Context of this question: This question follows from a post Decomposition of a function and chain rule. and discusses on something different. Using calculation of variation we can find critical points of a function of a vaiable curve $\gamma$ with its end points fixed at $a,b$ and therefore defining geodesic on a manifold. (The rest of the paragraph is unnecessary reading for the question; it's mainly for the purpose of arranging my several posts on a topic.) Along the geodesic, exponential maps on a manifold project a tangent vector at a point $p$ (locally approximately linearly) to another point, as discussed here What is exponential map in differential geometry (a related but different concept of exponential maps of Lie group is discussed here Relations between two definitions of Lie algebra). Geodesics have as such properties like the 'closed curve' {$\exp_p(v),\forall v$ of the same norm and belonging to $T_pM$} is perpendicular to all the geodesics passing through $p$, and is the shortest curve connecting $a,b$ (i.e. it's also a critical point for length). So we can say the 'closed curve' is very much resembles a circle, and a geodesic a radius or a straight line (We can perhaps even say that with geodesic and exponential maps we 'maps' projective geometry on to a manifold, similar to what we do when, with homeomorphism in definition of a manifold, we 'map' Euclidean space to a manifold). With the fact that geodesics is the shortest curve, we can define a metric (a measure of distance, NOT Riemannian metric which is an inner product and 2-tensor, as discussed here: Calculation of inner product for Riemannian metrics.) on a manifold. The metric is homeomorphic to the original metric of the manifold, as discussed here Comparison of metrics on a manifold..
My question is as follows:
A critical point of 'energy' (as Spivak calls it) $E(\gamma)=\int_a^b \langle \frac{d\gamma}{dt},\frac{d\gamma}{dt}\rangle dt$--where $\frac{d\gamma}{dt}$ is tangent vector along $\gamma$ at the point of $\gamma(t)$--is called geodesic. (I guess he uses the name 'energy' for in physics square of velocity is proportional to energy.)
Why we define critical point for energy, instead of critical point for length $L(\gamma)=\int_a^b\sqrt{\langle \frac{d\gamma}{dt},\frac{d\gamma}{dt}\rangle} dt$, to be geodesic?