If i can get back to the point i can show you a little math at this moment.
Using Christoffel symbols of second kind i can write system of partial differencial equations which describes geodesics.
So (is there any chance that i can input latex code?):
$$u^{\prime \prime} + \Gamma^{1}_{11} (u^\prime)^2 + 2\Gamma^1_{12} u^\prime v^\prime + \Gamma^{1}_{22} (v^\prime)^2 = 0$$
$$v^{\prime \prime} + \Gamma^{1}_{21} (u^\prime)^2 + 2\Gamma^2_{12} u^\prime v^\prime + \Gamma^{2}_{22} (v^\prime)^2 = 0$$
There are a few ways how to compute Christoffel symbols easiest one is from metric of a surface.
When i want to solve this type of eqaution systems a need to define some conditions (sorry i am not familiar with english therm).
In this case they are:
u[0] == u0, v[0] == v0, u'[0] == Cos[Theta], v'[0] == Sin[Theta]};
So i think that problem could be undefined the angle [Theta] that says the direction, because it's not in the input. I would like to ask if anyone could help me and try to run this on his computer.
First i define function:
solvegeo[x_, {u0_, v0_}, b_, m_] :=
Module[{su, e, so},
christoff2[x][u, v];
su = {u -> u[s], v -> v[s], p -> u'[s], q -> v'[s]};
e[j_] := e[j] = G[j, 1, 1]p^2 + 2G[j, 1, 2]p q + G[j, 2, 2]q^2 /. su;
eqic := {u''[s] + e[1] == 0, e[2] + v''[s] == 0,
u[0] == u0, v[0] == v0, u'[0] == Cos[Theta], v'[0] == Sin[Theta]};
so := NDSolve[eqic, {u, v}, {s, 0, b}];
Flatten[Table[so, {Theta, 2 Pi/m, 2 pi, 2 pi/m}], 1]
]
and apply it on:
hp:=paraboloid[-1,1], b:=2,m:=50 ,sg:=solvegeo[hp,{1,-1}/2,b,m]
or, you can try any of imported notebooks
Thank you for any good ideas
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