Dear Reza,

If you are also looking for a material with the curvature, besides the geodesic, of the torus, the following .pdf is very helpful and easy to read.

Cheers,

Ana Paula

Dear Reza

I once derived the equations of geodesic for torus as a byproduct when I was 

doing my graduate studies and a course in analytic mechanics (long time ago) . 

The task was to find equilibrium points  if a particle is restricted to move on a surface of a torus and show that the 'outer equator' of a torus is a stable equilibrium and inner equator is unstable equilibrium. 

As it happens : If a particle is restricted to move on a smooth surface with no external 

forces (only holonomic constraint forces keeping the particle on the surface) then 

the particle moves along the geodesics of the surface. So the solution to equations 

of motion are at the same time geodesics of the surface. 

I attach here a pdf about my solution. It is in Finnish, but the equations of motion 

(1), (2) , (3) and (4) are also equations for the geodesics if you want to analyze them yourself. The parameters I have used 

are shown in Fig.1 and I have plotted some figures about the trajectories / geodesics in Fig. 3, Fig.5 and Fig.7. 

I hope this kind of thinking helps you since you seem to be a student in physics. 

Cheers !

Samuli Piipponen

UN of Tampere, Finland

Please see the  attached paper.

Hi George

Yeah finnish is not the easiest language :-) But the main conclusions

in the pdf are that for initial velocities small enough the trajectories in the phase plane form closed curves if the particle starts from the outer equator and

the curves are never closed and \theta is increasing if it starts from inner equator. Hence outer equator is stable and inner equator unstable.

I am guessing its due to the fact that the (Gaussian) curvature in inner equator is neqative and in outer equator positive ?

regards

Samuli

Dear friends

Thank you for your excellent answers.

Reza

Uni of Tehran

Dear Reza,

The geodesic curves on a manifold are determined by the so-called Riemann metric tensor. A relatively simple intoductory material could be found in Spence and Moon Field Theory Handbook and in other related monographs.

Piotr Grabowski

Dear Reza,

Please find the attached papers.

http://www.rdrop.com/~half/math/torus/torus.geodesics.pdf

http://www.cmsim.eu/papers_pdf/january_2012_papers/25_CMSIM_2012_Pokorny_1_281-298.pdf

http://www.win.tue.nl/~rvhassel/Onderwijs/Tensor-ConTeX-Bib/Examples-diff-geom/Torus-diff-geom/geod-torus.pdf

http://arxiv.org/abs/1212.6206

http://homeweb.unifr.ch/parlierh/pub/shortmark.pdf

http://www.math.u-bordeaux1.fr/~pmounoud/articles/torgeod.pdf

Dear Reza. My article

Article On constructing generally periodic solutions of a complicate...

This is a good question with a number of possible answers.

In addition to the incisive answers already given, the following may prove interesting:

K.  Dankwart, On the large-scale geometry of flat surfaces, Ph.D. thesis, Rheinischen Friedrich-Wilhelms-Universitaet Bonn, 2010:

http://bib.math.uni-bonn.de/downloads/bms/BMS-401.pdf

See Section 2.1, starting on page 12, on metric spaces and geodesics.    An example branch cover of the torus is given in Ch. 6, starting on page 92.