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Complex Variables have been generalized in a natural way. They are used to solve 3rd order ODE just as ordinary complex variables are helpful when solving certain 2nd orderODE.Many engineering applications are possible.
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Third Order O.D.E.'s And Cubic Complex Numbers: Possible Applications By Ronald H. Brady
Table Of Contents
Topic Page No.
Abstract 2
Second and Third Order O.D.E.’s 3
Cubic Complex Numbers 5
The Cubic Complex Exponential Function 11
Norms and Pseudo Norms 22
Kinematics Of Simple Harmonic Motion And Damped
Harmonic Motion 23
Abstract
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, 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.
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
(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). Equations (eq.3) and (eq.4),
play indispensable roles in math, science,
engineering and theoretical physics. The reader will
recall that since eix is a linear combination of cos(x)
and sin(x), it is also a solution of the ordinary
differential equation (O.D.E), (eq.2).
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 choose units so that
C = B2
then the equation above reduces to
y’’’ – B3y = 0. (eq.7)
Before continuing, let’s note that a more general form
of (eq.2) is the following
y’’ + k2y = 0 (eq.8)
where k is a constant. This equation (eq.8) may be
referred to as the one (spatial) dimensional “un-damped
wave” equation.
When B ≠ 0 then (eq.1) may be thought of as the
one spatial dimensional damped wave equation.
Equation (eq.7) may be thought of as a 3rd order counter-
part of (eq.8).
The reader is probably familiar with many cases
in (math) physics where the units of physical quantities
and the units associated with constants of proportionality
have been chosen for the purpose of simplifying equations
and/or formulas. For example, in quantum field theory it is often
the case that units are chosen so that both c the speed of light
and Plancks’ constant h are set equal to 1 with no loss of
generality.
If we write y(n) to denote the nth derivative of y with
respect to x, then both (eq.7) and (eq.8) may be derived
from
y(n) + Lny = 0 (eq.9)
by letting n = 3 and L = -B we obtain (eq.7). By
setting n = 2 and L = k (or –k) we obtain (eq.8).
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 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)
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 a
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.
We will now show that the three fundamental functions are
independent solutions of
y’’’ + y = 0. (eq.19)
which may be obtained from (eq.7) by letting B = -1.
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
Now we will show that the three fundamental functions, F1 ,F2 and
F3 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 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. 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.
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.37a)
It can also be easily shown that
(f2)2 + (f2’)2 = 1 (eq.37b)
We will now derive the corresponding identities
for the third order case which involve concepts
from CCV theory. But first it will be helpful if
we recall some familiar relationships from ordinary
complex variables.
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.
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 (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.
Norms and Pseudo Norms
A review of eqs.(44) and (45) will reveal that algebraic
expressions of the form a3 – b3 + c3 + 3abc plays 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 defined 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
With a convenient choice of units, associated with the
relevant physical constants, 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
(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.
Let’s also make note of the fact that conventionally
the arguments of the sine and cosine functions, in
connection with simple harmonic motion, are often
written as ωt + β where ω is the angular frequency
and β is a dimensionless constant (initial angular
displacement). A typical representation of a function
describing simple harmonic motion is f(t) = A*cos(ωt + β)
where A is the amplitude. For convenience we are choosing
units that will set A = 1 and initial conditions such that
ω = 1 and β = zero. These specifications should not, in any way,
affect the generality of the principles discussed and proposed.
It will be recalled that previously we let y = y(x) and y’
denote dy/dx, etc., and demonstrated how the second order
equation
y’’ + By’ + Cy = 0
could be transformed into the third
order equation
y’’’ + y = 0.
By a similar procedure the equation
u’’ + Bu’ + Cu = 0 (eq.49)
can transformed into
u’’’ + u = 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 Damped
Harmonic Motion in one spatial dimension.
However, a relationship of form of (eq.48) does
not identically hold true for the function (and its
1st order derivative) that is a solution of the 2nd order
equation for damped harmonic motion. But,
alas, when (eq.49) is transformed into the form of (eq.50), such
a relationship does indeed exist.
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,
(f1)2 + (f1’)2 = 1
(f2)2 + (f2’)2 = 1
that are associated with simple harmonic motion and that were
recalled from eqs. 37a & 37b.
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 let u = F1 . We may easily calculate u’ and u’’ from (eq.20) &
(eq.23) that were re-written above. We have
u’ = F1’ = - F3
u’’ = F1’’ = - F2
so that
u’ = - F3
u’’ = - F2
Substituting u = F1 and
u’ = - F3
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. 20, 23 and 25a 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).
It is interesting to note that a corresponding
differential operator may be defined for the Simple
Harmonic Motion case. Let f = f(t) and let the differential
operator P be defined by
P(f) = (f)2 + (f’)2 (eq.56)
where f’ denotes df/dt. Then eq.37a & 37b may be
combined by writing
P(fn) = 1 (eq.57)
where n = 1 or 2 and f1 = cos(t) and f2 = sin(t).
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. In general the arguments of each of these
functions, which are solutions of (eq.50) and are realistic
candidates for representing the position of an object that is
executing damped simple harmonic motion in one spatial dimension,
is ωt + β. But as is customarily the case, convenient choices of
the initial conditions and the units of the constants of
proportionality have been adopted. More specifically we have
chosen to set ω = 1 and β = 0.
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.
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.
Some readers may consider the discussion, pertaining to the
potential application of CCV theory to damped simple 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.
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 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.