Coordonnées

Cartésienne

Position

$\overrightarrow{p}=a\cdot \overrightarrow{u_x}+b\cdot \overrightarrow{u_y}+c\cdot \overrightarrow{u_z}$

Vitesse

$\overrightarrow{v}=\frac{d\overrightarrow{p}}{dt}=\frac{da}{dt}\overrightarrow{u_x}+\frac{db}{dt}\overrightarrow{u_y}+\frac{dc}{dt}\overrightarrow{u_z}$

Accélération

$\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}=\frac{d\overrightarrow{p}}{d{t}^2}=\frac{da}{d{t}^2}\overrightarrow{u_x}+\frac{db}{d{t}^2}\overrightarrow{u_y}+\frac{dc}{d{t}^2}\overrightarrow{u_z}$

Polaire

Position

$\overrightarrow{p}=l\overrightarrow{u_r}$

Vitesse

$\overrightarrow{v}=\frac{d\overrightarrow{p}}{dt}=\frac{d(l\overrightarrow{u_r)}}{dt}=\frac{dl}{dt}\overrightarrow{u_r}+r\frac{d\overrightarrow{u_r}}{dt}=\frac{dl}{dt}\overrightarrow{u_r}+r\frac{d\theta }{dt}\overrightarrow{u_{\theta }}$

Accélération

$\overrightarrow{a}=\frac{d\overrightarrow{p}}{d{t}^2}=\frac{d\overrightarrow{v}}{dt}=(\frac{dl}{d{t}^2}-l\frac{d\theta }{dt})\overrightarrow{u_r}+(r\frac{d\theta }{d{t}^2}+2\frac{dl}{dt}\frac{d\theta }{dt})\overrightarrow{u_{\theta }}$

Astuces

$\frac{d\overrightarrow{u_r}}{dt}=\frac{d\theta }{dt}\overrightarrow{u_{\theta }}$ et $\frac{d\overrightarrow{u_{\theta }}}{dt}=-\frac{d\theta }{dt}\overrightarrow{u_r}$