Mathematics is just a method, and one of its main characteristics is abstractness.
Abstraction's perhaps the single most important word for appreciating
mathematical work, have it some natural phenomena as an amorphous and
rapidly abandoned spawn or not even that. Grammatically, the root form
is the adjectival, from the Latin abstractus = "drawn away". Some major (english) dictionary definitions of interest(1):
- "withdrawn/separated from matter/material embodiment/practice/particular examples;"
- "ideal, distilled to its essence."
Then,
for our purposes, abstract's basic adopted definition will be twofold:
a) (adjective) removed from or transcending concrete particularity,
sensuous experience; (b) (verb) abstract something = reducing it (e.g.,
a natural/fabricated phenomenum - industrial, biological, sociological,
metheorological, etc., in nature, at every conceivable level) to its
absolute skeletal essence(2), frequently as the only practical way to
produce some minimum understanding of an extremelly complex phenomenum
pretty sure (in)directly affected by a huge number of influences at
various different degrees.
Mathematicians and nowadays also a
significant proportion of biologists, sociologists, economists et al.,
as well as, you know, philosophers, spend a lot of time thinking
abstractly, or about abstractions, or both. Abstraction proceeds in
levels. For instance, let's say man meaning some particular man is Level One, Man meaning the species is Level Two, something like humanity/humanness
is Level Three, and so forth. Quoting Bertrand Russel: "That fact is, that in Algebra the mind is
first taught to consider general truths no asserted to hold only of
this or that particular thing, but of any one of a whole group of
things. It's in the power of understanding/discovering such truths that
the mastery of the intellect of the whole world of things actual &
possible resides; and ability to deal with the general as such is one
of the gifts a mathematical education should bestow."
In a more purposeful instance, consider a simple pendulum (Figure 1), an idealized (physical) system composed of an object of m units of mass and no particularly relevant shape + a perfectly rigid l-unit
long stem with almost 0 units of mass (i.e., "no mass" for pratical
purposes) who freely moves along a perfect, single vertical plane. Note
that the general variables, or simply noums, m for mass and l
for height, are only mathematical entities entertained by the mind,
general concepts divorced from particular instances. As well as motion.
What exactly do we mean when we talk about motion? In the particular
level of abstraction embedded in our example motion corresponds to
angular displacement, k, along a single (vertical) plane, of a general object of mass m, connected to the end of a fixed, perfectly rigid stem of height l.
If we abandon the object from a particular (angular) position, the
pendular motion starts. Consider that we have ideally assumed that this
is a free motion, meaning that the (supposed) air friction was
abstracted out (except the laws of gravitation). As time passes (and
let's not begin, given our current more modest purposes, a discussion
about the most important abstraction of all abstractions) on, the
object of mass m (henceforth simply m) moves along the
plane theoretically assuming all possible angular displacements,
provided that we consider infinitesimally small time steps (i.e.,
almost 0), until a cessation determined only by its mass and the stem
height. Let's call the abstract "mode" where we can monitor m's position (i.e., observe and perfectly measure and collect k) within almost 0 time steps a continuous mode, or simply continuous time.
Reality dictates that we can only register those angular displacements
within significant time steps determined by the technology we use to do
the measurements, who actually samples the first (idealized) set of
collected k's. Let's call this "mode" discrete time(3)(4).
Given all such idealized conditions we can perhaps mathematically
represent this specific, simple pendular movement by an equation such as
A:
m x
l2 x
(dk2/d2t) +
m x
g x
l x Sin(
k) = 0
where
k is the pendulum angular displacement,
dk2/d2t is the second (temporal)
derivative of
k, and
g is the well-known
local acceleration due to gravity or
standard gravity (value = 9.80665 m/s
2).
A
derivative is defined by the ratio of variation of one or more
variables in relation to the variation of another one or more
variables. In its most simple form, we have a derivative of a single
variable in relation to time, or, in our particular example, the
variation of a single variable,
k, in relation to only one
variable, time, considering the ideal case where time steps are
infinitesimally small, an operation we usually symbolically represent
by
dk/
dt, meaning the first temporal derivative of
k. (Abstractly) Repeating this same (mathematical) process n times we have an n-th order temporal derivative of
k.
