I posted this question here the other day but it didn't receive much attention. Perhaps because of the length of the post. So I shall take a minimalist approach here and link to the other question for all the extra details/hints.
Let γ$_s$ : I → R$^n$, s ∈ (−a, a), a > 0, be a variation with compact support K ⊂ I$^{'}$ of a regular curve γ = γ$_0$. Show that there exists some 0 < b ≤ a such that γs is a regular curve for all s ∈ (−b, b). Thus, we may assume w.l.o.g. that any variation of a regular curve consists of regular curves
So I really am unfamiliar with the calculus of variations but I know that: The variation $\gamma_s$ would be something like $\gamma(t + s\eta)$ with s ranging from (-a to a) with s = 0 corresponding to $\gamma$ and so presumably we must find an s (in the interval +/- b) such that the variation is regular?
I really need help on this so anyone who can offer some I would truly appreciate it.
Also the link to more info is in my previous post: Any Variation of a regular curve consists of Regular Curves