then (eq.37.N.10) is a statement of Newton’s 2nd law for the case
when the force is constant and the acceleration is uniform.
Third and Higher Order Newtonian Kinematics
Speculative is an adjective that could describe the theory of
strings but we can also speculate on other things!
We will now postulate that there exists a physical property of a
particle that is a measure of its resistance to changes in the
nth order derivative of the displacement of the particle with
respect to time. That property will be called the “nth order mass
of the particle”. In addition we will define (for the case of one
spatial dimension) the nth order force as follows:
Fn = (mn)*dnx/dtn. (eq.37.N.12a)
where Fn denotes the nth order force, mn denotes the nth order
mass and dnx/dtn is the nth order time rate of change of the
displacement x of the particle: or more briefly, the nth order
rate of motion. From (eq.37.N.12a) above it is clear that if
dnx/dtn vanishes then the nth order force also vanishes.
Let’s make one more observation before we consider the case of n
=3. Let n = 1 in (eq.37.N.12a) and we obtain
F1 = (m1)*dx/dt. (eq.37.N.12b)
In this case m1 will be defined as ordinary mass, dx/dt is
velocity and so F1 denotes momentum. For the case of n=2 we have
F2 = (m2)*d2x/dt2 (eq.37.N.13)
Where m2 = m1 is also ordinary mass, d2x/dt2 is acceleration and F2
ordinary Newtonian force. The fact that m2 = m1 = ordinary mass
does not indicate an inconsistency in the forthcoming theory
anymore than the fact that 0! = 1! = 1 indicates a deficiency in
the utility of (n!) factorial n. But the expectation is that
mn+k is different mn for n > or = 2 and k > or = 1.
We will now begin our discussion with the case of a particle
undergoing uniform jerk. The reader will recall that jerk is the
third derivative of displacement with respect to time.
Now let n = 3 in (eq.37.N.12a) and we obtain
F3 = (m3)*d3x/dt3. (eq.37.N.14)
We will now present what we propose will be a practical
method for determining the third order mass m3.
The spring in a simple harmonic motion system may be used to
determine the (ordinary) mass of an object. As is well known,
this can be accomplished via an application of Hooke’s Law.
Earlier we noted the well established fact that the governing
equation for an unforced simple harmonic motion system with
damping could be written as
y’’ + By’ + Cy = 0
which was labeled as (eq.1).
From the discussion, associated with equations (eq.5) through
(eq.7), it was determined that if C = B2 (the case for which the
quality factor Q is equal to 1) then one would arrive at
y’’’ – B3y = 0, (eq.7) which for convenience we will re-label as
follows:
y’’’ – B3y = 0 (eq.37.N.15)
We will now present a generalization of Hooke’s Law. Since the
original version of Hooke’s Law is associated with the second
derivative of displacement with respect to time, we will refer to
it as the “second order Hooke’s Law”. We propose that Hooke’s Law
can be generalized to the nth order in a straight forward fashion
but for now we will deal with the third order version.
Third Order (Postulate of)Hooke’s Law
It is postulated that the necessary third order force for causing an extension of or a compression of a spring by a displacement of y is given by
F3 = k3y (eq.37.N.16)
Where k3 is a constant of proportionality that is
characteristic of the spring.
But by definition
F3 = m3y’’’
So we therefore have
m3y’’’= k3y (eq.37.N.17)
m3y’’’- k3y = 0 (eq.37.N.17a)
y’’’- (k3/m3)y = 0 (eq.37.N.17b)
But for a damped simple harmonic oscillator
(with Q = 1) we have, from (eq.37.N.15)
y’’’ = B3y, therefore the above equation (eq.37.N.17a)may be
written as
m3(B3y)- k3y = 0 so that m3(B3) - k3 = 0, from which we
obtain the fundamental relationship between constants
(B3) = k3/m3 (eq.37.N.18) Now let the g’ denote the “jerk of gravity”. This third order
rate of change of displacement with respect to time can be
measured experimentally and will be assumed to be constant. The
prime is used in this case to merely distinguish the jerk of
gravity g’ from the acceleration g of gravity. It does not
indicate differentiation in this case.
Clearly when an object is dropped, to fall freely, from a point
above ground, the initial instantaneous acceleration is zero. We
will assume that the jerk of gravity will act for the short time
that it takes for the acceleration of the object to change from
zero to g = - (32 feet per second square). Now let y’’’ in
(eq.37.N.17) be equal to the jerk of gravity (y’’’ = g’). Note
that the jerk will be pointed in the negative (downward)
direction. Therefore (eq.37.N.17) may now be written as
m3g’= k3y (eq.37.N.19)
so that
m3 = k3y/g’ (eq.37.N.19a)
or
k3 = g’m3/y (eq.37.N.19b)
++++
A higher order concept should increase the level of the depth of our understanding of
the nature of physical interactions.
For concepts of a higher order will bring us closer to the border of future technological
Innovations or attractions.
But scientists should never be bored because mysteries will always remain. But a
simple truth is often ignored:
That the search for a true Theory of Everything will always be in vain.
RHB
++++
Experimental Procedure
The following experimental procedure should prove to be
practical. The jerk of gravity g’ can be determined
experimentally and then a standard damped simple harmonic
oscillator spring ( with Q = 1) can be chosen. And then a unit
for the “third order stiffness constant” k3 can then be assigned.
Then if a material object is chosen as a standard and is placed
at the end of the vertically aligned spring and it causes an
extension y of the spring in the negative (downward) direction,
then (eq.37.N.19a)
m3 = k3y/g’
may be utilized in the determination of the
third order mass m3 of the object.
This standard third order mass can then be used to determine
The third order spring stiffness constants for other SHO springs
with Q = 1.
