I posted about this problem a few days ago but it didn't seem to get too much play so I'm going to re-write it and hopefully get some assistance.
Consider the Kinetic Energy Functional...
E($\gamma$) = $\frac 12\int_I ||\gamma^{'}(t)||^{2}dt$
for the curve $\gamma: I \to R^{n}$
i) Show this functional is not Parameterization Invariant
ii) Show that γ is a critical point of E under compactly supported variations $γ_s$ if and only if $\gamma^{''}$ = 0, i.e., γ(t) = $x_0$ + tv is a constant speed straight line (notice, no assumptions on the regularity of the curve is needed).
iii) Find the Minima of E
My thoughts...
i) So for this I was going to basically argue that one could calculate E in a particular parameterization and then reparameterize by arc length and thus since ||$\gamma^{'}||$ = 1 for Arc-Length parameterized curves I would say that you can go from any arbitrary parameterization (then probably show an E that doesn't equal 1 -- i.e., parameterize like x(t) = acos(t) and y(t) = bcos(t)) to an arc length parameterization where E always equals t/2 when I runs from 0 to t.
ii) I just dont know how exactly do differentiate a functional and thus how to find the criitcal points of such a functional
iii) Again I just need assistance with differentiating and any other facets to finding the minima of a functional (if it's not in direct analogy with minima of functions)