Is it true that applying the Euler-Lagrange equation to the integral
$E(\gamma)=\int_{t_1}^{t_2} g_{\alpha\beta}(\gamma^{\alpha})'(\gamma^{\beta})'\operatorname{d}\!t$ rather than the arc length integral $L(\gamma)=\int_{t_1}^{t_2} \sqrt{g_{\alpha\beta}(\gamma^{\alpha})'(\gamma^{\beta})'}\operatorname{d}\!t$ is just a mathematical trick for simplifying computation when parameterizing by arc length?
Isn't it true that if you are parameterizing a curve on a surface by something other than a constant multiple of arc length (for example, a curve $\mathbf{C}(x)=(x,y(x))$ in the parameter space of the hemisphere surface mapping $z=\sqrt{1-x^2-y^2}$), the critical point of $E(\mathbf{C})$ will not be a geodesic in general?
Best Answer
The minimum for the length functional is badly non-unique: it is given by any parametrization of the geodesic, not necessarily the natural one. On the other hand, the energy functional is locally uniformly convex; therefore the minimizer is unique and the variational calculus is nice.
It is instructive to do the computation for the Euclidean plane. Suppose we are looking for the condition that a curve $(x(t),y(t))$ is a geodesic. Then, for the energy functional $ \int \left(x'(t)^2+y'(t)^2\right)dt, $ the Euler-Lagrange equations read $x''(t)=0$ and $y''(x)=0$, which is the straight line traced at constant speed. On the other hand, for the length functional $\int \sqrt{x'(t)^2+y'(t)^2}dt, $ one gets, after some algebra, $$ \frac{x''}{x'}=\frac{x'x''+y'y''}{x'^2+y'^2}\;\mathrm{and}\;\frac{y''}{y'}=\frac{x'x''+y'y''}{x'^2+y'^2}, $$ and from the equality between the left-hand sides we get $x'(t)\equiv Cy'(t)$, which also gives a straight line, but with arbitrary parametrization.
Added: It is worth to mention that the "energy functional" is a Lagrangian action for a free particle confined to the surface. Therefore, the law of conservation of energy implies that any solution $\gamma(t)$ will move at constant speed (which is of course also possible to see directly). And if one restricts the the class of curves to such solutions, the EL equations indeed take the same form for both functionals.