It should be noted at this point that if the basic assumptions
of third order Newtonian Kinematics hold true then third order
mass will have to be added to the conventional list of the
fundamental dimensions (mass,(2nd order), length, time and
charge). As an alternative (and perhaps more practical) method, for
initiating a procedure for measuring the third order mass, is to
simply select an object (in a standards lab) to have a third
order mass of one “third order kilogram”. And once the third
order spring constant has been determined for a given damped
SHO(with Q=1), that SHO may be used to determine 3rd order mass
via an application of (eq.37.N.19a)
We will now verify that (eq.37.N.14) is invariant under a
uniformly accelerated transformation of coordinates. Let the
(X,T) coordinate system be moving along the positive x-axis of
the fixed (x,t) coordinate system in such a way that
X = x – ((1/2)a*t2 + v*t + d ) (eq.37.N.20a)
T = t (eq.37.N.20b)
where the acceleration a, the initial velocity v, and the initial
displacement d are constant. Let (F3)# denote the third order
force with respect to the (X,T)frame of reference. Then since m3
is assumed to be constant, the counterpart of (eq.37.N.14) may be
written as
(F3)# = (m3)*d3X/dT3 = m3*d3/dt3(x –((1/2)a*t2 + v*t + d))= m3*d3x/dt3
(F3)# = m3*d3x/dt3 = F3 (eq.37.N.21)
Therefore the third order force is the same in both
coordinate systems even though they are undergoing uniform
acceleration with respect to each other. From relativistic
considerations we all know that a moving observer, such as one
who is at the origin of the (X,T)frame, carries a clock that
ticks at a different rate than a clock at the origin of the fixed
(x,t) frame. And that fact will be addressed in a more general
formulation, which will be discussed later. In that discussion T
= t, in (eq.37.N.20b) will be replaced by T = T(x,t), which
expresses T as a function of x and t.
Theoretically a fixed coordinate system has a clock and a
distance marker at each point and a moving observer could note
the instantaneous time as he passed each point of the fixed
frame. Consider a possible practical case. If a relatively short
straight stretch of highway has a distance marker and a
synchronized clock at each quarter mile post, then a uniformly
accelerated (or other type of) observer could clearly note the
time t at each distance marker of the fixed frame as he passed
by. Therefore he will be aware of the time in the fixed frame and
could compare it with the time T on the clock he carries with
him. We are assuming that the accelerated observer synchronized
his clock with a clock in the fixed frame before beginning his
journey.
Later on we will consider the relativistic ramifications of Third
(and higher) order Newtonian kinematics. We will refer to the
equations (eq.37.N.20a) and (eq.N.20b) as the Galilean Coordinate
transformations for uniformly accelerated motion.
Now consider an (x,t) and an (X,T) coordinate system in which
both the x and the X axis coincide and are positioned in a
vertical orientation. Now let equations (eq.N.20a) and
(eq.37.N.20b), which we have rewritten for easy reference below,
be the coordinate transformation equations.
X = x – ((1/2)a*t2 + v*t + d )
T = t
Let x = 0 when t = 0 then X = d when t = 0. Now let the symbol a
denote the acceleration of gravity g, that is a = g. Then an
observer at the origin of the (X, T) coordinate system will be
freely falling (neglecting air resistance, etc.) and his equation
of motion will be given
x = (1/2)a*t2 + v*t + d
which is obtained by setting X = 0. And since d3x/dt3 = 0, then by
(eq.37.N.21) no net third order force will act upon the uniformly
accelerating observer but since his acceleration is given by
d2x/dt2 = a = g, the force of gravity = mg, where m is his mass,
will act upon the observer in accordance with well established
fact.
Before we take up the next topic, the derivation of a fundamental
cubic complex variable theory identity (associated with
kinematics), we will note in passing, that one of the basic
motivations for the development of General Relativity (GR) and
other non-Newtonian theories of gravity, was the desire to find
physical laws that are invariant in form under all coordinate
transformations.
For the one spatial dimensional case, it is not difficult to show
that the equation (eq.37.N.12a), rewritten for convenience below,
Fn = (mn)*dnx/dtn.
is invariant in form under a coordinate transformation, from the
(x, t) coordinate system to the (X, T) coordinate system, which
is given by
X = x - (∑(1/k!)*ak*tk) (eq.37.N.22a)
T = t (eq.37.N.22b)
Where the summation is from (k = 0 to k = n-1) and where the ak
are constants and Fn and the mn are as previously defined: in
connection with (eq.37.N.12a).
The equation T = t will have to be replaced by a more general
time transformation function T = T(x,t) when relativistic effects
are taken into consideration. The specific form of this
transformation is to be determined.
A more general discussion for the case in which T = T(x,t) as
well as a higher spatial dimensional treatment of these concepts
will be presented later.
Most physicists are in general agreement with the notion that
the fundamental laws of physics should be invariant under general
coordinate transformations. Perhaps, as we have attempted to do
above, this can be accomplished via the search for laws of
physics that must be expressed in terms of ordinary differential
equations and/or partial differential equations that are of order
higher than the second. It is true that such equations are more
difficult to solve but the increase in generality may prove to be
more beneficial than the increase in the difficulties of
solution.
It should be noted that the experimental verification of the
third and higher order versions of Hooke’s Law is very much
easier than the experimental verification of the actual existence
of the strings that are postulated to exist in String Theory.
Derivation of a Fundamental CCV Theory Identity
We will now derive the third order counterpart to the
trigonometric identity stated in (eq.37j). This derivation will
involve concepts from cubic complex variable (CCV) theory. But
first it will be helpful if we recall some familiar relationships
from ordinary complex variables. Let a, b, x, y,
X and Y all be real numbers. Now let
X + iY = (a + ib)(x + iy)
then it is easy show that
X = ax – by (eq.38a)
Y = bx + ay. (eq.38b)
If we let a and b be constants and then consider
equas.(38a & b) as a system of linear equations in
x and y, then the matrix of coefficients of that
system may be denoted by M1 where M1 is given by
M1 = a -b (eq.39)
b a
where for typographical convenience we have omitted
the parenthesis that are conventionally used to
enclose the elements of matrices. We have,
writing the determinant of M1,
det(M1) = a2 + b2 (eq.40)
Now the reader will recall, or can easily verify,
that
X2 + Y2 = (a2 + b2)(x2 + y2) (eq.41)
where X and Y are defined as in equas.(38a & b).
It will also be noted that the left side and both
factors on the right side of (eq.41) are (2nd
degree) expressions that have the same algebraic
form as does the right side of (eq.40). Also if
a = cos(θ) and b = sin(θ), then of course
a2 + b2 = 1
and equas.(38a & b) represent an orthogonal coordinate
transformation.