Now note the second term in the left-hand side of equation A: Sin(
k) is the
sine of
k, the well-known
trigonometrical function - i.e., for each value of
k Sin(
k)
returns a particular value exactly equal to the sine of that angle.
Take note that usually we can also derivate a function in relation to
time or in relation to one (or more) of its constituent variables.
So A is a second-order
Ordinary Differential Equation
(henceforth O.D.E.), "ordinary" in the sense that it contains functions
of only one independent variable, and one or more of its derivatives
with respect to that variable. O.D.E.s are one of the most common and
simple building blocks for the mathematical representation of actual
physical/biological/sociological/economical/... phenomema. In fact,
there are an enormous number of these abstract representations
available to the analyst. Let's call them simply
mathematical models.
The most high level distinction between them is the following: Models
as in A are constructed based on
first-principles knowledge of the system's underlying physical laws, while there are models that are pure
unspecific black-boxes with several internal characteristics to be somewhat adjusted based on recorded information of system's behavior.
Despite
the very idealized conditions, there's no analytical way to solve A in
terms of elementary functions(5). However, we can hopefully identify
the main characteristics of those "unreachable" solutions,
qualitativelly scanning the possible movements inherent to this quite
idealized representation of the actual physical system.
The simple pendulum as in A is an
autonomous system,
since there's no explicit dependence on time. In a non-
autonomous
system, behavior explicitly depends on time. Think of a pendulum that's
periodically forced by an external stimuli, in which case we can
identify a system output - the angular displacement - and a system
input - the external force. I.e., there's a direct causal relation
between the external stimuli & the angular displacement. A system
is called causal if the output at any time depends only on values of
the input at the present time and/or in the past. Embedded in this
definition's another important one, that of a
dynamical system
- i.e., if a system's behavior - e.g., represented by its identified
output - also somewhat depends on the past values of this output and/or
past values of the input, that system is not a static, memoryless one
for which current inputs are all that's needed to determine the current
outputs. Roughly speaking, the concept of dynamics in a system
corresponds to the presence of an internal "mechanism" that
retains/stores information about input and/or output values at times
other than the current time. In some physical systems,
memory's
directly associated with the storage of some form of energy - e.g.,
think of a simple electrical circuit with a
capacitor who stores energy
by accumulating electrical charge. An automobile has memory stored in
its kinetic energy. In computers, memory's typically directly
associated with storage registers that retain values between clock
pulses. At this point is quite natural to recognize almost all
conceivable system or phenomenum as dynamic by nature. However, there's
some useful cases where this memory effect can be small enough to make
a simple static (mathematical) representation of the system all that's
needed for the majority of meaningful purposes.
There's also an inherently
nonlinear behavior embedded in the second term of the left-hand side of A, as
k
will depend of its own sine. A
linear system possesses the property of
superposition: If an input consists of the weighted sum of several
signals (i.e., recorded values of some variable evolving over time),
then the output is the weighted sum of the responses of the system to
each of those signals.
Nonlinear terms in the equations of a system can
involve algebraic or more complicated functions and variables, and
these terms may have a physical couterpart, such as forces of
inertia
that damp oscillations of a pendulum (e.g., air friction),
viscosity of
a fluid, or the limits of growth of a biological population, to name a
few.
