Elliptical functions or integrals

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Concerning K. Lee Lerner's question:

> Does anyone have any experience or thoughts regarding at what level it might
> be appropriate to introduce elliptical  functions or integrals to physics
> students.

It seems to me that the first natural places such functions arise in
contexts dealt with by physics students are in the motion of a simple
pendulum swinging at non-infinitesimal angles and in the Keplerian
relative orbital motion of a gravitationally bound two-body system (or,
equivalently, in the bound motion of a one body system interacting with
an attractive 1/r^2 central force).

> Assuming the absence of prerequisite math among high school students, what
> would be the clearest and most easily accessible, demonstration(s) or
> observation(s) (e.g., engineering applications)  of the power and scope of
> elliptical functions or integrals.

This is tougher.  Explaining higher math functions to students who don't
know the necessary calculus used in defining those functions in the first
place could be tricky.  My guess would be to demonstrate them via their
graphs as they are applied to the relevant physical situations.  For
instance, one could plot the graph of the total distance travelled by a
planet or comet along its orbital path as a function of time and/or
orbital longitude.  A graph of the ratio of the circumference of an
ellipse to its semi-major axis as a function of its eccentricity may
also be helpful in demonstrating elliptic integrals of the first kind.
Another possibility is a graph of the period of a simple pendulum as a
function of it maximum swing angle amplitude (or at least as a function
of the sine of half that angle) would demonstrate the behavior of the
complete elliptic integral of the second kind.  I doubt you could do much
for the understanding of such students beyond showing the graphs of the
relevant functions and admiring/noting their shapes, asymptotic
behaviors, and divergences.  I expect Maple or Mathematica or even a
spreadsheet package would be useful in generating such graphs.  They
could, presumably, even be animated, if so desired, when shown on a
computer.

David Bowman
David_Bowman@georgetowncollege.edu