X = x*cos(θ) – y*sin(θ) (eq.41a)
Y = x*sin(θ) + y*cos(θ). (eq.41b)
Now for the CCV counterparts of the foregoing, let u
v be defined as in equas.(12e & 12f) above. Then the
product of u and v is given by
uv = ax - bz- cy + j(ay +bx -cz) + j^2(az+cx+by) (eq.42a)
on account of the product rule for CCV expressed by
(eq.14). Since u and v are cubic complex variables,
their product is one also. Let the product uv be, the
cubic complex variable, given by
uv = X + jY + j2Z
then (eq.42a) may be written as
X + jY + j2Z = ax - bz- cy + j(ay +bx -cz) + j^2(az+cx+by)
(eq.42b)Then therefore, by the definition of the equality
of cubic complex numbers, we have
X = ax - bz- cy
Y = ay + bx - cz
Z = az + cx + by
It is convenient to rearrange the order of the terms
on the right sides of the three above equations as
follows:
X = ax – cy - bz (eq.43a)
Y = bx + ay - cz (eq.43b)
Z = cx + by + az (eq.43c)
Now if we consider equas.(43a, 43b & 43c)
be to be a system of three linear equations in x,
y and z, then the matrix of coefficients, which we
denote by M2 is given by
M2 = a -c -b
b a -c
c b a
the determinant of the matrix M2 is given by
det(M2) = a3 – b3 + c3 + 3abc (eq.44)
After a somewhat tedious calculation, it can be
shown that
X3 - Y3 + Z3+ 3XYZ =
(a3 – b3 + c3 + 3abc)(x3 – y3 + z3 + 3xyz) (eq.45)
where X,Y and Z are defined by equas.(43a,43b &
43c). So then (eq.45) is the CCV theory analog to
(eq.41) which is fundamental in ordinary complex
theory.
It can be shown, in a straight forward fashion,
that
(F1)3 – (F2)3 + (F3)3+ 3(F1)(F2)(F3) = 1 (eq.46)
where F1, F2 and F3 , each a function of x, are defined by
equas.(17a, 17b and 17c). The relationship holds true for all
real x. So (eq.46) corresponds to the identity of (eq.3)
which is associated with complex variable theory.
Fortunately there should exist software that makes the above
computations easy.
Norms and Pseudo Norms
A review of eqs.(44) and (45) will reveal the fact that
algebraic expressions of the form a3 – b3 + c3 + 3abc play an
integral part in CCV theory. In fact, for real a, b and c, the
mapping
(a, b, c) ==> a3 – b3 + c3 + 3abc
is the counterpart of the mapping, from ordinary complex
variables
(a, b) ==> a2 + b2
for real components a and b.
We are therefore motivated to define a “restricted pseudo
norm” N on the set of Cubic Complex Numbers (with real components
a, b and c) as follows:
N(a, b, c) = |a3 – b3 + c3 + 3abc |1/3
so that
[N(a, b, c)]3 = |a3 – b3 + c3 + 3abc|
The word “restricted” is placed in the definition of N to
indicate that N may not always satisfy the triangle inequality.
However, as it easily may be shown, N satisfies the other
condition(s) for the definition of pseudo norms or semi-norms.
Kinematics Of Simple Harmonic Motion And Damped
Harmonic Motion
Simple Harmonic Motion may be described by the equation
u’’ + u = 0. (eq.47)
Where u = u(t) and (in this section)
the primes indicate differentiation with respect to t.
A particular solution of this equation is
u = cos(t), so that u’ = -sin(t), therefore
(u)2 + (u’)2 = (cos(t))2 + (-sin(t))2
= (cos(t))2 + (sin(t))2 = 1
so that also
(u)2 + (u’)2 = 1 (eq.48)
It is easily seen that the same result would have been
obtained if we had set u(t) equal to sin(t) which is the other
independent solution of (eq.47).
So then, for the case of Simple Harmonic Motion, it can be
stated that (eq.48) is an identity that involves the position
u = u(t) and the velocity u’ of an object that is in
Simple Harmonic Motion in one spatial dimension. It will be
recalled that the governing equation for one spatial dimensional
simple harmonic motion may be written in this form.
Y’’ + ω2y = 0
where ω is the angular frequency and where y now denotes
displacement from equilibrium. The (general) solution of this
equation may be written in the form
y = A*cos(ωt + β)
where A and β are constants that are determined by the initial
conditions. More specifically A is the amplitude and β is the
phase angle. The angular frequency ω is dimensionless.
For convenience we will initially set ω = 1 and choose initial
conditions such that A = 1 and β = zero. More general
specifications will be given later.
It will be recalled that we previously let y = y(x) and we
denoted dy/dx by y’, etc. And we considered the special case in
which C = B2 so that equation
y’’ + By’ + Cy = 0
could be transformed into the third order equation
y’’’ – B3y = 0.
By a similar procedure the equation
u’’ + Bu’ + Cu = 0 (eq.49)
can transformed into
u’’’ - B3u = 0. (eq.50)
where now u = u(t) and u’ = du/dt, etc.,
It will be recalled that the second order Equation
(eq.49) may be interpreted as representing unforced Damped
Harmonic Motion in one spatial dimension. However, if u = u(t) is
a solution of (eq.49), then (eq.48) is not an identity. But a
differential identity does exist for the solution of
(eq.50). And for the important special case when C = B2 (when the
quality factor Q =1), (eq.49) can be transformed into the form of
(eq.50).
We will now return our attention to the three functions F1, F2 and
F3 that were defined by equas.(17a, 17b and
17c)respectively).These functions were referred to as the
fundamental functions (of a single variable) of Cubic Complex
Variable (CCV) theory. They correspond to the cosine and sine
functions of ordinary complex variables.
For notational convenience replace x by t in
equas.(17a, 17b and 17c) and obtain
F1 = 1 – (1/3!)(t^3) + (1/6!)(t^6) + ... (eq.51a)
F2 = t - (1/4!)(t^4) + (1/7!)(t^7) +... (eq.51b)
F3 = (1/2!)(t^2)-(1/5!)(t^5) +(1/8!)(t^8) + ... (eq.51c)
where F1, F2 and F3 are now denoting functions of t.
For convenience we will now recall (eq.20), (eq.23)
and (eq.25a) where the primes will now indicate
differentiation with respect to t.
