Beschleunigung

Die Beschleunigung ist in kartesischen Koordinaten

$$\vec{a}=\frac{d^{2}\vec{r}}{dt^{2}}= \begin{pmatrix}\frac{d^{2}x}... ...t{z} \end{pmatrix} =\ddot{x}\vec{e}_{x}+\ddot{y}\vec{e}_{y}+\ddot{z}\vec{e}_{z}$$ (H..974)

Wir verwenden die Beziehungen

$$x = r\sin(\theta)\cos(\phi)$$ (H..975)

$$y = r\sin(\theta)\sin(\phi)$$ (H..976)

$$z = r\cos(\theta)$$ (H..977)

und leiten sie zweimal ab. Wir erhalten aus

$$\dot{x} = \dot{r}\sin(\theta)\cos(\phi)+r\cos(\theta)\cos(\phi)\dot{\theta }-r\sin(\theta)\sin(\phi)\dot{\phi}$$

$$\dot{y} = \dot{r}\sin(\theta)\sin(\phi)+r\cos(\theta)\sin(\phi)\dot{\theta }+r\sin(\theta)\cos(\phi)\dot{\phi}$$

$$\dot{z} = \dot{r}\cos(\theta)-r\sin(\theta)\dot{\theta}$$

die Gleichungen

$$\ddot{x}= \ddot{r}\sin(\theta)\cos(\phi)+\dot{r}\cos(\theta)\cos(\phi )\dot{\theta}-\dot{r}\sin(\theta)\sin(\phi)\dot{\phi}$$ (H..978)

$$+\dot{r}\cos(\theta)\cos(\phi)\dot{\theta}-r\sin(\theta)\cos(\phi... ...s(\theta)\sin(\phi)\dot{\phi}\dot{\theta}+r\cos (\theta)\cos(\phi)\ddot{\theta}$$

$$-\dot{r}\sin(\theta)\sin(\phi)\dot{\phi}-r\cos(\theta)\sin(\phi)\... ...heta}-r\sin(\theta)\cos(\phi)\dot{\phi}^{2}-r\sin(\theta)\sin (\phi)\ddot{\phi}$$

$$= \ddot{r}\sin(\theta)\cos(\phi)$$

$$+\dot{r}\dot{\theta}\left[ \cos(\theta)\cos(\phi)+\cos(\theta)\cos (\phi)\right]$$

$$+\dot{r}\dot{\phi}\left[ -\sin(\theta)\sin(\phi)-\sin(\theta)\sin (\phi)\right]$$

$$+r\dot{\theta}^{2}\left[ -\sin(\theta)\cos(\phi)\right]$$

$$+r\dot{\phi}\dot{\theta}\left[ -\cos(\theta)\sin(\phi)-\cos(\theta )\sin(\phi)\right]$$

$$+r\ddot{\theta}\left[ \cos(\theta)\cos(\phi)\right]$$

$$+r\dot{\phi}^{2}\left[ -\sin(\theta)\cos(\phi)\right]$$

$$+r\ddot{\phi}\left[ -\sin(\theta)\sin(\phi)\right]$$

$$= \ddot{r}\sin(\theta)\cos(\phi)+2\dot{r}\dot{\theta}\cos(\theta)\c... ...dot{r}\dot{\phi}\sin(\theta)\sin(\phi)-r\dot{\theta}^{2}\sin (\theta)\cos(\phi)$$

$$-2r\dot{\phi}\dot{\theta}\cos(\theta)\sin(\phi)+r\ddot{\theta}\co... ...\phi)-r\dot{\phi}^{2}\sin(\theta)\cos(\phi)-r\ddot{\phi} \sin(\theta)\sin(\phi)$$

und

$$\ddot{y} = \ddot{r}\sin(\theta)\sin(\phi)+\dot{r}\cos(\theta)\sin(\phi )\dot{\theta}+\dot{r}\sin(\theta)\cos(\phi)\dot{\phi}$$ (H..979)

$$+\dot{r}\cos(\theta)\sin(\phi)\dot{\theta}-r\sin(\theta)\sin(\phi... ...s(\theta)\cos(\phi)\dot{\theta}\dot{\phi}+r\cos (\theta)\sin(\phi)\ddot{\theta}$$

