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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?

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    $\begingroup$ It seems so. 'energy' minimization will give a curve along which a particle 'moves' at constant speed and will give a unique solution. I will look into it further. $\endgroup$ Jul 27, 2020 at 14:36
  • $\begingroup$ The critical point for length is not unique since length is independent of parameterization, while critical point for energy is dependent on that. Basically parameterization is time-distance/position relation, and gives us information about speed, it's easy to visualize that if we consider velocity and 'travel distance'. $\endgroup$ Jul 27, 2020 at 15:06

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The quantity $$ {\displaystyle E(\gamma )~{\stackrel {\text{def}}{=}}~{\frac {1}{2}}\int _{a}^{b}\left\|\gamma '(t)\right\|^{2}~\mathrm {d} {t}}{\displaystyle E(\gamma )~{\stackrel {\text{def}}{=}}~{\frac {1}{2}}\int _{a}^{b}\left\|\gamma '(t)\right\|^{2}~\mathrm {d} {t}} $$ is sometimes called the energy or action of the curve; this name is justified because the geodesic equations (derive from) the same Euler–Lagrange equations of motion for this action.

So a a unit speed can include geodesics with orbits minimizing energy as well as lengths ... the concepts extending to Hamiltonian and Relativity formulations.

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  • $\begingroup$ Do you have links to articles or books about it's relation to Hamiltonian mechanics and Relativity? $\endgroup$ Jul 27, 2020 at 15:08
  • $\begingroup$ There seem many, e.g. macs.hw.ac.uk/~simonm/mechanics.pdf with a good preface. $\endgroup$ Jul 27, 2020 at 15:18
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    $\begingroup$ Yes. More reading convinces gradually more. The earliest connection I saw was the Brachistochrone problem. It is was surprising that $\int (U-K) d.. $ invariance formulation has the same result as $U+K$ energy conservation. The identity of Lagrangian/ Hamiltonian approaches reveals so much of the scientific identity among the laws. At first when a projectile motion was understood as a dynamic geodesic it was so exciting. $\endgroup$
    – Narasimham
    Jul 27, 2020 at 16:03

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