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, ooguri@theory.caltech.edu

++++

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.