$$+\dot{r}\sin(\theta)\cos(\phi)\dot{\phi}+r\cos(\theta)\cos(\phi)\... ...\phi}-r\sin(\theta)\sin(\phi)\dot{\phi}^{2}+r\sin(\theta)\cos(\phi )\ddot{\phi}$$

$$= \ddot{r}\sin(\theta)\sin(\phi)$$

$$+\dot{r}\dot{\theta}\left[ \cos(\theta)\sin(\phi)+\cos(\theta)\sin (\phi)\right]$$

$$+\dot{r}\dot{\phi}\left[ \sin(\theta)\cos(\phi)+\sin(\theta)\cos (\phi)\right]$$

$$-r\dot{\theta}^{2}\sin(\theta)\sin(\phi)$$

$$+r\dot{\theta}\dot{\phi}\left[ \cos(\theta)\cos(\phi)+\cos(\theta)\cos (\phi)\right]$$

$$+r\cos(\theta)\sin(\phi)\ddot{\theta}$$

$$-r\dot{\phi}^{2}\sin(\theta)\sin(\phi)$$

$$+r\sin(\theta)\cos(\phi)\ddot{\phi}$$

$$= \ddot{r}\sin(\theta)\sin(\phi)+2\dot{r}\dot{\theta}\cos(\theta)\s... ...dot{r}\dot{\phi}\sin(\theta)\cos(\phi)-r\dot{\theta}^{2}\sin (\theta)\sin(\phi)$$

$$+2r\dot{\theta}\dot{\phi}\cos(\theta)\cos(\phi)+r\ddot{\theta}\co... ...\phi)-r\dot{\phi}^{2}\sin(\theta)\sin(\phi)+r\ddot{\phi} \sin(\theta)\cos(\phi)$$

sowie

$$\ddot{z} = \ddot{r}\cos(\theta)-\dot{r}\sin(\theta)\dot{\theta }$$ (H..980)

$$-\dot{r}\sin(\theta)\dot{\theta}-r\cos(\theta)\dot{\theta}^{2}-r\sin (\theta)\ddot{\theta}$$

$$= \ddot{r}\cos(\theta)-2\dot{r}\sin(\theta)\dot{\theta}-r\cos(\theta )\dot{\theta}-r\sin(\theta)\ddot{\theta}$$

Wir setzen in die Gleichung H.24 die Gleichungen H.8, H.9, H.10, H.28, H.29 und H.30 ein und ordnen nach $\vec{e}_{r}$, $\vec{e}_{\theta}$ und $\vec{e}_{\phi}$.

$$\vec{a}= \ddot{x}\vec{e}_{x}+\ddot{y}\vec{e}_{y}+\ddot{z}\vec{e}_{z}$$ (H..981)

$$= \ddot{x}\left[ \sin(\theta)\cos(\phi)\vec{e}_{r}+\cos(\theta)\cos (\phi)\vec{e}_{\theta}-\sin(\phi)\vec{e}_{\phi}\right]$$

$$+\ddot{y}\left[ \sin(\theta)\sin(\phi)\vec{e}_{r}+\cos(\theta)\sin (\phi)\vec{e}_{\theta}+\cos(\phi)\vec{e}_{\phi}\right]$$

$$+\ddot{z}\left[ \cos(\theta)\vec{e}_{r}-\sin(\theta)\vec{e}_{\theta }\right]$$

$$= \left[ \ddot{x}\sin(\theta)\cos(\phi)+\ddot{y}\sin(\theta)\sin(\phi )+\ddot{z}\cos(\theta)\right] \vec{e}_{r}$$

$$+\left[ \ddot{x}\cos(\theta)\cos(\phi)+\ddot{y}\cos(\theta)\sin(\phi )-\ddot{z}\sin(\theta)\right] \vec{e}_{\theta}$$

$$+\left[ -\ddot{x}\sin(\phi)+\ddot{y}\cos(\phi)\right] \vec{e}_{\phi }$$

Der Übersichtlichkeit halber berechnen wir nun die drei Komponenten $\vec{e}_{r}$, $\vec{e}_{\theta}$ und $\vec{e}_{\phi}$ getrennt. Wir beginnen mit $\vec{e}_{r}$.

