(1) To try out various harmonic and anharmonic oscillators,
you may change the parameters and in the text panels.
(2) To get a feeling for the limitations of the Verlet algorithm,
play around with .
EXERCISE:
a) Write a program to tabulate and/or draw the analytical
solution to the HO equation. (You may achieve a very concise visualization
by displaying the trajectory in phase space, i.e. in the
coordinate system
; where for the approximation
may be used.) Choose
values of
, and
,
and use these to determine the exact value of . Starting
with and , employ the above algorithm to compute
the further path
. Test the performance of
your program by varying and
.
b) Now apply your code to the anharmonic oscillator
To start the algorithm you may use the approximate value given by
EXERCISE:
The planar pendulum is described by the equation of motion
Solution strategies vary between the three regions:
very small; small; arbitrary.
Very small : Here
, and the e.o.m. is that
of a harmonic oscillator, with the usual analytic solution.
Small : We may put
,
adding an anharmonic term to the e.o.m. Again, an analytical solution
may be found,
but it is more involved than in the harmonic case; see
Landau-Lifshitz, Mechanics.
Any :
The exact solution is known in implicit form:
One may clumsily invert this equation for regular times , using
Newton-Raphson.
Instead, one may choose to simulate the pendulum, using the Stoermer-Verlet
algorithm:
As a starting value for one may use the Taylor approximation
EXERCISE:
Let us divide the rod into pieces of equal
length, with node points ,
and assume the boundary conditions
and
. The values for the temperature
at time (the initial values) are
and
(step function).
Next:2. Linear AlgebraUp:1. Finite Difference Calculus Previous:1.3.4 Second derivatives in Franz J. Vesely Oct 2005
See also: "Computational Physics - An Introduction," Kluwer-Plenum 2001