Geodesics on the Torus and other Surfaces of Revolution
Clarified Using Undergraduate Physics Tricks with Bonus:
Nonrelativistic and Relativistic Kepler Problems
by bob jantzen
article pdf [final update December 27, 2012]
Related talks and Maple worksheets:
Geodesics in the NYTimes! "Think Globally," by
See the beautiful video of a tiny bike riding straight ahead on the double torus [8 function graphs, no continuous symmetry].
Context: Most discussions (okay, a few discussionsthey are hard to find—in particular see the very interesting article by Mark L. Irons, a math enthusiast who actually went to my high school and by coincidence learned some differential geometry from a friend of mine in the GR community) that you find of this problem of geodesics on the torus are by mathematicians who have no clue about the physics approach to it, which gives much more insight into this dynamical system. [Clairaut's theorem? In over 3 decades of using differential geometry in GR I never heard of this. Who needs it? But then I never studied the classical differential geometry of curves and surfaces in ordinary space—my aims were for GR instead.] This article is aimed at providing enough background to the uninitiated (to differential geometry) to explain these ideas, and introduce Maple worksheets that can be played with to experiment with the behavior of actual geodesics. Finding the closed geodesics might even be considered a fun computer game, for some of us math lovers anyway.
The same ideas immediately extend to the Kepler problem of nonrelativistic or
relativistic planar orbits in spherically symmetric gravitational fields. The
effect below includes a photon` pressure drag.
Abstract: In considering the mathematical problem of describing the geodesics on a torus or any other surface of revolution, there is a tremendous advantage in conceptual understanding that derives from taking the point of view of a physicist by interpreting parametrized geodesics as the paths traced out in time by the motion of a point in the surface, identifying the parameter with the time. Considering energy levels in an effective potential for the reduced motion then proves to be an extremely useful tool in studying the behavior and properties of the geodesics. The same approach can be easily tweaked to extend to both the nonrelativistic and relativistic Kepler problems. The spectrum of closed geodesics on the torus is analogous to the quantization of energy levels in models of atoms.
Geodesics on Surfaces with Helical
A 3-parameter family of helical tubular surfaces obtained by screw revolving a circle provides a useful pedagogical example of how to study geodesics on a surface that admits a 1-parameter symmetry group, but is not as simple as a surface of revolution like the torus which it contains as a special case. It serves as a simple example of helically symmetric surfaces which are the generalizations of surfaces of revolution in which an initial plane curve is screw-revolved around an axis in its plane. The physics description of geodesic motion requires a slightly more involved effective potential approach than the torus case due to the nonorthogonal coordinate grid necessary to describe this problem. Amazingly this discussion allows one to very nicely describe the geodesics of the surface of cavatappi pasta.
see Cavatappo 1.0 [cavatappi-geos.mw]
Cavatappo 2.0: More of the Same but Better
Innocent musing on geodesic motion dynamics on the surface of helical pasta shapes leads to a single continuous 4-parameter family of surfaces invariant under at least a 1-parameter symmetry group and which contains as various limits spheres, tori, helical tubes, and cylinders, all useful for illustrating various aspects of geometry in a visualizable setting that are important in general relativity.
see Cavatappo 2.0 [cavatappitilted-geos.mw, ]
With a little help from general relativity, the general relativistic Poynting-Robertson effect can be studied in the same way.
and our published article:
The general relativistic Poynting-Robertson effect