F1’ = - F3 (eq.20)
F1’’ = - F2 (eq.23)
F1’’’ = - F1 (eq.25a)
Now since we are denoting the displacement variable,
of a physical system undergoing damped harmonic motion,
by u =u(t), we desire to find “differential identities”,
involving u and its derivatives, that correspond to the
“differential identities” that are associated with simple
harmonic motion and that were expressed by
(f1)2 + (f1’)2 = 1
(f2)2 + (f2’)2 = 1
Where
f1 = cos(t) and f2 = sin(t).
Motivated by the algebraic form of the right side of (eq.44)
we are led to define the differential operator L(u) as follows:
L(u) = (u’’)3 - (u’)3 + (u)3 + 3(u’’)(u’)(u) (eq.52)
Now (in this discussion) let u = F1(t).We may easily calculate u’
and u’’ from (eq.20) and (eq.23) that were re-written above. The
primes indicate differential with respect to t. We have
u’ = F1’ = - F3
u’’ = F1’’ = - F2
so that
u’ = - F3
u’’ = - F2
Substituting u = F1, u’ = - F3 and u’’ = - F2 into (eq.52), we
obtain
L(F1) = (-F2)3 - (-F3)3 + (F1)3 + 3(-F2)(-F3)(F1)
or
L(F1) = (F1)3 - (F2)3 + (F3)3 + 3(F1)(F2)(F3) (eq.53)
But the right side of (eq.53), as a consequence of
(eq.46), is equal to unity. So we have
L(F1) = 1 (eq.54a)
Now we will calculate L(F2). We may do this by
now setting u = F2 and then calculate u’ and u’’
by making use of eqs.(24, 25a and 20)above. We obtain
u’ = F1
u’’ = - F3
Substituting the above results (along with u = F2) into
(eq.52) will give us
L(F2) = - ((F1)3 - (F2)3 + (F3)3 + 3(F1)(F2)(F3))
and as a consequence of (eq.46) we have
L(F2) = - 1 (eq.54b)
And if we set u = F3 , then by a process similar to
that above it can be shown that
L(F3) = 1 (eq.54c)
Actually eqs.(54a, 54b & 54c) may be combined by writing
(L(Fn))2 = 1 (eq.55)
for n = 1, 2 or 3 and where the differential operator L
is defined by (eq.52). Equation (eq.55) is a fundamental
identity that involves the three fundamental functions of
Cubic Complex Variable Theory.
It is interesting to note that a corresponding
differential operator may be defined for the Simple
Harmonic Motion case. Let fn = fn(t) and let the differential
operator P be defined by
P(fn) = (fn)2 + (fn’)2 (eq.56)
where fn’ denotes dfn/dt. Then it is easy to show that
P(fn) = 1 (eq.57)
where f1 = cos(t) and f2 = sin(t) and where n = 1 or 2.
Thus (eq.55) holds identically true, when Fn = Fn(t)
(for n = 1, 2 or 3), are the functions defined by eqs.(51a, 51b
and 51c) respectively. For more general results, the argument of
each of these functions may be changed from t to ωt. They are all
realistic candidates for representing the position of an object
that is executing damped simple harmonic motion in one spatial
dimension with a quality factor of q = 1.
Returning our attention to (eq.52), the starting point for
the identities expressed in (eq.55), the instantaneous
acceleration, velocity and position are given by u’’, u’ and u
respectively. It is strikingly noteworthy to observe that (u’)3,
the cube of u’, which is also associated with the dissipative
force of friction, [see (eq.49), the 2nd order O.D.E. governing
damped simple harmonic motion], is preceded by a negative sign in
(eq.52). This fact adds weight to the assertion that (eq.55),
which is defined by way of (eq.52), is the damped simple harmonic
motion counterpart of (eq.48): which is a fundamental
identity associated with Simple Harmonic Motion.
We will now derive the CCV theory counterpart of (eq.37m)
Which has been re-written below for convenience.
(1/2)*m*(x’)2 + (k/2)*x2 = (k/2)*A2
The above equation simply asserts that the total energy of a
Simple harmonic oscillator is constant. The total energy of the
SHO has two constituents. These are the potential energy which is
proportional to the square of the displacement x and the kinetic
energy which is proportional to the square of the velocity x’.
The differential operator L(u), where u = u(t), is defined by
(eq.52). From (eq.55) we have
(L(Fn))2 = 1
for n = 1, 2 and 3 and where the functions F1, F2 and F3 are
defined by equations (eq.51a), (eq.51b) and (eq.51c)
respectively. The primes indicate differentiation with respect to
t. Making use of (eq.52) which defines the differential operator
L, the last equation above may be expanded as follows:
((Fn’’)3 - (Fn
’)3 + (Fn)3 + 3(Fn’’)( Fn
’)(Fn))2 = 1 (eq.58)
for n =1, 2 and 3. In (eq.58), Fn = Fn(t) and the primes indicate
differentiation with respect to t. Now for notational convenience
we will define the functions (of t), Hn = Hn(t)
as follows: let Hn = Fn(w), where w = ωt and ω is a constant, for
n = 1, 2 and 3.
For the present time we will restrict our attention to the case
for which n =1. We have H1 = F1(w). Note that the functional form
of the function F1 is defined by (eq.17a). Differentiating with
respect to t and using the chain rule results in,
dH1/dt = (dF1/dw)(dw/dt) = (dF1/dw)ω
dH1/dt = (dF1/dw)ω (eq.58a)
Similarly it can be shown in a straight forward fashion
That
d2H1/dt2 = (d2F1/dw2)ω2 (eq.58b)
d3H1/dt3 = (d3F1/dw3)ω3 (eq.58c)
From the above three equations (eq.58a, 58b and 58c) it can be
easily seen that
dF1/dw = (dH1/dt)/ω (eq.58d)
d2F1/dw2 = (d2H1/dt2)/ω2 (eq.58e)
d3F1/dw3 = (d3H1/dt3)/ω3 (eq.58f)
It will be recalled that the functions Fn = Fn(w), for n =1, 2 and
3, have been defined by merely changing the name of the
independent variable, in equas.(51a, b and c), from t to w.
Therefore, since (equa.58) is an identity associated with equas.
