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Équations d'évolution

Le paramètre $ \alpha$ est indépendant du temps mais il n'en est pas de même pour l'abscisse curviligne s. Avant d'écrire les équations d'évolution pour $ \theta$ et $ \kappa$, nous devons alors évaluer le commutateur $ \left[\vphantom{ \frac{\partial}{\partial s} , \frac{\partial}{\partial t} }\right.$$ {\frac{\partial}{\partial s}}$,$ {\frac{\partial}{\partial t}}$$ \left.\vphantom{ \frac{\partial}{\partial s} , \frac{\partial}{\partial t} }\right]$.

$\displaystyle \left[\vphantom{ \frac{\partial}{\partial s},\frac{\partial}{\partial t} }\right.$$\displaystyle {\frac{\partial}{\partial s}}$,$\displaystyle {\frac{\partial}{\partial t}}$$\displaystyle \left.\vphantom{ \frac{\partial}{\partial s},\frac{\partial}{\partial t} }\right]$ = $\displaystyle {\frac{\partial}{\partial s}}$ $\displaystyle {\frac{\partial}{\partial t}}$ - $\displaystyle {\frac{\partial}{\partial t}}$ $\displaystyle {\frac{\partial}{\partial s}}$  
  = $\displaystyle {\frac{\partial}{\partial s}}$ $\displaystyle {\frac{\partial}{\partial t}}$ - $\displaystyle {\frac{\partial}{\partial t}}$$\displaystyle \left[\vphantom{ \frac{1}{g_f} \, \frac{\partial}{\partial \alpha} }\right.$$\displaystyle {\frac{1}{g_f}}$ $\displaystyle {\frac{\partial}{\partial \alpha}}$$\displaystyle \left.\vphantom{ \frac{1}{g_f} \, \frac{\partial}{\partial \alpha} }\right]$  
  = $\displaystyle {\frac{\partial}{\partial s}}$ $\displaystyle {\frac{\partial}{\partial t}}$ - $\displaystyle {\frac{1}{g_f}}$ $\displaystyle {\frac{\partial}{\partial t}}$ $\displaystyle {\frac{\partial}{\partial \alpha}}$ + $\displaystyle {\frac{1}{g_f^2}}$ $\displaystyle {\frac{\partial g_f}{\partial t}}$ $\displaystyle {\frac{\partial}{\partial \alpha}}$  
  = $\displaystyle {\frac{\partial}{\partial s}}$ $\displaystyle {\frac{\partial}{\partial t}}$ - $\displaystyle {\frac{1}{g}}$ $\displaystyle {\frac{\partial}{\partial \alpha}}$ $\displaystyle {\frac{\partial}{\partial t}}$ + $\displaystyle {\frac{1}{g_f}}$ $\displaystyle {\frac{\partial g_f}{\partial t}}$ $\displaystyle {\frac{1}{g_f}}$ $\displaystyle {\frac{\partial}{\partial \alpha}}$  
  = $\displaystyle {\frac{\partial}{\partial s}}$ $\displaystyle {\frac{\partial}{\partial t}}$ - $\displaystyle {\frac{\partial}{\partial s}}$ $\displaystyle {\frac{\partial}{\partial t}}$ + $\displaystyle {\frac{1}{g_f}}$ $\displaystyle {\frac{\partial g_f}{\partial t}}$ $\displaystyle {\frac{\partial}{\partial s}}$  
  = $\displaystyle {\frac{1}{g_f}}$ $\displaystyle {\frac{\partial g_f}{\partial t}}$ $\displaystyle {\frac{\partial}{\partial s}}$  . (E.11)

