Let's consider a problem from a previous class.
Problem 1:
A tunnel is dug along a chord through the earth. The purpose
is to allow for high-speed, mass transit train travel with little
energy usage (see figure 7). Show that the time of transit for
a train car which is "dropped" into the tunnel is the same no
matter where the chord is located. Find this time of transit.
Assume no friction is acting.
Solution 1:
In looking at figure 1, we can see that the location of the
train car is specified by x which gives the distance of
the train car from the center of the tunnel. ME and
RE are the mass and radius of the earth and r is
the position vector for the car. If no friction is acting,
then the force acting on the car is
| (3.7.2) |
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Recall that the equation of motion for a system undergoing simple harmonic oscillation is always defined as
| (3.7.5) |
The constant, c is a magnitude so as to always maintain the negative sign on the right-hand side (rhs) of this equation. If the quantity on the rhs is not negative, then the motion does not repeat endlessly and hence is not simple harmonic. Equation 3.7.5 describes motion which moves about the equilibrium position at regular intervals of time. In solving the above equation (easily done with Maple), we find that
| (3.7.6) |
As a review of the connection between circular motion and oscillation, consider the Maple file that produces an animation showing the connection (as depicted below).
In the case of the tunnel through the earth, we have the time of travel from one end to the other as one-half of a cycle of oscillation. If we label the period as T, then
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noun.
What scientists and mathematicians mean by chaos is very much related to the spirit of the definitions given above. We state that systems are chaotic if they:
This may seem to be a strange definition before we've motivated it, but it becomes clearer as we consider examples. We can see many examples using Maple (you knew that was coming, didn't you.) What follows below is an animation of one of the curves generated by the 1994 Pre-Freshman class. You can see a Java-animated version of the same effect at this site on molecular modelling.
###################################################### # Begin by loading plot library with(plots): # # Animate the curve animate([sin(u*t*2)+cos(u*t*3),sin(u*t)-cos((u*t)^2), t=0..1], u=0..2*Pi, frames=60,numpoints=100); ######################################################The path traced out by end of the line as the animation proceeds certainly looks chaotic (in the general meaning of that word) as shown in figure 3.
However, the general meaning of chaos implies unpredictability of the path. Certainly as you trace the animation out frame-by-frame you would be hard put to guess the next move of the end of the line, but, since the curve is generated from a mathematical formula, you can in fact predict exactly what the curve will do next in the animation and what its final shape will be. So what might appear to the naked eye as infathomably complex is, in reality, governed by a relatively simple mathematical expression. In essence, there is no difference between the complex path shown above and, say, a path along a straight line or sinusoidal curve. Its simply that a straight line path leads the eye in an obvious way from the start point to the end point.
The trick, in practice, of predicting the path of something or the future course of an evolving system described by a complex algorithm comes in the sensitivity of the algorithm to small changes in the initial point. Let's take the case of a straight line and the complex curve we looked at previously. If the line is described by
| (3.7.1) |
##################################################################### # Just add in the shift, in this case 0.1 animate([sin(0.1+u*t*2)+cos(0.1+u*t*3),sin(0.1+u*t)-cos((0.1+u*t)^2), t=0..1], u=0..2*Pi, frames=60,numpoints=100); #####################################################################
Now we note that our animated curve generally follows the same loopy pattern we saw before, BUT it ends up considerably further from the previous endpoint than the 0.1 shift might have led us to believe (compare the curves in figures 3 and 4 if you didn't do the Maple animation).
The essence of chaos in science is just that: a relatively complex behavior which is strictly governed by a mathematical algorithm, but, is nonetheless unpredictable due to sensitivity to initial conditions. So, although in principle we can predict how a system will behave to an arbitrary level of precision, in practice we can't find the initial starting point of the system accurately enough to be able to predict in detail what will happen beyond a short period of time. Small mismeasurements eventually add up to a big discrepancy between calculated and observed behavior.
The surefire way to have a system described by an algorithm that exhibits chaotic behavior is to have it be nonlinear. The importance of studying chaotic behavior lies in the fact that most systems encountered in the real world are nonlinear to some extent and either exhibit chaotic behavior or can be made to exhibit it. Prime examples are weather prediction, population kinetics (i.e. fluctuations in populations from generation to generation), fluid flow, mechanical and electrical oscillatory phenomena (e.g. heart beats or the electrical activity of your brain), the tumbling motion of the moon Hyperion in its orbit around the planet Saturn, economic systems, and many, many other phenomena. In fact, chaos is observed in so many systems in the real world that some scientists rank the understanding of chaos as being as important as the theories of relativity and quantum mechanics in that its ramifications stretch into every aspect of scientific study.
Nonetheless, how do you know nonlinear behavior when you see it? Let's start by looking at an example. A relatively simple one is shown in figure 5.
The picture depicts a pendulum with a magnet (blue) as the bob on the end of a rigid rod which is free to swing on a supporting bar. Two other magnets (red) are fixed in position on either side of the equilibrium position (in the absence of the red magnets) of the blue magnet. The red magnets must be placed so that pendulum can rest in equilibrium with the blue magnet directly above either of the fixed magnets. Hence we have a system with two possible equilibrium positions. If the pendulum is pulled aside along the y axis and released, it quickly begins to execute an extremely complicated motion. Since the magnetic force is a strong function of the relative distance between the magnets and the magnetic forces can provide acceleration, deceleration, and damping (the damping force comes about because of induced currents, a topic you will learn more about in Physics 2 or 151). There is also damping due to friction of the pendulum support on the supporting rod.
The damping eventually causes the blue magnet to move to one of the equilibrium points, BUT which one? That depends very much on the initial condition. If the pendulum starts its motion ever so slightly closer to one magnet than the other, then its motion will eventually become highly perturbed. The complexity of the motion is high enough so as to make it nearly impossible to determine which equilibrium position the pendulum will choose given certain starting positions. Of course the motion of the pendulum is strictly governed by Newton's Laws so we have a mathematical description of the motion. Certainly the motion is complex and sensitive to initial conditions. Therefore, this is a perfect example of chaotic motion!
Figure 6 shows the path of the pendulum magnet as projected on the plane with the fixed magnets. The white dot represents the position of one fixed magnet while the blue dot marks the position of the other fixed magnet. The colored path shows that the pendulum follows a complicated path to its eventual equilibrium position around one of the fixed magnets.
Figure 7 displays which equilibrium position the magnet winds up on according to its initial position as projected on the plane containing the fixed magnets. The blue and white regions show initial positions which correspond to the magnet coming to equilibrium around either the blue or white fixed magnet.
If we could blow up the region around the boundaries between blue and white areas, we would find that they are not infinitely sharp. Instead, we would see a complex structure which is termed a fractal. Fractals have fractional dimensions and the unique property of self-similarity to all levels of magnification. If you magnify any part of a fractal, you see a minature recreation of the overall fractal structure repeated on the small scale. Magnify a small piece of any part of the small structure and you see the overall structure repeated again and again, ad infinitum. The coastlines of continents and the structure of snowflakes are just two of the many examples of fractals found in nature. The most famous shape among the fractals is the Mandelbrot set shown in figure 8.
You can find out more about fractals on the Web. Some starting points are:
Chaos enters into cutting-edge research into phenomena like convection. To see chaos for a system as simple as a pendulum, check out this applet of a pendulum with a driver for the support point and damping.
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