(51a, 51b and 51c), the following equation
((d2Fn/dw2)3 - (dFn/dw)3 + (Fn)3 + 3(d2Fn/dw2)(dFn/dw)Fn)2 = 1
(eq.59)
is also an identity since we merely changed the name of the
independent variable from t to w and we changed the notation
for the derivative from F’ = dF/dt to dF/dw, etc.
We will first illustrate how (eq.59) may be converted into a
(higher order) kinematic identity for the case for n = 1 and
then we will consider the more general case for any value for
integer n between 1 and 3.
By setting n = 1 in (eq.59) and making use of (eq.58d) and
(eq.58e), (eq.59) becomes
[((d2H1/dt2)/ω2)3 - ((dH1/dt)/ω)3 + (H1)3 + 3((d2H1/dt2)/ω2)*
((dH1/dt)/ω)*H1]2 = 1
(eq.59a)
Where the symbol * is used to indicate multiplication in the
above equation. [Of course, as is usual, AB also indicates the
product of A and B].
Now for typographical convenience let u = H1(t)denote the one
spatial dimensional displacement for the damped harmonic
oscillator.
Then of course u’ = dH1/dt and u’’ = d2H1/dt2 where the primes
indicate differentiation with respect to t. Therefore (eq.59a)
may be written as
[((u’’)/ω2)3 - ((u’)/ω)3 + (u)3 + 3((u’’)/ω2)*
((u’/ω)*u]2 = 1 (eq.59a.1)
The reader will recall that we previously showed that z = F1(u)
is a solution of (eq.36a) (which is now also designated by
(eq.59a.2) for easy reference
z’’’ + k3z = 0 (eq.59a.2)
where u = kx for constant k and where the primes indicated
differentiation with respect to x. The equation above is
(eq.36a) re-written for convenience. The function
F1 = F1(x) is defined by (eq.17a).
Now by setting k = - ω0, where ω0 is angular frequency, and by
replacing the independent variable x by t (so that it represents
time) we may transform (eq.36a)into the same differential form
as that of (eq.37b.d.6): which is rewritten
below for convenience
y’’’ – ω03y = 0 (eq.37b.d.6)
Since both of the equations (eq.36a) and (eq.37b.d.6) are third
order linear O.D.E.’s of the same form with constant coefficients, it is easy to see that by
simply renaming the dependent variable (of (eq.59a.2) ) from z to y, renaming the
independent variable from x to t and by letting the constant k = - ω0, we will then obtain
(eq.37b.d.6) where the primes now indicate differentiation with respect to t.
Furthermore, since u = kx, k = - ω and x = t, the solution z = F1(u) of (eq.36a)
becomes
y = F1(u)
or with u = - ωt
y = F1(-ωt)
which is a solution of (eq.37b.d.6)
y’’’ – ω03y = 0
This equation, the reader will recall, was derived in
connection with damped harmonic motion with a quality factor of
Q = 1.
Since y= F1(u)is a solution of the above linear differential
equation (eq.37b.d.6), it also follows that
y = AF1(u) (eq.59b)
where A is constant,
is also a solution. We therefore may write
F1(u) = y/A (eq.59c)
where u = - ωt
Differentiation of (eq.59c) with respect to t, via the chain
rule, will result in
(dF1/du)(-ω) = y’/A
Where the primes indicate differentiation with respect to t,
and where we made use of the fact that u = - ωt and du/dt = -ω.
We therefore have (dF1/du) = - y’/(Aω) (eq.60a)
And it is also easy to show that
d2F1/du2 = y’’/(Aω2) (eq.60b)
From (eq.59), after changing the name of the independent variable
from w to u, for convenience, we have
((d2Fn/du2)3 - (dFn/du)3 + (Fn)3 + 3(d2Fn/du2)(dFn/du)Fn)2 = 1
Now let n = 1 and we will obtain
((d2F1/du2)3 - (dF1/du)3 + (F1)3 + 3(d2F1/du2)(dF1/du)F1)2 = 1
(eq.60c)
We will now substitute equations (59c), (60a), (60b) into
(eq.60c) and we will arrive at
[(y’’/(Aω 2))3 + (y’/(Aω))3 + (y/A)3 - 3(y’’/(Aω 2))*
(y’/(Aω))(y/A)]2 = 1
so that
[(y’’)3/(A3ω6) + (y’)3/(A3ω3) + (y/A)3 - 3(y’’)(y’)y/(A3ω3)]2
= 1 (eq.61a)
[(1/(A3ω6)){(y’’)3 + (y’)3ω3 + y3ω6 - 3(y’’)(y’)yω3}]2
= 1
(1/(A6ω12)){(y’’)3 + (y’)3ω3 + y3ω6 - 3(y’’)(y’)yω3}2
= 1
Multiplying both sides of the above equation by A6ω12 Will result in
{((y’’)3 + (y’)3ω3 + y3ω6 - 3(y’’)(y’)yω3)}2
= A6ω12 (eq.61b)
+++
He who discovered the molecule probably encountered much ridicule when he
presented his idea.
But there will always be some who are creatively dumb who chooses unfounded
criticism as part of their career.
RHB
+++
Recalling (eq.37.N.18) we have
(B3) = k3/m3 If we let B = ω in the above equation we obtain
(ω3) = k3/m3 (eq.62a)
Therefore
ω6 = (k3/m3)2 (eq.62b)
ω12= (k3/m3)4 (eq.62c)
Substituting (eq.62a, 62b and 62c)into(eq.61b) will result in
{((y’’)3 + (y’)3(k3/m3) + y3(k3/m3)2 - 3(y’’)(y’)y(k3/m3)}2
= A6(k3/m3)4 (eq.62d)
It will be convenient, for purposes of the definitions to be
presented below, to multiply both sides of the equation above by
(1/6)2(m3)2.