Évaluons maintenant la dérivée temporelle de la métrique gf. On a, par définition de gf :  gf2 = | f'|2 = $ \langle$$ {\frac{\partial \vec{r}}{\partial \alpha}}$,$ {\frac{\partial \vec{r}}{\partial \alpha}}$$ \rangle$ ; alors :
$\displaystyle {\frac{\partial g_f}{\partial t}}$ = $\displaystyle {\frac{1}{g_f}}$ $\displaystyle {\frac{\partial \vec{r}}{\partial \alpha}}$ . $\displaystyle {\frac{\partial}{\partial t}}$$\displaystyle \left(\vphantom{ \frac{\partial \vec{r}}{\partial \alpha} }\right.$$\displaystyle {\frac{\partial \vec{r}}{\partial \alpha}}$$\displaystyle \left.\vphantom{ \frac{\partial \vec{r}}{\partial \alpha} }\right)$  
  = $\displaystyle {\frac{1}{g_f}}$ $\displaystyle \left(\vphantom{ \frac{\partial \vec{r}}{\partial s} \, \frac{\partial s}{\partial \alpha} }\right.$$\displaystyle {\frac{\partial \vec{r}}{\partial s}}$ $\displaystyle {\frac{\partial s}{\partial \alpha}}$$\displaystyle \left.\vphantom{ \frac{\partial \vec{r}}{\partial s} \, \frac{\partial s}{\partial \alpha} }\right)$ . $\displaystyle \left(\vphantom{ \frac{\partial}{\partial \alpha} \, \frac{\partial \vec{r}}{\partial t} }\right.$$\displaystyle {\frac{\partial}{\partial \alpha}}$ $\displaystyle {\frac{\partial \vec{r}}{\partial t}}$$\displaystyle \left.\vphantom{ \frac{\partial}{\partial \alpha} \, \frac{\partial \vec{r}}{\partial t} }\right)$  
  = $\displaystyle {\frac{1}{g_f}}$ $\displaystyle \left(\vphantom{ \vec{\tau} \, g_f }\right.$$\displaystyle \vec{\tau}\,$ gf$\displaystyle \left.\vphantom{ \vec{\tau} \, g_f }\right)$ . $\displaystyle \left(\vphantom{ g_f \, \frac{\partial}{\partial s} \, \vec{v} }\right.$gf $\displaystyle {\frac{\partial}{\partial s}}$ $\displaystyle \vec{v}\,$$\displaystyle \left.\vphantom{ g_f \, \frac{\partial}{\partial s} \, \vec{v} }\right)$  
  = gf $\displaystyle \left(\vphantom{ \vec{\tau} \cdot \frac{\partial \vec{v}}{\partial s} }\right.$$\displaystyle \vec{\tau}\,$ . $\displaystyle {\frac{\partial \vec{v}}{\partial s}}$$\displaystyle \left.\vphantom{ \vec{\tau} \cdot \frac{\partial \vec{v}}{\partial s} }\right)$  
  = gf $\displaystyle \left(\vphantom{ \frac{\partial}{\partial s} \left[ \vec{\tau} \c... ...\vec{v} \right] - \vec{v} \cdot \frac{\partial \vec{\tau}}{\partial s} }\right.$$\displaystyle {\frac{\partial}{\partial s}}$$\displaystyle \left[\vphantom{ \vec{\tau} \cdot \vec{v} }\right.$$\displaystyle \vec{\tau}\,$ . $\displaystyle \vec{v}\,$$\displaystyle \left.\vphantom{ \vec{\tau} \cdot \vec{v} }\right]$ - $\displaystyle \vec{v}\,$ . $\displaystyle {\frac{\partial \vec{\tau}}{\partial s}}$$\displaystyle \left.\vphantom{ \frac{\partial}{\partial s} \left[ \vec{\tau} \c... ...\vec{v} \right] - \vec{v} \cdot \frac{\partial \vec{\tau}}{\partial s} }\right)$  ; (E.12)

soit :

$\displaystyle {\frac{1}{g_f}}$$\displaystyle {\frac{\partial g_f}{\partial t}}$ = $\displaystyle {\frac{\partial v_{\tau}}{\partial s}}$ + $\displaystyle \kappa$ vn  . (E.13)

Il s'en suit que :

$\displaystyle \left[\vphantom{ \frac{\partial}{\partial s},\frac{\partial}{\partial t} }\right.$$\displaystyle {\frac{\partial}{\partial s}}$,$\displaystyle {\frac{\partial}{\partial t}}$$\displaystyle \left.\vphantom{ \frac{\partial}{\partial s},\frac{\partial}{\partial t} }\right]$ = $\displaystyle \left(\vphantom{ \frac{\partial v_{\tau}}{\partial s} + \kappa \, v_n }\right.$$\displaystyle {\frac{\partial v_{\tau}}{\partial s}}$ + $\displaystyle \kappa$ vn$\displaystyle \left.\vphantom{ \frac{\partial v_{\tau}}{\partial s} + \kappa \, v_n }\right)$ $\displaystyle {\frac{\partial}{\partial s}}$  . (E.14)


L'équation d'évolution de $ \theta$ est obtenue en appliquant le commutateur précédent sur le vecteur position $ \vec{r}\,$ :

$\displaystyle \left[\vphantom{ \frac{\partial}{\partial s},\frac{\partial}{\partial t} }\right.$$\displaystyle {\frac{\partial}{\partial s}}$,$\displaystyle {\frac{\partial}{\partial t}}$$\displaystyle \left.\vphantom{ \frac{\partial}{\partial s},\frac{\partial}{\partial t} }\right]$$\displaystyle \vec{r}\,$ = $\displaystyle {\frac{\partial \vec{v}}{\partial s}}$ - $\displaystyle {\frac{\partial \vec{\tau}}{\partial t}}$  
  = $\displaystyle \left(\vphantom{ \frac{\partial v_{\tau}}{\partial s} + \kappa \, v_n }\right.$$\displaystyle {\frac{\partial v_{\tau}}{\partial s}}$ + $\displaystyle \kappa$ vn$\displaystyle \left.\vphantom{ \frac{\partial v_{\tau}}{\partial s} + \kappa \, v_n }\right)$ $\displaystyle \vec{\tau}\,$  . (E.15)

