I was confused with two different ways to derive the gradient flow of curve evolution. Given curve $C(p)$, where $p$ is an arbitrary parameter of curve, and there is a relationship between it and arc length parameter, $ds = ||C_p||dp$. Given an energy function $$E(C)=\int_C ds$$
There are two methods to derive gradient descent flow.
(1) If I want to get the gradient descent flow $\nabla_C E$, below is the derivation. $$\dfrac{d}{dt}E(C)=\langle\nabla_C E, C_t\rangle = \int_C \nabla_C E\cdot C_t ds=\dfrac{d}{dt}\int_Cds$$ replace $ds$ with $||C_p||dp$, it becomes $$\dfrac{d}{dt}E(C)=\int_0^1\dfrac{d}{dt}||C_p||dp = \int_0^1\dfrac{C_{pt}\cdot C_p}{||C_p||}dp$$ Since $p$ and $t$ are independent, $C_{pt}$=$C_{tp}$. With relationship of $ds = ||C_p||dp$, $$\dfrac{d}{dt}E(C)=\int_C C_{ts}\cdot C_sds$$ Take integral by parts, $$\dfrac{d}{dt}E(C)=\int_C C_t\cdot (-C_{ss})ds$$ Therefore, $\nabla_C E=-C_{ss}$.
(2) Another way is that we know $\nabla_C E$ equals to Euler-Lagrange criterion. So How about directly computing it from $E(C)=\int_0^1 ||C_p||dp$ ? And what is the difference between these two ways?.
Method (2) seems much simplier.