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

As an example let us get the equations in polar coordinates for motion in a plane

Here

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

$\displaystyle \dot{x}^{2}+\dot{y}^{2}=v^{2}=\dot{r}^{2}+r^{2}\dot{\phi}^{2} $

and

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

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

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

$\displaystyle \delta_{r}W=m[\ddot{r}-r\dot{\phi}^{2}]\delta r=R\delta r $
if $R$ is the impressed force resolved along the radius vector.

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

$\displaystyle \frac{\partial T}{\partial \phi}=0. $

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

if $\Phi$ is the impressed force resolved perpendicular to the radius vector.

In a more familiar form

$\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. $



"polar coordinate motion example of generalized coordinates" is owned by bloftin.

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Cross-references: radius vector, motion

This is version 2 of polar coordinate motion example of generalized coordinates, born on 2008-07-18, modified 2008-07-21.
Object id is 287, canonical name is PolarCoordinateMotionExampleOfGeneralizedCoordinates.
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Physics Classification: 45.20.Jj (Lagrangian and Hamiltonian mechanics)
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