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Next: Atwood Machines Up: Lagrangian Dynamics Previous: Lagrange's Equation


Motion in a Central Potential

Consider a particle of mass $m$ moving in two dimensions in the central potential $U(r)$. This is clearly a two degree of freedom dynamical system. As described in Section 5.5, the particle's instantaneous position is most conveniently specified in terms of the plane polar coordinates $r$ and $\theta $. These are our two generalized coordinates. According to Equation (223), the square of the particle's velocity can be written
\begin{displaymath}
v^2 = \dot{r}^{\,2} + (r\,\dot{\theta})^2.
\end{displaymath} (614)

Hence, the Lagrangian of the system takes the form
\begin{displaymath}
L = \frac{1}{2}\,m\,(\dot{r}^{\,2} + r^2\,\dot{\theta}^{\,2}) - U(r).
\end{displaymath} (615)

Note that
$\displaystyle \frac{\partial L}{\partial\dot{r}} = m\,\dot{r},$ $\textstyle ~~~~~~$ $\displaystyle \frac{\partial L}{\partial r} = m\,r\,\dot{\theta}^{\,2} - \frac{dU}{dr},$ (616)
$\displaystyle \frac{\partial L}{\partial \dot{\theta}} = m\,r^2\,\dot{\theta},$   $\displaystyle \frac{\partial L}{\partial\theta} = 0.$ (617)

Now, Lagrange's equation (613) yields the equations of motion,
$\displaystyle \frac{d}{dt}\!\left(\frac{\partial L}{\partial\dot{r}}\right)- \frac{\partial L}{\partial r}$ $\textstyle =$ $\displaystyle 0,$ (618)
$\displaystyle \frac{d}{dt}\!\left(\frac{\partial L}{\partial\dot{\theta}}\right)- \frac{\partial L}{\partial \theta}$ $\textstyle =$ $\displaystyle 0.$ (619)

Hence, we obtain
$\displaystyle \frac{d}{dt}\!\left(m\,\dot{r}\right) - m\,r\,\dot{\theta}^{\,2} + \frac{dU}{dr}$ $\textstyle =$ $\displaystyle 0,$ (620)
$\displaystyle \frac{d}{dt}\!\left(m\,r^2\,\dot{\theta}\right) = 0,$     (621)

or
$\displaystyle \ddot{r} - r\,\dot{\theta}^{\,2}$ $\textstyle =$ $\displaystyle - \frac{d V}{dr},$ (622)
$\displaystyle r^2\,\dot{\theta}$ $\textstyle =$ $\displaystyle h,$ (623)

where $V = U/m$, and $h$ is a constant. We recognize Equations (622) and (623) as the equations we derived in Chapter 5 for motion in a central potential. The advantage of the Lagrangian method of deriving these equations is that we avoid having to express the acceleration in terms of the generalized coordinates $r$ and $\theta $.


next up previous
Next: Atwood Machines Up: Lagrangian Dynamics Previous: Lagrange's Equation
Richard Fitzpatrick 2011-03-31