$$a_{r} = \ddot{x}\sin(\theta)\cos(\phi)+\ddot{y}\sin(\theta)\sin(\phi )+\ddot{z}\cos(\theta)$$ (H..982)

$$= \left[ \ddot{r}\sin(\theta)\cos(\phi)+2\dot{r}\dot{\theta}\cos(\t... ...dot{\phi}\sin(\theta)\sin(\phi)-r\dot{\theta}^{2} \sin(\theta)\cos(\phi)\right.$$

$$-2r\dot{\phi}\dot{\theta}\cos(\theta)\sin(\phi)+r\ddot{\theta}\cos (\theta)\cos(\phi)$$ (H..983)

$$\left. -r\dot{\phi}^{2}\sin(\theta)\cos(\phi)-r\ddot{\phi}\sin(\theta )\sin(\phi)\right] \sin(\theta)\cos(\phi)$$

$$+\left[ \ddot{r}\sin(\theta)\sin(\phi)+2\dot{r}\dot{\theta}\cos(\theta )\sin(\phi)+2\dot{r}\dot{\phi}\sin(\theta)\cos(\phi)\right.$$

$$-r\dot{\theta}^{2}\sin(\theta)\sin(\phi)+2r\dot{\theta}\dot{\phi}\cos (\theta)\cos(\phi)$$

$$\left. +r\cos(\theta)\sin(\phi)\ddot{\theta}-r\dot{\phi}^{2}\sin ... ...ta)\sin(\phi)+r\ddot{\phi}\sin(\theta)\cos(\phi)\right] \sin(\theta )\sin(\phi)$$

$$+\left[ \ddot{r}\cos(\theta)-2\dot{r}\dot{\theta}\sin(\theta)-r\dot{\theta }^{2}\cos(\theta)-r\ddot{\theta}\sin(\theta)\right] \cos(\theta)$$

$$= \ddot{r}\left[ \sin(\theta)\cos(\phi)\sin(\theta)\cos(\phi)+\sin (\theta)\sin(\phi)\sin(\theta)\sin(\phi)+\cos(\theta)\cos(\theta)\right]$$

$$+2\dot{r}\dot{\theta}\left[ \cos(\theta)\cos(\phi)\sin(\theta)\co... ...)+\cos(\theta)\sin(\phi)\sin(\theta)\sin(\phi)-\sin(\theta)\cos (\theta)\right]$$

$$+2\dot{r}\dot{\phi}\left[ -\sin(\theta)\sin(\phi)\sin(\theta)\cos (\phi)+\sin(\theta)\cos(\phi)\sin(\theta)\sin(\phi)\right]$$

$$+r\dot{\theta}^{2}\left[ -\sin(\theta)\cos(\phi)\sin(\theta)\cos(... ...)-\sin(\theta)\sin(\phi)\sin(\theta)\sin(\phi)-\cos(\theta)\cos(\theta )\right]$$

$$+2r\dot{\theta}\dot{\phi}\left[ -\cos(\theta)\sin(\phi)\sin(\theta )\cos(\phi)+\cos(\theta)\cos(\phi)\sin(\theta)\sin(\phi)\right]$$

$$+r\ddot{\theta}\left[ \cos(\theta)\cos(\phi)\sin(\theta)\cos(\phi )+\cos(\theta)\sin(\phi)\sin(\theta)\sin(\phi)-\sin(\theta)\cos(\theta )\right]$$

$$+r\dot{\phi}^{2}\left[ -\sin(\theta)\cos(\phi)\sin(\theta)\cos(\phi )-\sin(\theta)\sin(\phi)\sin(\theta)\sin(\phi)\right]$$

$$+r\ddot{\phi}\left[ -\sin(\theta)\sin(\phi)\sin(\theta)\cos(\phi )+\sin(\theta)\cos(\phi)\sin(\theta)\sin(\phi)\right]$$

$$= \ddot{r}\left[ \sin^{2}(\theta)\cos^{2}(\phi)+\sin^{2}(\theta)\sin ^{2}(\phi)+\cos^{2}(\theta)\right]$$

$$+2\dot{r}\dot{\theta}\left[ \cos(\theta)\sin(\theta)\cos^{2}(\phi )+\cos(\theta)\sin(\theta)\sin^{2}(\phi)-\sin(\theta)\cos(\theta)\right]$$

$$+2\dot{r}\dot{\phi}\left[ -\sin^{2}(\theta)\sin(\phi)\cos(\phi)+\sin ^{2}(\theta)\cos(\phi)\sin(\phi)\right]$$