For the sake of simplicity, we can or must dismember the
second-order O.D.E. in a set of two coupled first-order O.D.E.s by
introducing a new variable
v =
dk/
dt. Actually,
the first derivative (over time) of the variable "position" is
(abstractly) another well-know variable called "velocity". So now we
have an (still) autonomous, 2-variable, 3-dimensions, 2-state
mathematical model(6). Usually, the
state of a system is a pair
of variables (or a triple, whatever) who somewhat represents how the
system behavior evolves over time. As a bidimensional representation,
we can simply put
k (position) and
dk/
dt (velocity) on the axes of a diagram on the plane and qualitatively sketch the
phase paths,
flow lines or
orbits which represent A's possible solutions. Each pair (k,
dk/
dt) in the plane corresponds to a particular solution of A. Let's call the set of all these curves the system's
phase portrait, see
Figure 2. Note that point that the central point in the bottom part of the figure corresponds to A's trivial solution:
k = 0 (then
dk/
dt = 0); physically, we put the object at
k = 0 and the pendulum remains there indefinetelly, what we usually call an
equilibrium point (or
fixed point). There's another notable
equilibrium point in Figure 2, b:
k = pi, 3xpi, ... and
k = -pi, -3xpi, ... and
k
= 0 (simply meaning that the pendulum is rotating clockwise or
anti-clockwise), which physically corresponds to the object suspended
above its stem, remaining there indefinetelly, obviously a pretty
counter-intuitive condition.
Let's make a more detailed look at the phase portrait in Figure 2. The
set of closed curves around fixed points represents all the supposedly
periodical movements the simple pendulum can perform - i.e., as some
time passes, the system returns to the same state. Fixed points where
each curve intersects axis
k, corresponds to the oscillations amplitude. The wave-like curves at the top and at the bottom represent movements where
k always increase or always decrease (conventionally, it's assumed that
k increases clockwise) - i.e., movements where the pendulum rotates. The two curves who actually intersect axis
k
spatially define two separated regions with quite different
(qualitative) behaviors: Inside these two curves, the movement is
periodical & bounded; outside, the movement is unbounded. As we
abstracted out air friction and any other possible external influence
besides
gravitation, the simple pendulum keeps its total energy
indefinetelly - i.e., this hyper-idealized but yet somewhat useful
(e.g., at least in an educational sense) mathematical representation of
an actual physical system (i.e., a pendulum) can be called a
convervative or Hamiltonian system.
Let's go now to a more
interesting yet idealized system, naturally ocurring or artifically
built, known as the damped harmonic
oscillator, who we can
mathematically idealize by the following second-order, autonomous,
linear O.D.E.
B:
(dx2/
d2t) +
y x
(dx/
dt) +
o2 x
x = 0
This oscillator is a dissipative system, and note that, according to
its
phase portrait shown in
Figure 3, there's a single point for which
all the trajectories converge provided that a sufficient long time has
passed - i.e., the movement is damped and finally comes to a halt. This
very important construct, only available for dissipative systems, is
known as the
attractor of the system, which in this particular
case is only a point, because B is a linear equation, and coincides
with B's trivial solution, or the system's trivial equilibrium point -
i.e. (
x,
dx/
dt) = (0,0). That is, movements
approach a fixed/equilibrium point, that attracts the closest
orbits.
However, as we'll see in the next parts of this "essay",
attractors
may have any dimension, provided that this dimension is smaller than
the dimension of its system's equilibrium points (also known as the
number of
degrees of freedom).
--
Footnotes:
(1) E.g., the O.E.D.
(2) As the
abstract of an article or research paper.
(3)
Because of their speed, computational power, and flexibility, modern
digital processors are used to implement many practical systems,
ranging from digital auto-pilots to digital audio systems. Such systems
require the use of discrete time sequences representing sampled
versions of continuous time variations - e.g., aircraft position and
velocity for the auto-pilot, and speech & music the audio system.
(4)
However,
discrete time recorded data may represent a phenomenum for
which the variables are inherently discrete, such as in demographic
studies in which, e.g., average budget, crime rate, or pounds of fish
caught, are tabulated against family size, total population, or type of
fishing vessel, respectively. Also, printed pictures actually consist
of a very fine grid of points, each of these representing a sample of
the brightness of the corresponding point in the original image.
(5) A solution for A comprises all values of
k accross the time. Surely, if we arbitrate specific values for
m and
l, there are a number of
computational methods we can use to simulate the pendulum behavior.