--- Henry Baker <hbaker1@pipeline.com> wrote:
I'm curious as to why elliptic functions seem to be relegated to the if-we-have-time-this-semester sections of textbooks.
The proper place to introduce elliptic functions is in a course in complex variables, after the machinery has been constructed. It then has to compete with other applications of the theory. But there has been a tremendous revival of interest in the last couple of decades. Go to your local university library, and you will find more books on that subject than you could ever hope to read. When I pick up a new math book, I like to read the preface. If the author states that the subject matter has been restricted to fit within a course of so many semesters, then that tells me that the book has been dumbed down and is not worth my time or money. I prefer books suitable for self-learning. For an excellent introduction to elliptic functions for someone approaching the subject for the first time, I highly recommend Whittaker and Watson, "Modern Analysis", Chapters 20-22. This is where I learned the material. Elliptic functions are defined as meromorphic doubly periodic functions. This means that elliptic functions are periodic with two distinct periods whose ratio is a nonreal complex number, and that their only singularities are poles. An immense number of elegant results then follow from applying Liouville's theorem and the residue theorem. In your first foray into elliptic functions, you must eschew any development which proceeds entirely in the real variable domain.
1. Are elliptic functions useful for closed form solutions of things other than tumbling bricks?
Typically elliptic functions arise in physical problems as solutions to differential equations. The equation (dx/dt)^2 = (1 - x^2)(1 - k^2 x^2), or its trigonmetric variants, arises in the solution for the motion of the pendulum when one does not make the approximation that the swing angle is small. This same equation arises in frequency doubling. When a collimated laser beam propagates through a nonlinear crystal under suitable "phase matched" conditions, the optical power is converted into a beam at twice the frequency. When the phase matching is not perfect, the powers at each of the two frequencies are given as a function of distance by elliptic functions. The green laser pointers that are becoming popular work this way. A semiconductor laser diode produces light at 808 nm which is absorbed by and pumps a neodymium doped YAG (yttrium aluminum garnet) or yttrium vanadate crystal. The neodymium lases at 1064 nm, and that light is frequency-doubled in a crystal of KTP (potassium titanyl phosphate) to green 532 nm. These are just two physical applications of elliptic functions that come to mind immediately; there is much more. In the case of the pendulum it is more than just coincidence that the motion is given by an elliptic function. One can see by inspection that the motion must be described by a doubly periodic function. The motion for imaginary time corresponds to reversing the potential energy, that is, inverting the pendulum and allowing it to swing along the complementary arc. Therefore there must be both real and imaginary periods. I picked up this little tidbit from Whittaker's "Analytical Dynamics". It applies to any system whose kinetic energy is quadratic in its velocity variables, and whose potential energy is a function of position variables only.
2. What specific feature about tumbling bricks would give a clue that elliptic functions were needed in its solution?
The Euler equations for the components of angular velocity in body-fixed coordinates, and along the principal axes, are (modulo a possible sign error) dw /dt = (I - I ) w w , 1 2 3 2 3 dw /dt = (I - I ) w w , 2 3 1 3 1 dw /dt = (I - I ) w w . 3 1 2 1 2 The three fundamental Jacobian elliptic functions (with fixed modulus k) satisfy differential equations (d/dx) sn x = (cn x)(dn x), (d/dx) cn x = -(sn x)(dn x), (d/dx) dn x = - k^2 (sn x)(cn x). So it's a natural. In the case of the tumbling ridid body, elliptic functions get you only the angular velocities. To obtain the orientation, say Euler angles, one needs bare theta functions, and not merely the ratios of theta functions that constitute the elliptic functions.
At 09:34 AM 7/4/03 -0700, Eugene Salamin wrote:
I've written a paper which reviews the exact mathematical solution for the tumbling motion of torque-free rigid body whose three principal moments of inertia are different. This has all been known for over 100 years, and is adapted from Whittaker's "Analytical Dynamics". It's an orgy of elliptic functions and theta functions. I'll be delighted to email the paper to anyone who wants it. It is a Microsoft Word document.
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