(1/6)2(m3)2{((y’’)3 + (y’)3(k3/m3) + y3(k3/m3)2 - 3(y’’)(y’)y(k3/m3)}2
= (1/6)2[(k3)4/(m3)2]A6 (eq.62e)
From which we have
[((1/6)(m3)){(y’’)3 + (y’)3(k3/m3) + y3(k3/m3)2 -
3(y’’)(y’)y(k3/m3)}]2 = (1/6)2[(k3)4/(m3)2]A6
{(1/6)m3(y’’)3 + (1/6)(y’)3(k3) + (1/6)y3(k3)2/(m3) -
(1/2)(y’’)(y’)y(k3)}2 = (1/6)2[(k3)4/(m3)2]A6 (eq.63)
It will now be convenient to define the quantities inside of the
Curly braces, on the left side of (eq.63), as follows:
Quantity Name (description) Symbol
[(1/6)m3((y’’)3 Dynamic Cubic Energy EDC
(1/6)(y’)3k3 Kinetic Cubic Energy EKC
(1/6)y3((k3)2/m3) Potential Cubic Energy EPC
(1/2)(y’’)(y’)y(k3) Interactive Cubic Energy EIC
Table 3
Making use of the definitions presented in Table (3), (eq.63) may
be re-written as follows:
[EDC + EKC + EPC - EIC]2 = (1/36)A6(k3)4/(m3)2 (eq.64)
For the special case when the quality factor is Q =1, equations
(eq.63) and (eq.64) are the Cubic Complex Variable Theory damped
harmonic oscillator counterparts, respectively, of (eq.37m) and
(eq.37m.1), which are associated with the simple harmonic
oscillator.
While more details of the present theory will be presented in
later papers, it should be noted that these concepts can be
generalized to higher orders and degrees. The author has also
developed “canonical form” techniques for the mathematical
treatment of anharmonic wave phenomena: also to be presented in
subsequent papers.
Harmonic Oscillator Electrical Circuits
The governing equations for many important applications for
harmonically oscillating electrical circuits may be also written
in the form of second order linear ordinary differential
equations with constant coefficients. In electric circuit theory
the circuit elements are resistance, inductance and capacitance
and are denoted by R, L and C respectively.
For the case when the circuit elements are connected in series
and the source is a constant voltage the governing equation is
(eq.65a)
which can be expressed in the more useful form, (per Wikipedia
RLC circuit article) ,
as follows:
(eq.65b)
Where i = i(t) is the time varying current. And where α and ωo are constants: and where α
is the neper frequency and ωo is the angular resonance frequency. The parameters
α and ωo may also be expressed as follows:
(eq.66a)
(eq.66b)
The presence of a resistance element R in a circuit not surprisingly will have a damping effect.
Consequently (eq.65a) is the governing equation for a damped harmonic electrical oscillatory
circuit. If R = 0, a theoretical possibility but practically impossible, then the result will be simple
harmonic oscillations. If we let
B = 2α (eq.66c)
C = (ωo)2, (eq.66d)
then (eq. 65b) may be written in the form
d2i(t)/dt2 + B*di(t)/dt + C*i(t) = 0 (67a)
from which we have
d2i(t)/dt2 = -B*di(t)/dt - C*i(t) (67b)
Differentiation of (67b) with respect to t results in
d3i(t)/dt3 = (-B)d2(t)/dt2 - (C)di(t)/dt (eq.68)
Substituting (eq.67b)into (eq.68) and simplifying
results in
d3i(t)/dt3 + (C – B2)di(t)/dt – (BC)i(t) = 0. (eq.69)
In the special case in which C = B2, (eq.69) becomes
d3i(t)/dt3 – (B3)i(t) = 0 (eq.70)
If C = B2 then making use of (eq.66c & d) we have
(ωo)2 = 4α2 or
ωo = 2|α|, or for α > 0
ωo = 2α (eq.71)
From (eq.66a) we have
L = R/(2α) (eq. 72)
Now for a resonant RLC series circuit, the quality factor Q is given by (per Wikipedia),
Q = ωoL/R (eq.73)
If we now substitute (eq.71) and (eq.72) into (eq.73) we
Obtain
Q = [(2α)( R/(2α)]/R from which we obtain
Q =1.
Therefore if C = B2 , where C and B are defined as in (eq.66c) and (eq.66d), then the quality
factor Q is equal to unity as it also is, when the conditions are similar, in the case of the
mechanical damped harmonic oscillator.
The special case in which the quality factor Q is unity is
important for several reasons. We will mention a few of them
below. But first we will upgrade the governing equation for
(unforced) damped harmonic motion from the second order to the
third order by designating (eq.69) as the governing equation.
Recall from (eq.66d) that C = (ωo)2 and is therefore positive. So if C > B2
then C - B2 is a positive number which we may denote by
K1 and then (eq.69) may be written as
d3i(t)/dt3 + (k1)di(t)/dt – (k2)i(t) = 0. (eq.74)
where k2 is a constant. If C = B2 we have (eq.70) rewritten
below
d3i(t)/dt3 – B3i(t) = 0
And if C < B2 then C - B2 is a negative number and (eq.69)
May be written as
d3i(t)/dt3 - (k3)di(t)/dt – (k4)i(t) = 0. (eq.75)
where k3 and k4 are positive numbers.
So it follows that there are three families of third order
governing equations and the one that applies is determined by
whether C < B2, C = B2 or C > B2.
In third order theory, when (eq.69) is adopted as the governing
equation, one can examine the 3D phase space diagram for a damped
harmonic oscillator system. We are referring to a 3D phase space
diagram in which the three mutually perpendicular axes, for the
series RLC circuit case, corresponds to i(t), di(t)/dt and
d2i(t)/dt2. Clearly each of these quantities is a function of t
and are linearly independent. Now let the parameter C vary over
the interval
-B2 < C ≤ B2
then one should observe a marked change in the qualitative
description of the trajectories when C = B2
There are many potentially interesting applications of the
theoretical framework being set forth herein. In this paper,
however, we will only briefly discuss one: Deep Level Transient
Spectroscopy (DLTS).
“An important goal in semiconductor technology is the reduction of intrinsic and process-
induced defects in the crystalline, polycrystalline and amorphous layers which comprise all
semiconductor devices. Defects arising from impurities, grain boundaries, interfaces, etc.
result in the creation of traps which capture free electrons and holes. Even at very low
concentrations these trapping centers can dramatically alter device performance.
