I am using the following definition of geodesic:
Let $S$ be a regular surface. A proper variation of a parametrized curve $\gamma: [a,b] \to S$ is a smooth map $$\Gamma:(-\epsilon,\epsilon)\times [a,b] \to S$$ satisfying $\Gamma(0,s)=\gamma(s), \Gamma(t,a)=\gamma(a), \Gamma(t,b)=\gamma(b).$
A parametrized curve $\gamma:I \to S$ is a geodesic on $S$ if for all proper variations $\Gamma$ of $\gamma$, we have $$\frac{d}{dt}\ell(\Gamma(t,s))|_{t=0}=0.$$
By this definition, if we take two separate non-antipodal points on a 2-sphere, there are two geodesics connecting these points, both are parts of the great circle passing through these points. It is easy to see that the shorter geodesic is the minimal point of the length function above. Some people say that the longer geodesic is the saddle point of the length function but I could not see why. It seems to me that the longer geodesic is a local maximal point by looking at smooth deformation (proper variation) nearby it. Am I missing something?