Geodesic Equation: Generalizing for Functions F

[tom.stoer]

The geodesic equation follows from vanishing variation $\delta S = 0$ with

$S[C] = \int_C ds = \int_a^b dt \sqrt{g_{ab}\,\dot{x}^a\,\dot{x}^b}$

In many cases one uses the energy functional with $\delta E = 0$ instead:

$E[C] = \int_a^b dt \, {g_{ab}\,\dot{x}^a\,\dot{x}^b}$

Can this be generalized for other functions f with $\delta F = 0$ and

$F_f[C] = \int_a^b dt \, f\left(\sqrt{g_{ab}\,\dot{x}^a\,\dot{x}^b}\right)$

 

[martinbn]

If the function f is monotone.

 

[tom.stoer]

That was my idea as well, but I don't see how to generalize the proofs used for S and E. They rely partially on the L2 norm, special case of inner product etc.

Wikipedia writes "The minimizing curves of S ... can be obtained by techniques of calculus of variations ... One introduces the energy functional E ... It is then enough to minimize the functional E, owing to the Cauchy–Schwarz inequality ... with equality if and only if |dγ/dt| is constant"