Deep Level Transient Spectroscopy (DLTS) is an extremely versatile technique for the
determination of virtually all parameters associated with traps including density, thermal
cross selection, energy level and spacial profile.“ (Per Sula Technologies)
As mentioned above a quality factor of Q = 1, which occurs in
when C = B2 in (eq.69), may serve as qualitative boundary between
different classes of solutions of (eq.69) or trajectories in 3D
phase space. The physical or dynamic implications of this, in
connection with DLTS, may be seen in the following:
“A factor of Q = 1 cancels out the DLTS signal and this last has its signal reversed for Q > 1. Signal reversal can lead to confusion between majority and minority carrier traps.” (Per Vasco Matias)
For the interested reader who would like to search the
literature, via the internet, for specific cases when the quality
factor Q = 1, for unforced damped harmonic motion, it should be
recalled that when Q = 1 then the damping factor
ζ = 0.5
++++
If Shakespeare was here (?)
Greetings O readers of math and science! Now take a small step in an act of defiance
of the rigidity of the status quo.
And step beyond equations of the second order and draw near to the intellectual border
of the limit of the knowledge that man is allowed to know.
RHB
Please accept my apologies to Shakespeare, the master of English Lit, who gave to the world much wisdom and wit.
++++
Tetrahedronomy
Trigonometry is the study of the relationships between the sides and the angles
of triangles. It is an understatement to say that triangles are very important in
engineering and science. A couple of decades ago the present author envisioned the
possibility of a three (and higher) dimensional version of trigonometry. Some very
preliminary new concepts are presented below. But it definitely should be noted that the
literature is already rich in analytical accounts of the properties of tetrahedrons. The
treatments of this subject are both abstract and non-abstract in style. Nevertheless, it is
strongly felt that the rich symmetries of Cubic Complex Variable theory will highlight
the existence of more powerful potential engineering and scientific applications of the subject.
Let us consider the following informal correspondences. Let θ and Φ be real
valued parameters.
Cos(θ) ----- exp(i θ) ----- Complex Variables: A commutative algebra
defined in the plane.
F1(Φ) ) ----- exp(jΦ) -----Cubic Complex Variables: A commutative algebra
defined in 3D space.
Where the function F1 is defined by (eq.17a)
Now consider the following:
Cos(θ) ----- Trigonometry: the study of the relationships between the sides and
angles of triangles which are 3 sided plane figures
F1(Φ) ----- “Tetrahedronomy” : the study of the relationships between the areas
of the faces and the solid angles of the vertices of a tetrahedron which is an object with
four faces embedded in 3D space.
Let us recall the matrix M1 of (eq.39)
M1 = a -b
b a
Note that the determinant of M1 is
det (M1) = a2 + b2
Now let the rows of the matrix M1 be denote by the vectors
A = (a, -b) and B = (b, a).
And let the parallelogram generated by the vectors A and B be denoted by P. Then the
volume of the parallelogram P is given by
V = det (M1) = a2 + b2
The quantity a2 + b2, incidentally, is also the length of the hypotenuse of a right
triangle with legs a and b. Also the diagonal of P splits it into two congruent triangles
each with area (1/2!)( a2 + b2). = (1/2)( a2 + b2).
Now let us consider the corresponding concepts in 3D. Recall the matrix M2 and its
determinant (which was given by (eq. 44) ).
M2 = a -c -b
b a -c
c b a
det(M2) = a3 – b3 + c3 + 3abc
Now let the rows of this matrix M2 be denoted by
A = ( a, -c, -b)
B = (b, a, -c)
C = (c, b, a)
And let T denote the tetrahedron generated by the 3D vectors A, B and C. Then it can
be shown that the volume of T is given in terms of a scalar triple product
V = (1/3!) Aˑ(BxC)
or equivalently by
V = (1/3!)(det (M2)) = ( 1/6) (a3 – b3 + c3 + 3abc)
(See Wikipedia: Tetrahedron, General Properties)
Initially the more elaborate concepts of Tetrahedronomy will be introduced in a fashion
that corresponds to the presentation of trigonometry: that is in a manner that is
accessible to engineers with the focus being on potential applications. Later, a more
abstract treatment, using algebraic topology tools such as homology groups, will be
given.
Before we leave the topics of trigonometry and Tetrahedronomy, let us briefly
note the following:
“Efimov Trimers” were named after Vitaly Efimov who in 1970
“was manipulating the equations of quantum mechanics in an attempt to calculate the
behavior of sets of three particles, such as the protons and neutrons that populate
atomic nuclei, when he discovered a law that pertained not only to nuclear ingredients
but also, under the right conditions, to any trio of particles in nature.”
http://www.wired.com/2014/05/physicists-rule-of-threes-efimov-trimers/
Since that time, what was once regarded as outlandish
theoretical speculation on the behavior of trios of particles,
is gaining a measure of acceptance:
“experimentalists have reported strong evidence that this bizarre state of matter is
real.”
The web site reference above gives an excellent account of what may very well turn out
to be a very important scientific discovery. It turns out, however, that a mysterious
numerical scaling factor plays a recurrent role in the theory. The article notes the fact
that: “The ultimate proof [of the theory] would be an observation of consecutive Efimov
trimers, each enlarged by a factor of 22.7. “
Well it just so happens that the present author, who has a great appreciation for the role
that symmetry plays in physics, has a possible explanation for the appearance of the
number 22.7 in the theory.
(13 + 21 +34)/3 = 22.7 approximately.
In other words 22.7 is the mean of three consecutive Fibonacci Numbers. Just
something to wonder about, since we are discussing third order equations , etc., in this
paper.
Some Introductory Notes On Alternative Approaches To String Theory
In non technical terms string theory associates the vibrations of tiny sub microscopic strings with the energy and/or momentum of elementary particles. In the everyday world strings do not vibrate forever. Their vibrations attenuate or decay. At the quantum level hadrons or quarks are known to decay or transition into other particles. Could the (possible) attenuation of the vibrations of these postulated sub microscopic strings play a part in the decay of the hadrons? In such interactions the four momentum is conserved for relativistic particles.
In the effort to push the boundary of knowledge toward a theory of everything it may potentially be productive if the fact, that even sub microscopic strings probably don’t vibrate perpetually, is taken into account. Their vibrations must attenuate at some point in time and this decay in amplitude of vibration may in some way be associated with the decay or transition of a given elementary particle into other elementary particles.
