Examples for Free Motion

(a) Polar Coordinates in a Plane

Let us get the equations in polar coordinates for motion in a plane. Here

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

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

and

$\displaystyle T=\frac{m}{2}\left(\dot r^{\,2}+r^{2}\dot\phi^{\,2}\right).$

Then

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

If $R$ is the impressed force resolved along the radius vector,

$\displaystyle \delta_r W = m\left(\ddot r-r\dot\phi^{\,2}\right)\delta r = R\delta r.$

Also

$\displaystyle \frac{\partial T}{\partial\dot\phi}=mr^{2}\dot\phi, \qquad \frac{\partial T}{\partial\phi}=0,$

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

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

In more familiar form,

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

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

(b) Cylindrical Coordinates

In cylindrical coordinates, where

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

$\displaystyle T=\frac{m}{2} \left( \dot r^{\,2}+r^{2}\dot\phi^{\,2}+\dot z^{\,2} \right).$

We have

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

$\displaystyle \frac{\partial T}{\partial\dot\phi}=mr^2\dot\phi,\qquad \frac{\partial T}{\partial\phi}=0,$

$\displaystyle \frac{\partial T}{\partial\dot z}=m\dot z,\qquad \frac{\partial T}{\partial z}=0.$

Hence

$\displaystyle m\left(\ddot r-r\dot\phi^{\,2}\right)\delta r=R\delta r,$

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

$\displaystyle m\ddot z\,\delta z=Z\delta z,$

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

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

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

(c) Spherical Coordinates

In spherical coordinates where

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

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

Then

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

$\displaystyle \frac{\partial T}{\partial r} = mr\left(\dot\theta^{\,2}+\sin^2\theta\,\dot\phi^{\,2}\right),$

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

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

$\displaystyle \frac{\partial T}{\partial\dot\phi} = mr^2\sin^2\theta\,\dot\phi, \qquad \frac{\partial T}{\partial\phi}=0.$

Thus

$\displaystyle \delta_r W = m\left[ \ddot r -r\left(\dot\theta^2+\sin^2\theta\,\dot\phi^2\right) \right]\delta r = R\delta r,$

$\displaystyle \delta_\theta W = m\left[ \frac{d}{dt}(r^2\dot\theta) -r^2\sin\theta\cos\theta\,\dot\phi^2 \right]\delta\theta = \Theta r\,\delta\theta,$

and

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

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

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

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

Source

William Elwood Byerly, An Introduction to the Use of Generalized Coördinates in mechanics and Physics, Ginn and Company, 1916. Chapter I, “Introduction.”

The 1916 source work is in the public domain in the United States.

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