Consider the kinetic energy functional:
$$ E(\alpha) = \frac 12\int_I ||\alpha^{'}(t)||^{2}dt$$
for the curve $\alpha$
(i) Show $E$ is not Parameterization invariant
(ii) Show that $\alpha$ is a critical point of $E$ under compactly supported variations $\alpha_s$ iff $\alpha^{''} = 0$. i.e., $\alpha(t) = x_0 + tv$ is a constant speed straight line (notice no assumptions on the regularity of the curve is needed.)
(iii) What are the minima of $E$?
For..
(i) I just am unsure as to whether to perform this in a specific case (i.e., explicitly state the parameterization and reparameterization I'm doing (say just $t$ then $-t$)) or in the general case (i.e., use properties about the energy functional in order to prove it in the abstract) and if in the general case how exactly I should start that.
(ii) I know I'm supposed to use the calculus of variations here but I'm really unfamiliar with it especially the terms "compactly supported variations $\alpha_s$" I am unfamiliar with the term but I imagine the s denotes particular "variations" on the curve but if someone could help with me whats being varied and how to represent that mathematically I would really appreciate it.
(iii) On this I just imagine you find $E^{'}(t) = 0$ then perform whatever test you like to verify its a minima?