A curve $\gamma$ is parametrized by the parametric equations $x(t), y(t)$ and $z(t)$ where $t$ is the parametrization parameter (not a normal parametrization).
$x(t), y(t)$ and $z(t)$ are unkown. What we do know is their differentials with respect to the curvilinear abscissa $s$ :
$\frac{dx}{ds} = \frac {B_x}{B}$
$\frac{dy}{ds} = \frac {B_y}{B}$
$\frac{dz}{ds} = \frac {B_z}{B}$
Where : $ds = \sqrt{ dx^2 + dy^2 + dz^2 }$ is the curvilinear abscissa.
and $B_x, B_y, B_z$ are functions of $x(t), y(t)$ and $z(t)$.
and $B = \sqrt{B_x^2 + B_y^2 + B_z^2}$
On other hand, the curvilinear abscissa caracteristic regarding a parametrization is :
$s(t)= \int_{t_0}^{t} \left\| \gamma ^\prime (u) \right\| \, \mathrm{d}u $
I want to solve the differential equation system $\frac{dx}{ds}$, $\frac{dy}{ds}$, $\frac{dz}{ds}$
The problem is that $x(t), y(t)$ and $z(t)$ are functions of the parameter $t$, but the differential equation system is function of the curvilinear abscissa $s$.
So, I used the relation $s(t)= \int_{t_0}^{t} \left\| \gamma ^\prime (u) \right\| \, \mathrm{d}u $, to calculate $s(t)$ :
$\frac {dx}{du} = \frac {dx}{ds}\frac {ds}{du}$
$s(t)= \int_{t_0}^{t} \sqrt {\left(\frac {dx}{du}\right)^2 + \left(\frac {dy}{du}\right)^2 + \left(\frac {dz}{du}\right)^2} \, \mathrm{d}u $
$s(t)= \int_{t_0}^{t} \sqrt {\left(\frac {dx}{ds}\right)^2 + \left(\frac {dy}{ds}\right)^2 + \left(\frac {dz}{ds}\right)^2} \frac {ds}{du} \, \mathrm{d}u $
$s(t)= \int_{t_0}^{t} \sqrt {\left(\frac {B_x}{B}\right)^2 + \left(\frac {B_y}{B}\right)^2 + \left(\frac {B_z}{B}\right)^2} \, \mathrm{d}s $
$s(t)= \int_{t_0}^{t} \sqrt {\frac {B_x^2 + B_y^2 + B_z^2}{B^2}} \, \mathrm{d}s $
$s(t)= \int_{t_0}^{t} 1 \, \mathrm{d}s = [s]_{t_0}^{t} = t - t_0$
So, with $s(t) = t - t_0 \Rightarrow ds = dt$, can I write :
$\frac{dx}{ds} = \frac {dx}{dt}$
$\frac{dy}{ds} = \frac {dy}{dt}$
$\frac{dz}{ds} = \frac {dz}{dt}$
Can I solve the differential equation system $\frac{dx}{dt}$, $\frac{dy}{dt}$, $\frac{dz}{dt}$ instead?
Do I have to replace $x(t)$ by $x(s)$ or something like that? Something is not right what do you think?
My knowledge about the curvilinear abscissa $s$ is limited. Please if you have any idea help!
Thank you all
S.