$$+r\dot{\theta}^{2}\left[ -\sin^{2}(\theta)\cos^{2}(\phi)-\sin^{2} (\theta)\sin^{2}(\phi)-\cos^{2}(\theta)\right]$$

$$+r\ddot{\theta}\left[ \cos(\theta)\sin(\theta)\cos^{2}(\phi)+\cos (\theta)\sin(\theta)\sin^{2}(\phi)-\sin(\theta)\cos(\theta)\right]$$

$$+r\dot{\phi}^{2}\left[ -\sin^{2}(\theta)\cos^{2}(\phi)-\sin^{2}(\theta )\sin^{2}(\phi)\right]$$

$$+r\ddot{\phi}\left[ -\sin^{2}(\theta)\sin(\phi)\cos(\phi)+\sin^{2} (\theta)\cos(\phi)\sin(\phi)\right]$$

$$= \ddot{r}\left[ \sin^{2}(\theta)+\cos^{2}(\theta)\right]$$

$$+2\dot{r}\dot{\theta}\left[ \cos(\theta)\sin(\theta)-\sin(\theta )\cos(\theta)\right]$$

$$+r\dot{\theta}^{2}\left[ -\sin^{2}(\theta)-\cos^{2}(\theta)\right]$$

$$+r\ddot{\theta}\left[ \cos(\theta)\sin(\theta)-\sin(\theta)\cos (\theta)\right]$$

$$+r\dot{\phi}^{2}\left[ -\sin^{2}(\theta)\right]$$

$$= \ddot{r}-r\dot{\theta}^{2}-r\sin^{2}(\theta)\dot{\phi}^{2}$$

und

$$a_{\theta} = \ddot{x}\cos(\theta)\cos(\phi)+\ddot{y}\cos(\theta)\sin (\phi)-\ddot{z}\sin(\theta)$$ (H..984)

$$= \left[ \ddot{r}\sin(\theta)\cos(\phi)+2\dot{r}\dot{\theta}\cos(\t... ...dot{\phi}\sin(\theta)\sin(\phi)-r\dot{\theta}^{2} \sin(\theta)\cos(\phi)\right.$$

$$\left. -2r\dot{\phi}\dot{\theta}\cos(\theta)\sin(\phi)+r\ddot{\th... ...ta)\cos(\phi)-r\ddot{\phi} \sin(\theta)\sin(\phi)\right] \cos(\theta)\cos(\phi)$$

$$+\left[ \ddot{r}\sin(\theta)\sin(\phi)+2\dot{r}\dot{\theta}\cos(\... ...dot{\phi}\sin(\theta)\cos(\phi)-r\dot{\theta}^{2} \sin(\theta)\sin(\phi)\right.$$

$$\left. +2r\dot{\theta}\dot{\phi}\cos(\theta)\cos(\phi)+r\ddot{\th... ...ta)\sin(\phi)+r\ddot{\phi} \sin(\theta)\cos(\phi)\right] \cos(\theta)\sin(\phi)$$

$$-\left[ \ddot{r}\cos(\theta)-2\dot{r}\dot{\theta}\sin(\theta)-r\dot{\theta }\cos(\theta)-r\ddot{\theta}\sin(\theta)\right] \sin(\theta)$$

$$= \ddot{r}\left[ \sin(\theta)\cos(\phi)\cos(\theta)\cos(\phi)+\sin (\theta)\sin(\phi)\cos(\theta)\sin(\phi)-\cos(\theta)\sin(\theta)\right]$$

$$+2\dot{r}\dot{\theta}\left[ \cos(\theta)\cos(\phi)\cos(\theta)\co... ...)+\cos(\theta)\sin(\phi)\cos(\theta)\sin(\phi)+\sin(\theta)\sin (\theta)\right]$$

$$+2\dot{r}\dot{\phi}\left[ -\sin(\theta)\sin(\phi)\cos(\theta)\cos (\phi)+\sin(\theta)\cos(\phi)\cos(\theta)\sin(\phi)\right]$$

$$+r\dot{\theta}^{2}\left[ -\sin(\theta)\cos(\phi)\cos(\theta)\cos(... ...)-\sin(\theta)\sin(\phi)\cos(\theta)\sin(\phi)+\cos(\theta)\sin(\theta )\right]$$

