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[parent] cylindrical coordinate motion example of generalized coordinates (Example)

As an example let us get the equations in cylindrical coordinates

$\displaystyle x=r\cos\phi, \,\,\,\,\,\, y=r\sin\phi, \,\,\,\,\,\, z=z, $

$\displaystyle T=\frac{m}{2} \left[\dot{r}^{2}+r^2\dot{\phi}^{2}+\dot{z}^{2} \right]. $

$\displaystyle \frac{\partial T}{\dot{r}}=m\dot{r}, $

$\displaystyle \frac{T}{\partial r}=m r\dot{\phi}^{2}, $

$\displaystyle \frac{\partial T}{\partial\dot{\phi}}=mr^{2}\dot{\phi}, $

$\displaystyle \frac{\partial T}{\partial \dot{z}}=m\dot{z}. $

$\displaystyle \delta_{r}W=m \left[\ddot{r} - r\dot{\phi}^{2} \right] \delta r=R\delta r, $

$\displaystyle \delta_{\phi}W=m\frac{d}{dt} \left(r^{2}\dot{\phi}\right)\delta\phi=\Phi r\delta\phi, $

$\displaystyle \delta_z W= m \ddot{z} \delta z = Z \delta z; $

or

$\displaystyle m \left[ \frac{d^{2}r}{dt^{2}}-r \left(\frac{d\phi}{dt}\right)^{2}\right]=R,$

$\displaystyle \frac{m}{r}\frac{d}{dt}\left(r^{2}\frac{d\phi}{dt}\right)=\Phi, $

$\displaystyle m\frac{d^{2}z}{dt^{2}}=Z. $



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This is version 2 of cylindrical coordinate motion example of generalized coordinates, born on 2008-07-18, modified 2008-07-21.
Object id is 288, canonical name is CylindricalCoordinateMotionExampleOfGeneralizedCoordinates.
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Physics Classification: 45.20.Jj (Lagrangian and Hamiltonian mechanics)
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