- #1
Demon117
- 165
- 1
Homework Statement
Let [itex]\alpha : I =[a,b]→R^{2}[/itex] be a rgular curve parametrized by arc length [itex]s[/itex]. Let [itex]f:I→R[/itex] be a differentiable function with [itex]f(a)=f(b)=0[/itex]. For small values of [itex]\epsilon[/itex]
[itex]\alpha_{\epsilon}:=\alpha (s) + \epsilon f(s) N(s)[/itex]
defines a one parameter family of regular curves having endpoints [itex]\alpha(a)[/itex] and [itex]\alpha(b)[/itex]. The length of these curves defines a function
[itex]L(\epsilon):=L_{a}^{b}(\alpha_{\epsilon})=\int_{a}^{b} \left|\frac{d}{ds}\alpha_{\epsilon}\right| ds[/itex]
By differentiating under the integral sign, show that
[itex]L'(0)=-\int_{a}^{b}k_{\alpha}f(s)ds[/itex]
where [itex]k_{\alpha}[/itex] is the curvature of the function [itex]\alpha[/itex].
The Attempt at a Solution
My attempt at the solution is somewhat limited to purely differentiating the term inside the integrand. I merely want to check that I am approaching this the right way. What I have so far is the following. Since the term [itex]k_{\alpha}f(s)[/itex] is continuous we can differentiate under the integral sign. We have therefore,
[itex]\frac{d}{ds}\alpha_{\epsilon}=\alpha'(s)+\epsilon [f'(s)N(s)+f(s)N'(s)][/itex]
[itex]=\alpha'(s)+\epsilon [f'(s)N(s)+f(s)(-k_{\alpha}(s)T(s))][/itex]
Where T(s) is the tangent to the curve [itex]\alpha[/itex]. Furthermore, this gives
[itex]\alpha'(s)+\epsilon [f'(s)(\frac{1}{k_{\alpha}(s)}T'(s)+f(s)(-k_{\alpha}(s)\alpha'(s))][/itex]
[itex]=\alpha'(s)+\epsilon [f'(s)(\frac{1}{k_{\alpha}(s)}\alpha''(s)+f(s)(-k_{\alpha}(s)\alpha'(s))][/itex]
Now this is where it gets tricky. I want to show
[itex]L'(0)=-\int_{a}^{b}k_{\alpha}f(s)ds[/itex]
which implies that I look at the derivative of the function at [itex]\epsilon=0[/itex]. But how would one differentiate [itex]\left|\frac{d}{ds}\alpha_{\epsilon}\right|[/itex] with the "modulus" present?
Either I am over complicating this or I have simply forgotten this sort of exercise from previous work. . . Suggestions?