$$+2r\dot{\phi}\dot{\theta}\left[ -\cos(\theta)\sin(\phi)\cos(\theta )\cos(\phi)+\cos(\theta)\cos(\phi)\cos(\theta)\sin(\phi)\right]$$

$$+r\ddot{\theta}\left[ \cos(\theta)\cos(\phi)\cos(\theta)\cos(\phi )+\cos(\theta)\sin(\phi)\cos(\theta)\sin(\phi)+\sin(\theta)\sin(\theta )\right]$$

$$+r\dot{\phi}^{2}\left[ -\sin(\theta)\cos(\phi)\cos(\theta)\cos(\phi )-\sin(\theta)\sin(\phi)\cos(\theta)\sin(\phi)\right]$$

$$+r\ddot{\phi}\left[ -\sin(\theta)\sin(\phi)\cos(\theta)\cos(\phi )+\sin(\theta)\cos(\phi)\cos(\theta)\sin(\phi)\right]$$

$$= \ddot{r}\left[ \sin(\theta)\cos(\theta)\cos^{2}(\phi)+\sin(\theta )\cos(\theta)\sin^{2}(\phi)-\cos(\theta)\sin(\theta)\right]$$

$$+2\dot{r}\dot{\theta}\left[ \cos^{2}(\theta)\cos^{2}(\phi)+\cos^{2} (\theta)\sin^{2}(\phi)+\sin^{2}(\theta)\right]$$

$$+r\dot{\theta}^{2}\left[ -\sin(\theta)\cos(\theta)\cos^{2}(\phi )-\sin(\theta)\cos(\theta)\sin^{2}(\phi)+\cos(\theta)\sin(\theta)\right]$$

$$+r\ddot{\theta}\left[ \cos^{2}(\theta)\cos^{2}(\phi)+\cos^{2}(\theta )\sin^{2}(\phi)+\sin^{2}(\theta)\right]$$

$$+r\dot{\phi}^{2}\left[ -\sin(\theta)\cos(\theta)\cos^{2}(\phi)-\sin (\theta)\cos(\theta)\sin^{2}(\phi)\right]$$

$$= \ddot{r}\left[ \sin(\theta)\cos(\theta)-\cos(\theta)\sin(\theta)\right]$$

$$+2\dot{r}\dot{\theta}\left[ \cos^{2}(\theta)+\sin^{2}(\theta)\right]$$

$$+r\dot{\theta}^{2}\left[ -\sin(\theta)\cos(\theta)+\cos(\theta)\sin (\theta)\right]$$

$$+r\ddot{\theta}\left[ \cos^{2}(\theta)+\sin^{2}(\theta)\right]$$

$$-r\dot{\phi}^{2}\left[ \sin(\theta)\cos(\theta)\right]$$

$$= 2\dot{r}\dot{\theta}+r\ddot{\theta}-r\sin(\theta)\cos(\theta)\dot{\phi} ^{2}$$

und schliesslich

$$a_{\phi} = -\ddot{x}\sin(\phi)+\ddot{y}\cos(\phi)$$ (H..985)

$$= -\left[ \ddot{r}\sin(\theta)\cos(\phi)+2\dot{r}\dot{\theta}\cos (... ...dot{\phi}\sin(\theta)\sin(\phi)-r\dot{\theta} ^{2}\sin(\theta)\cos(\phi)\right.$$

$$\left. -2r\dot{\phi}\dot{\theta}\cos(\theta)\sin(\phi)+r\ddot{\th... ...{2}\sin(\theta)\cos(\phi)-r\ddot{\phi} \sin(\theta)\sin(\phi)\right] \sin(\phi)$$

$$+\left[ \ddot{r}\sin(\theta)\sin(\phi)+2\dot{r}\dot{\theta}\cos(\... ...dot{\phi}\sin(\theta)\cos(\phi)-r\dot{\theta}^{2} \sin(\theta)\sin(\phi)\right.$$

$$\left. +2r\dot{\theta}\dot{\phi}\cos(\theta)\cos(\phi)+r\ddot{\th... ...{2}\sin(\theta)\sin(\phi)+r\ddot{\phi} \sin(\theta)\cos(\phi)\right] \cos(\phi)$$

$$= \ddot{r}\left[ -\sin(\theta)\cos(\phi)\sin(\phi)+\sin(\theta)\sin (\phi)\cos(\phi)\right]$$

