Consider the energy functional $$ E(\gamma) = \int_I ||\gamma^{'}||^{2}dt$$ on regular curves $\eta$ : I $\rightarrow$ S$^{2}$
i) Show that if γ$_s$ : I → S $^2$ is any variation of γ = γ$_0$ by curves on S$^{2}$ then its variational vector field V = γ' satisfies < V, γ >= 0.
i) So for this part we basically show that if we have a variation, $\gamma_s$, which is just $\gamma(t + \eta*s)$ (with s=0 yields $\gamma$) then the field is orthogonal to the curve? Not really sure how to represent V such that I can show when it's dotted with $\gamma$ it's equal to zero. Or is it some property about the system?
ii) Let V : I → R$^3$ be smooth with < V, γ >= 0. Show that there is a variation γ$_s$ of γ by curves on S$^2$ . Moreover, if V is compactly supported so will be γs.
ii) This is essentially the converse of the previous statement right? So that if the inner product of (essentially) $\gamma$ and $\gamma^{'}$ is equal to zero then we much have a variation $\gamma_s$ on $\gamma$. How do we go about this?
iii) Characterize the critical points of E under compactly supported variations γ$_s$ by curves on S$^2$
iii) So for this part we employ the Euler-Lagrange equations to find the critical points then we employ some tactic which is analogous to the 2nd derivative test in order to classify the critical points correct? And if so can anyone describe the process? I think it's something like showing the 2nd variation is always positive or negative but I don't know what or how to find the 2nd variation.