PHYS 305 - Assignment #5
Due: Monday, February 18, 2013

Make sure your name is listed as a comment at the beginning of all
your work.  Please make sure that every graph and chart you turn in is
well-labeled with axes and a title.

Purpose: Develop a physical intuition about Chaotic scattering.
Complete all exercises on chaotic scattering in the course web page
(reproduced below):

Chaotic Scattering
==================

Use the program we wrote in class to solve this scattering
problem. Use a scaled mass m = 1.0.  Use a time step parameter dt =
0.01.  Check that the code compute each trajectory until the distance
of the particle from the origin exceeds a distance of 4.1 in scaled
units.  Choose v₀ = 0.15, k = 1.0, b = 0.25, and compute the
trajectory (y(t) versus x(t)) for y₀ = 0.0 and y₀ = 0.2.  Use x₀ =
-3.9.  Plot the two trajectories.  Plot E(t) as a check.  Use the code
potential.c as a guide to compute the latter.  Check the of the
scattering angle θ and the scattering time T.

* Compute the trajectories for y₀ = 0, …, 0.8 in 1000 equal
  increments.  Plot separately the scattering angle and the scattering
  time for these trajectories versus y₀.  You should find that, for
  some ranges in y₀, both θ and T vary smoothly with y₀.  However, you
  should also find certain irregular regions where both quantities
  change greatly from one trajectory to the next.  This sensitive
  dependence of the outcome on small changes in the initial conditions
  is the hallmark of chaos.  The behavior of the system in the
  irregular regions is known as chaotic scattering.

* Choose one of the irregular regions in the previous question and
  “zoom in” on it by successively narrowing the range in y₀ and
  covering the new range in the same number (1000) of steps.  Do you
  see regular regions within the chaotic band? Continue to zoom in on
  the new chaotic regions until you have decreased the range of y₀ by
  a factor of 1000 below that used initially.  You should find
  self-similar structure–the same pattern of regularity and
  irregularity keeps recurring on smaller and smaller scales.

* Identify some of the impact parameter y₀ values that yield the
  longest scattering times and plot the corresponding trajectories.
  You may want to draw small circles centered on the three repellor
  positions to guide the eyes.  See for instance: draw circles.c.
  Better yet is to draw the interception of a plane at the appropriate
  energy E with the potential surface.  See for instance: potential
  contour.c

* Can you identify any critical trajectories that seem to separate
  regular from irregular motion?

Energy Dependence
=================

The scattering functions, scattering angle and escape time, in reality
depend on both the impact parameter and the energy of the incoming
particle.

Form two color images (1024x1024) of the scattering angle and escape
time, having the incoming kinetic energy (vertical axis) and the
impact parameter (horizontal axis) as labels.  Use an impact parameter
range b = 0.0, …, 0.8 and an energy range corresponding to 0.1 to 1.2
the hill height.  Write a small feeder code setup scatter.c to
generate initial conditions evenly distributed within these
ranges. These are to be fed in the differential equation solver code.