geodesics on the torus
Geodesics on the Torus and other Surfaces of Revolution
Clarified Using Undergraduate Physics Tricks with Bonus:
Nonrelativistic and Relativistic Kepler
Problems
(in progress Fall 2009, nearly complete)
Context:
Most discussions (okay, a few discussions---they 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) 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 periodic 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 light 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 periodic geodesics on the torus is analogous to quantization of energy levels in models of atoms.
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
- For an overview of general relativity see:
http://en.wikipedia.org/wiki/Introduction_to_general_relativity
and
http://en.wikipedia.org/wiki/General_relativity
- Wolfram MathWorld:
http://mathworld.wolfram.com/SphericalCoordinates.html
discusses the mathematics and physics conventions for spherical coordinates.
- Wolfram MathWorld describes the torus:
http://mathworld.wolfram.com/Torus.html
see also:
http://en.wikipedia.org/wiki/Standard_torus
(Wiki and MathWorld can't seem to agree on spindle and horn tori!)
- For more details than you might want to face, see
dr bob's elementary differential geometry:
a slightly different approach based on elementary undergraduate linear
algebra, multivariable calculus and differential equations
- For the Kepler problem, the radial oscillations about the outer equator
are directly analogous to those about a stable circular orbit, where the
frequency of this oscillation is called the epicyclic frequency. This is
derived in the final equation (B.11) of the Poynting-Robertson effect
article linked above. An article which discusses this frequency in terms of
geodesic motion is
Marek Biesiada, General Relativity and Gravitation, Vol 35, 1503 (2003).
- I finally did find one brief discussion in some course notes by a mathematician(?) of
geodesics on the torus using a Lagrangian approach, discussing the constants
of the motion and the effective potential but no details about the actual
geodesics (browsing his website shows he appears to be a
physicist-mathematician):
Joel Feldman, Dept of Mathematics, University of British Columbia,
Vancouver, Canada
Course Notes:
Geodesics on the Torus (2007)
http://www.math.ubc.ca/~feldman/m428/torus_geodesics.pdf
-
by John Oprea - 2007 - Mathematics - 469
pages
Geodesic on the inside parallel of
torus
Figure 5.14
Geodesic on the inside parallel of
torus:
an example of round-off error ...
books.google.com/books?isbn=0883857480...
[I actually have this book,
but never found time to browse through it since I
ordered it. It has a Maple routine to start with
coordinate initial data and numerically solve and
visualize the geodesics on a surface, including the
torus, but not in terms of the more refined initial
angle that is geometrically obvious to interpret.]
- .From page 14 of Google hits (10/2009), comparing different numerical
integration schemes for geodesics on tori and other surfaces:
File Format: PDF/Adobe
Acrobat -
View as HTML
FINDING
GEODESICS ON SURFACES. 5 path. This method has
the same effect as looping a string around the
torus following the original path given,
...
cs.stanford.edu/people/jbaek/18.821.paper2.pdf -
Similar
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-
File Format: PDF/Adobe Acrobat -
Quick View
from the
geodesic equations, so sinφ = c cosv . This is a
special case of a result known as Clairaut's relation.
Example 5.4 The
Torus.
Take the
torus
with
...
education.uncc.edu/droyster/courses/fall98/.../geodesiceqn.pdf
-
Similar
- This book actually talks about the moving particle approach and conservation of
angular momentum as Clairaut's Theorem:
Andrew Pressley (2001).
Elementary Differential Geometry. Springer. p. 183.
ISBN 1852331526.
http://books.google.com/books?id=UXPyquQaO6EC&pg=PA185&dq=%22Clairaut%27s+theorem%22&lr=&as_brr=0&sig=ACfU3U214J0zkWQcRXLTohVjdHUD3Fuk2A#PPA183,M1.