$$+2\dot{r}\dot{\theta}\left[ -\cos(\theta)\cos(\phi)\sin(\phi)+\cos (\theta)\sin(\phi)\cos(\phi)\right]$$

$$+2\dot{r}\dot{\phi}\left[ \sin(\theta)\sin(\phi)\sin(\phi)+\sin(\theta )\cos(\phi)\cos(\phi)\right]$$

$$+r\dot{\theta}^{2}\left[ \sin(\theta)\cos(\phi)\sin(\phi)-\sin(\theta )\sin(\phi)\cos(\phi)\right]$$

$$+2r\dot{\phi}\dot{\theta}\left[ \cos(\theta)\sin(\phi)\sin(\phi )+\cos(\theta)\cos(\phi)\cos(\phi)\right]$$

$$+r\ddot{\theta}\left[ -\cos(\theta)\cos(\phi)\sin(\phi)+\cos(\theta )\sin(\phi)\cos(\phi)\right]$$

$$+r\dot{\phi}^{2}\left[ \sin(\theta)\cos(\phi)\sin(\phi)-\sin(\theta )\sin(\phi)\cos(\phi)\right]$$

$$+r\ddot{\phi}\left[ \sin(\theta)\sin(\phi)\sin(\phi)+\sin(\theta)\cos (\phi)\cos(\phi)\right]$$

$$= +2\dot{r}\dot{\phi}\left[ \sin(\theta)\sin^{2}(\phi)+\sin(\theta)\cos ^{2}(\phi)\right]$$

$$+2r\dot{\phi}\dot{\theta}\left[ \cos(\theta)\sin^{2}(\phi)+\cos(\theta )\cos^{2}(\phi)\right]$$

$$+r\ddot{\phi}\left[ \sin(\theta)\sin^{2}(\phi)+\sin(\theta)\cos^{2} (\phi)\right]$$

$$= +2\dot{r}\dot{\phi}\sin(\theta)+2r\dot{\phi}\dot{\theta}\cos(\theta )+r\ddot{\phi}\sin(\theta)$$

$$= \left[ r\ddot{\phi}+2\dot{r}\dot{\phi}\right] \sin(\theta)+2r\dot{\phi }\dot{\theta}\cos(\theta)$$

Zusammenfassend haben wir

$$\vec{a}= a_{r}\vec{e}_{r}+a_{\theta}\vec{e}_{\theta}+a_{\phi}\vec{e}_{\phi }$$ (H..986)

$$= \left[ \ddot{r}-r\dot{\theta}^{2}-r\sin^{2}(\theta)\dot{\phi}^{2}\right] \vec{e}_{r}$$

$$+\left[ 2\dot{r}\dot{\theta}+r\ddot{\theta}-r\sin(\theta )\cos(\theta)\dot{\phi}^{2}\right] \vec{e}_{\theta}$$

$$+\left[ \left( r\ddot{\phi}+2\dot{r}\dot{\phi}\right) \sin(\theta)+2r\dot{\phi}\dot{\theta }\cos(\theta)\right] \vec{e}_{\phi}$$

Interpretation

Wir teilen die Beschleunigung in drei Komponenten auf

$$\vec{a}= \vec{a}_p+\vec{a}_z+\vec{a}_c$$ (H..987)

Dies ist in der angegebenen Reihenfolge die Parallelbeschleunigung, die den Betrag der Geschwindigkeit erhöht, die Zentripetalbeschleunigung und die Coriolis-Beschleunigung.

Im Einzelnen haben wir

$$\vec{a}_p = \ddot{r}\vec{e}_r+r\ddot{\theta}\vec{e}_\theta+r\sin(\theta)\ddot{\phi}\vec{e}_\phi$$ (H..988)

$$\vec{a}_z = -r\left[\dot{\theta}^{2}+\sin^{2}(\theta)\dot{\phi}^{2}\right]\vec{e}_r -r\sin(\theta )\cos(\theta)\dot{\phi}^{2}\vec{e}_\theta$$ (H..989)

$$\vec{a}_c = 2\dot{r}\dot{\theta}\vec{e}_\theta+2\left[\dot{r}\sin(\theta)+r\dot{\theta }\cos(\theta)\right]\dot{\phi}\vec{e}_\phi$$ (H..990)