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

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1. Feb 16, 2017

### MxwllsPersuasns

1. The problem statement, all variables and given/known data

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'

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

3. 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!

2. Feb 23, 2017