In order to determine the geodesics between two points, one must solve the geodesic differential equations, which are as following
\begin{align}
p &= u'(s)\\
q &= v'(s)\\
p' + \Gamma^0\_{00}p^2 + 2\Gamma^0\_{01}pq + \Gamma^0\_{11}q^2 &= 0\\ 
q' + \Gamma^1\_{00}p^2 + 2\Gamma^1\_{01}pq + \Gamma^1\_{11}q^2 &= 0
\end{align}
where $\Gamma^i\_{jk}$ is the Christoffel symbol of second kind, and the initial conditions are $u(s\_0) = u\_0,\, \, v(s\_0) = v\_0,\, \, u'(s\_0) = p\_0,\, \, v'(s\_0) = q\_0$.


In order to solve these equations, one can only provide one point $(u,v)$, and the direction $(u',v')$. How is it possible to determine the geodesics between two points ?


For a sphere, both points will lie on the great circle. So here it is sufficient to provide the initial point and direction. But is it possible to accomplish this by only providing both points, if information such as direction is not known a priori ?



Determining geodesics between two points in curved space
I assume that  [imranal](https://mathoverflow.net/users/80965/imranal)  asks how to find numerically a geodesic connecting two given points if the connection is given. 
One way to do it is to implement the solution of the ODE system he wrote in his question numerically (there are many effective ways for it) and then use this implimintation to find the correct initial velocity vector $(u', v')$ such that the geodesic starting from $(u, v)$ in the direction $(u', v')$ hits the second point. One can slighly improve this search but the principle remains the same. 


Alternative method which actually needs that the connection you have comes from a Riemannian metric 
would be to use the curve shortening flow: start with any curve connecting these two points (say, a straight line) and then
 start to deform the curve such that the velocity vector of the deformation is orthogonal to the velocity vector of the curve and its length is the geodesic curvature times some fixed function which vanishes at the first and last points of the curve. 
This deformation should normally converge to a geodesic. 


I expect that there should be effective implimentation of this idea as a numerical scheme but I am not an expert in these questions. 


Of course geodesic does not always exist (as explained by Igor) and may be is not unique (as explained by Narasimham) or/and the second procedure may converge to a geodesic which is not a minimal one. 