On prend alors le produit scalaire de cette expression avec $ \vec{n}\,$ :

$\displaystyle \vec{n}\,$ . $\displaystyle {\frac{\partial \vec{\tau}}{\partial t}}$ = $\displaystyle \vec{n}\,$ . $\displaystyle {\frac{\partial \vec{v}}{\partial \vec{\tau}}}$  ; (E.16)

soit :

$\displaystyle \vec{n}\,$ . $\displaystyle {\frac{\partial \vec{\tau}}{\partial \theta}}$ $\displaystyle {\frac{\partial \theta}{\partial t}}$ = $\displaystyle {\frac{\partial}{\partial s}}$$\displaystyle \left[\vphantom{ \vec{v} \cdot \vec{n} }\right.$$\displaystyle \vec{v}\,$ . $\displaystyle \vec{n}\,$$\displaystyle \left.\vphantom{ \vec{v} \cdot \vec{n} }\right]$ - $\displaystyle \vec{v}\,$ . $\displaystyle {\frac{\partial \vec{n}}{\partial s}}$  . (E.17)

Et, en utilisant le fait que $ {\frac{\partial \vec{n}}{\partial s}}$ = $ {\frac{\partial \vec{n}}{\partial \theta}}$ $ {\frac{\partial \theta}{\partial s}}$ = - $ \vec{\tau}\,$ $ \kappa$, on obtient l'équation d'évolution désirée pour $ \theta$ :

$\displaystyle {\frac{\partial \theta}{\partial t}}$ = $\displaystyle \kappa$ v$\scriptstyle \tau$ - $\displaystyle {\frac{\partial v_n}{\partial s}}$  . (E.18)

De la même manière, en appliquant le commutateur [[*]] à $ \theta$, on obtient l'équation d'évolution de la courbure :

$\displaystyle {\frac{\partial \kappa}{\partial t}}$ = v$\scriptstyle \tau$ $\displaystyle {\frac{\partial \kappa}{\partial s}}$ - $\displaystyle {\frac{\partial^2 v_n}{\partial s^2}}$ - $\displaystyle \kappa^{2}_{}$ vn  . (E.19)

À partir de la définition de l'abscisse curviligne : $ \bar{s}$($ \alpha$) = $ \int_{\alpha=0}^{\alpha}$gf d$ \alpha$, on obtient :

$\displaystyle {\frac{\partial {\bar s}}{\partial t}}$ = $\displaystyle \int_{0}^{\alpha}$$\displaystyle {\frac{\partial g_f}{\partial t}}$ d$\displaystyle \alpha$  
  = $\displaystyle \int_{0}^{\alpha}$$\displaystyle \left(\vphantom{ \frac{\partial v_{\tau}}{\partial s} + \kappa \, v_n }\right.$$\displaystyle {\frac{\partial v_{\tau}}{\partial s}}$ + $\displaystyle \kappa$ vn$\displaystyle \left.\vphantom{ \frac{\partial v_{\tau}}{\partial s} + \kappa \, v_n }\right)$ gf d$\displaystyle \alpha$  
  = $\displaystyle \int_{0}^{\alpha}$$\displaystyle \left(\vphantom{ \frac{\partial v_{\tau}}{\partial s} + \kappa \, v_n }\right.$$\displaystyle {\frac{\partial v_{\tau}}{\partial s}}$ + $\displaystyle \kappa$ vn$\displaystyle \left.\vphantom{ \frac{\partial v_{\tau}}{\partial s} + \kappa \, v_n }\right)$ds  ; (E.20)

soit :

$\displaystyle {\frac{\partial {\bar s}}{\partial t}}$ = v$\scriptstyle \tau$($\displaystyle \bar{s}$($\displaystyle \alpha$)) - v$\scriptstyle \tau$($\displaystyle \bar{s}$(0)) + $\displaystyle \int_{{\bar s}(0)}^{{\bar s}(\alpha)}$$\displaystyle \kappa$ vnds  . (E.21)

Et pour $ \alpha$ = 1 (s(1) = L ; longueur de la courbe $ \cal {C}$) :

$\displaystyle {\frac{\partial L}{\partial t}}$ = v$\scriptstyle \tau$(L) - v$\scriptstyle \tau$($\displaystyle \bar{s}$(0)) + $\displaystyle \int_{{\bar s}(0)}^{L}$$\displaystyle \kappa$ vnds  . (E.22)

next up previous contents
suivant: Reparamétrisation dynamique et « monter: Représentation « intrinsèque » précédent: Coordonnées « intrinsèques »   Table des matières
fred 2001-07-02