The partial differential equation below is the one dimensional
wave equation
(eq.76)
http://www.superstringtheory.com/basics/basic4a.html
(Patricia Schwarz Ph.D, creator of the Official String Theory Website)
plays an important role in (non relativistic) String Theory. But
this equation describes the motion of a string that has no
damping and no attenuation. A more realistic wave equation is the
one dimensional damped wave equation
Utt + 2cUt = (β2)Uxx (eq.77)
(Ref.) http://uhaweb.hartford.edu/noonburg/m344lecture16.pdf
Where c (a measure of the level of damping) and β are constants.
As is well known, the PDE (eq.77)that governs the motion of a
damped string may be solved, with the specification of the
appropriate initial and boundary conditions, by a process known
as separation of variables. This process results in two O.D.E.’s
one of which is of the form
H’’ + BH’ + CH = 0 (eq.78)
Where the primes indicate differentiation with respect to the
time t, H = H(t) and B and C are constants.
When B is different from zero (eq.78) describes a damped
harmonic system. When C = B2, in a fashion entirely analogous to
that outlined above, (eq.78) may be transformed into the third
order O.D.E.
H’’’ - (B3)H = 0 (eq.79)
When C = B2 then the Quality factor Q is unity and the
theoretical speculation, from here, is that, in the special case
when Q = 1, the identities and symmetries associated with Cubic
Complex Variable Theory may throw more light upon the mysteries
and intricacies associated with the fundamental dynamics of the
elementary particles.
Abraham Lorentz Force
The Abraham Lorentz force “is the recoil force on an accelerating charged particle
caused by the particle emitting electromagnetic radiation. It is also called the radiation
reaction force or the self force” per Wikipedia. The Abraham Lorentz force Frad is
proportional to jerk (the third order time derivative of displacement ) and is given in cgs
units by
(eq.80a)or by
Frad = (2/3)(q2/c3)(d3y/dt3) (eq.80b)
Multiplying both sides by the third order mass m3
(Frad )m3 = (2/3)(q2/c3)(d3y/dt3)m3 (eq.80c)
But (d3y/dt3)m3 = F3 , the third order force by definition.
Therefore (eq.80c) may be written as
(Frad)m3 = (2/3)(q2/c3)F3 (eq.81)
Solving the above equation for F3 will give us
F3 = (3/2)(Frad )(m3)(c3/q2) (eq.82a)
We may also write
(F3/m3) = (3/2)(Frad)(c3/q2) (eq.82b)
The fact that the ratio (F3/m3) of fundamental quantities
from third order Newtonian Mechanics can be solved in terms
of previously defined physical quantities clearly
does not detract from the possibility of the validity
of the new theory.
It should be noted that the last equation above can
also be written as
(q2) (F3/m3) = (Frad )(3/2)( c3)
Which implies that if q = 0 then Frad = 0 which is just as expected.
Philosophical Remarks:
Ordinary mass can be informally defined as a measure of the
intrinsic resistance of a material body to acceleration or
changes in its velocity. Suppose that an object high above the
surface of the earth (where the air resistance is negligible) is
falling under the almost constant acceleration of gravity. To a
high degree of accuracy we may assume that the jerk acting upon
that body is zero. Now suppose that a physical agent of some sort
(outside of the material body in question) tries to change the
acceleration of that freely falling object (or to impart upon it
a non-zero jerk). Would we be surprised if that body offered an
intrinsic resistance to that effort? As defined above, third
order mass was the name given to that innate resistance to
changes in acceleration.
We must not discount the (remote) possibility that the third
order mass of a given object may turn out to be proportional to
the ordinary (second order) mass of the object. But even in that
unlikely circumstance the theories that have been set forth in
this paper potentially have great merit.
Some readers may consider the discussion, pertaining to the
potential application of CCV theory to damped harmonic motion, as
pure mathematical speculation or mathematical fiction. The author
has coined the phrase “mathematical phiction” to characterize
that view point. Nevertheless, it must be remembered that much of
the science fiction of yesterday is responsible for many of the
engineering wonders of today.
Let’s hope that an effort will be made for a more rapid
review of worthy science innovations and/or math phiction
formulations in order to more quickly determine the merits of
potentially productive conceptualizations.
A Note From The Author
This is a preliminary draft. More specific and detailed
descriptions of the applications of CCV theory will be presented
in a later paper. In that Paper the “renaming of variables for
convenience” will be minimized. But if the reader will keep in
mind the fact that for a given function f = f(x), f(x) = f(t) and
df(x)/dx = df(t)/dt etc., whenever x = t, then no
misunderstanding should ensue.
++++
A math model formulation is not needed for music appreciation but
both are often associated with some kind of vibration.
Math is the language used by scientists and engineers but music
is the language for all human ears.
RHB
++++
“The Universe is a United Verse” is not just a play on words.
For if a play is accidental or not well rehearsed it is probably for the birds!
But a bird is a wonder in its majestic flight and is not an accidental blunder that escapes
into sight.
And the birds have always sang with beauty and great passion
just as the aftermath of the Big Bang unfolded in a precise but accidental fashion(?!)
Music is to math as art is to science.
RHB
++++
References:
1.) Article: “Simple Harmonic Motion”
http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html
*) Sula Technologies: Deep Level Transient Spectroscopy (DLTS)
http://sulatech.com/what-is.shtml
retrieved on 5-23-14
*) Vasco Matias
https://lirias.kuleuven.be/bitstream/1979/1835/5/Thesis_19_06_2008_final_version.pdf
*) optional ref for non relativistic string theory
http://cds.cern.ch/record/465908/files/0009181.pdf
Non-Relativistic Closed String Theory
Jaume Gomis and Hirosi Ooguri
California Institute of Technology 452-48, Pasadena, CA 91125
and
Caltech-USC Center for Theoretical Physics, Los Angeles, CA
gomis, [email protected]
++++
Footnotes:
1.) Manuscripts explaining the fundamentals of Cubic Complex
Numbers were informally distributed (for comment and review)
among the grad students and faculty of Northwestern University
and the University of Minnesota between 1986 and 1993.
2.)
The author has also generalized the concept of the cubic
complex unit to the nth order. The nth order complex unit
jn has been defined by
jn = (-1,0,0,...)^(1/n)
where (1,0,0,...) is an n dimensional unit vector.
3.)
The author outlined in an informally distributed 1980 paper, a
generalization of the method used above for finding Taylor Series
solutions of nth order ordinary differential equations: subjected
to a set of n auxiliary conditions.