Harold Brochmann
When we draw graphs of functions we
usually plot a dependent variable vertically versus an independent
variable horizontally. In this graph the horizontal axis represents
degrees in the range 0 to 360 and the vertical axis shows the
corresponding sine(angle) value. The graph of the sine function
is associated with a variety of physical relationships, for example
the relationship between displacement and time in the case of
a freely swinging pendulum.
This graph was produced by a computer program. Such programs
are readily available from many sources. I prefer to write my
own computer programs because then I can modify them to investigate
things that interest me. The first modification I made creates
a graph that illustrates displacement for a dampened, or attenuated,
pendulum.
Another modification to the program allows me to plot dependent
variables on both axes. In the following illustration the independent
variable is also angle in the range 0 to 360 degrees; but it
is not indicated on the graph. The axes in this case are both
dependent variables . Such a graph is associated with displacement
versus velocity for a pendulum. Graphs of physical phenomena
in which both axes represent time-dependent variables are called
phase diagrams.
From high school physics we know that the familiar formula:
allows us to calculate the period of a pendulum. The formula
works only for very small displacements. It is also true that
circular sin vs cos phase diagrams represent velocity vs displacement
of pendula only for very small displacement angles.
In this last example the points on the curve were calculated using trigonometric functions. But there is another way.
Consider an a pendulum bob initially
placed at some displacement. Gravity applies a vertical force
which because of the influence of the pendulum's rigid support
results in a force tangential to the curved path that the pendulum
follows.
From this one can calculate momentary acceleration and the resulting
distance covered along the curve over some very short time period.
Repeating the calculation very many times using very small time
periods allows us to simulate the motion of a friction free pendulum.
Calculations of this type are called iterative and, provided
the time increments are small enough, yield quite accurate results.
The specifics of this calculation are deal with in another article
called "Interative Calculations on a Pendulum".
If the computer program is modified
so that values for velocity and displacement at various times
are calculated iteratively with an initial displacement of 5,
10, 22.5,.45, 67.5 and 90 degrees, we get the following.
We see that at small displacements, oval phase diagrams are produced.
These grow more 'squared' at greater amplitudes. At even greater
initial amplitudes the phase diagrams take on a different shape
altogether. Here we have phase diagrams for a friction free penduluma
with displacements of 90, 135 and 165 degrees.
The pendula we have considered so far are assumed to be frictionless.
But what does the phase diagram of a real pendulum which is subject
to air resistance look like? Here is the result produced by my
program for an initial displacement of 165 degrees, using an
empirically chosen damping factor.
The phase diagram of a dampened pendulum stabilizes into an oval
that spirals towards the origin, also known as the diagram's
'attractor'.
Phase diagrams reveal much information about physical phenomena, yet they are quite unfamiliar to most of us. Why is this so?
We have seen how computer based iterative calculations may be used to create graphs that illustrate aspects of motion. The field that studies motion is sometimes called Dynamics.
Another variation of my computer program
allows me to calculate and plot the trajectories of orbiting
objects. In the illustration, the large dot at the origin represents
some massive object - a star perhaps - and the ellipse is the
locus of positions occupied by a much smaller orbiting body like
a planet.
Each iteration of the calculations also provides data on instantaneous
velocity and distance from the central object. Plotting these
numbers produces a phase state diagram for an orbiting object:
A variation of the program allows plotting the orbits of objects
around two 'stars', Here is an example:
The relationship between instantaneous velocity and distance
from one of the stationary objects looks like this:
Varying the initial conditions; relative positions of the three
objects, and the relative velocity of the orbiting objects allows
the creation of phase state diagrams of incredible variety. One
of the phenomena that I like to explore are the shapes of the
attractors produced by stable versus unstable orbits.
An outcome of such experimentation is the observation that orbits involving three or more objects appear inherently unstable, and that the critical limit of initial conditions that yields unstable orbits is extremely difficult to determine. Phase state diagrams of unstable orbits reveal 'strange attractors'. The study of strange attractors has proved to be useful in a variety of fields including cardiology and meteorology. The attractors found in phase state diagrams of human heart beats, for example, are an indication of the heart's state of health. In the case of meteorology, phase state diagrams of data as temperature and wind velocity can reveal the reliability of the predictions made on the basis of the data.