Show existence of arc-length parameterized period p'....

[MxwllsPersuasns]

Homework Statement

Let γ : R → Rⁿ be a regular (smooth) closed curve with period p. Show that there exist an orientation preserving diffeomorphism ϕ: R → R, a number p' ∈ R such that ϕ(s + p') = ϕ(s) + p and γ' = γ ◦ ϕ is an arclength parametrized closed curve with period p'

Homework Equations

Arc-length Parameterized Curve: Length = ||dγ/dt|| = 1
Orientation preserving: dΦ/dt > 0 for all t

The Attempt at a Solution

So essentially this question is asking for the existence of a number p' such that when we reparameterize by arc-length we have a new (arclength parameterized) period. Is this correct? And if so, how should I go about proving the existence of such a number? Thanks so much in advance folks, I'm looking forward to the lively discussion!

 

[mighty2000]

Yes, your understanding of the problem is correct. To prove the existence of p', we can use the periodicity of γ and the fact that it is a smooth curve.

First, let's consider the case where p is the length of the curve γ. In this case, we can define ϕ(s) = s/p, which is an orientation preserving diffeomorphism from R to R. We can easily verify that ϕ(s + p) = ϕ(s) + p, and that dϕ/ds = 1/p, so ϕ is indeed an arclength parametrization.

Now, if p is not equal to the length of γ, we can use the fact that γ is a smooth curve to construct a new parametrization that has the desired period. Let p' = p/||dγ/dt||, which is a positive number since ||dγ/dt|| = 1 for an arclength parametrized curve. Then, we can define ϕ(s) = ∫_0^s ||dγ/dt|| dt, which is an orientation preserving diffeomorphism from R to R. We can verify that ϕ(s + p') = ϕ(s) + p, and that dϕ/ds = ||dγ/dt||, so ϕ is also an arclength parametrization.

Finally, we can compose ϕ with γ to get the desired arclength parametrized curve with period p'. That is, γ' = γ ◦ ϕ is an arclength parametrized curve with period p'. This completes the proof of the existence of p'.