Problem
I want to calculate the time it takes for a particle living in a spherical spiral to fall under de force of gravity down to the bottom. So far I've sketched the procedure but when I tried to solve the equations, they've seemed too complicated to solve analytically so I'm stucked. Let me introduce my attempt:
First I've calculated the expression of the curve in parametric form $\alpha(u)=(x(u),y(u),z(u))$: $$x(u)=\frac{R \cos(u)}{\sqrt{1+\kappa^2u^2}}\\ y(u)=\frac{R \sin(u)}{\sqrt{1+\kappa^2u^2}}\\ z(u)=\frac{-R\kappa u }{\sqrt{1+\kappa^2u^2}}$$ where $R$ is the radius of the sphere and $\kappa$ is some constant which encodes how much the spiral curls. I've used $u$ as the parameter of the curve. Now I want to solve the Euler-Lagrange equations, and I wanted to re parameterize the curve with respect to the length parameter($s(u)=\int_{u_0}^u |\alpha'(t)|dt$) to make use of the fact that our problem reduces to 1-D. The Lagrangian would take the form $\mathcal{L}=\frac{m}{2}\dot{s}^2-mgz(u)=\frac{m}{2}\dot{s}^2-mgz(s)$, where in the last step we simply reverse $s(u)\rightarrow u(s)$. Therefore the Euler-Lagrange equations would be: $$\frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{s}}-\frac{\partial \mathcal{L}}{\partial s}=0=\ddot{s}-g\frac{\partial z(s)}{\partial s}\rightarrow s(t)$$ And finally substituting $s(t)$ in $\alpha(s)$, I would have the equation of motion.
Where I'm stucked
Since the equations are so complicated I tried using python to solve them, but I can't even solve for $s(u)$, and even if I could, I don't think I could find the inverse $u(s)$.
What I'm asking
It´s been a long time since I used Python or Euler-Lagrange equations, so maybe I did something wrong in the procedure or even in the coding.
Is my attempt correct?
If it is, is there a way I can solve the problem numerically?
Should I tackle the problem in any other way?
Thanks