| 释义 |
GeodesicGiven two points on a surface, the geodesic is defined as the shortest path on the surface connecting them. Geodesicshave many interesting properties. The Normal Vector to any point of a Geodesic arc lies along the normalto a surface at that point (Weinstock 1974, p. 65).
Furthermore, no matter how badly a Sphere is distorted, there exist an infinite number of closedgeodesics on it. This general result, demonstrated in the early 1990s, extended earlier work by Birkhoff, who proved in 1917that there exists at least one closed geodesic on a distorted sphere, and Lyusternik and Schnirelmann, who proved in 1923 that thereexist at least three closed geodesics on such a sphere (Cipra 1993).
For a surface given parametrically by  ,  , and  , the geodesic can be found by minimizing theArc Length
 | (1) |
and similarly for  and  . Plugging in,
This can be rewritten as
where
and
Taking derivatives,
so the Euler-Lagrange Differential Equation then gives
 | (14) |
In the special case when  ,  , and  are explicit functions of  only,
 | (15) |
 | (16) |
 | (17) |
 | (18) |
Now, if and are explicit functions of only and  ,
 | (19) |
 | (20) |
 only, then
 | (21) |
 | (22) |
 | (23) |
 | (24) |
 | (25) |
 | (26) |
 | (27) |
 | (28) |
 is rotated about the  -axis so that the equation of the surface is
 | (29) |
The equation of the geodesics is then
 | (33) |
References
Cipra, B. What's Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer. Math. Soc., pp. 27, 1993.Weinstock, R. Calculus of Variations, with Applications to Physics and Engineering. New York: Dover, pp. 26-28 and 45-46, 1974. |
