September 25, 2007 --- Class 9 --- Oscillatory Motion, Script writing,
			Conservation of Energy in the Euler Algorithm

Activities:

        Chapter 4---Oscillatory Motion

                We used the code ~sg/chap4/sho2.c and executable sho2 to 
		calculate the motion of a simple harmonic oscillator.  
		This code is based on the Euler method.  We found that energy 
		is not conserved.  In fact, we found that the energy grows
		exponentially with time, no matter how small the 
		integration step size.

		With the Euler-Cromer algorithm, the energy
                varies during the motion of the body, but it is not
                monotonically increasing as it is for the Euler method.
		It was very easy to demonstrate this by making a few changes
		to the script used to run the euler code.

		The error is order dt for these methods, but the 
		Euler method is unstable, i.e., the errors
		grow with time.

                For homework #3, the Euler-Richardson and Runge-Kutta methods
		will be used.

	A Script to Find the Position as a Function of Step Size
	
		We wrote a script to run sho2 for different step sizes and to
		find the value of the position after one or two seconds.  We
		were then able to plot the results to see that the error is 
		linear for small dt.  Here is a script x_vs_dt.csh that you
		can find in ~sg/chap4

#!/bin/csh
# script to plot position at t= 1s vs dt
  foreach dt ( .1 .05 .01 .005 .001 .0005 )
   	echo -n $dt  "  " 
   	sho2 <<EOF |awk '{ if($1 == 2.0) print $2}'
1.
9.
2.
$dt
2.
EOF
end

		File pos_dt.ax contains a plot of the position.  
		It is important to practice writing shell scripts, because
		becoming proficient in this skill will save you a lot of time
		in the long run.

	Conservation of Energy in Euler algorithm

		By calculating the energy after a single step for the
		harmonic oscillator, we were able to show that for the Euler
		algorithm the change in energy is proportional to the
		current energy, thus it is increasing exponentally.  For
		the Euler-Cromer algorithm, the change in energy is
		proportional to the difference between potential and
		kinetic energy.  (This should be derived by the student.)
		Since for a harmonic oscillator the time
		average of kinetic and potential energies is the same, the
		Euler-Cromer algorithm conserves energy on average, i.e.,
		the energy ocsillates around the correct mean value.