gibson:teaching:spring-2016:math445:lecture:pendulum
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gibson:teaching:spring-2016:math445:lecture:pendulum [2016/04/27 13:23] gibson |
gibson:teaching:spring-2016:math445:lecture:pendulum [2016/04/27 13:24] (current) gibson [Changing a second-order ODE into a system of first-order ODEs] |
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| \end{eqnarray*} | \end{eqnarray*} | ||
| - | For example, to transform the linear undamped pendulum equation $\d^2\theta/dt^2 + g/\ell \theta = 0$, | + | For example, to transform the linear undamped pendulum equation $d^2\theta/dt^2 + (g/\ell) \theta = 0$, |
| let $x_1 = \theta$ and $x_2 = d\theta/dt$. Differentiate those two equations to get | let $x_1 = \theta$ and $x_2 = d\theta/dt$. Differentiate those two equations to get | ||
| - | $d x_1/dt = d\theta/dt = x_2$ and $d x_2/dt = d^\theta/dt^2$. Now note that, according to the linear undamped pendulum equation, $d^2\theta/dt^2 = -g/\ell \theta = -g/\ell x_1$. Putting all this together, we can write | + | $d x_1/dt = d\theta/dt = x_2$ and $d x_2/dt = d^2\theta/dt^2$. Now note that, according to the linear undamped pendulum equation, $d^2\theta/dt^2 = -(g/\ell) \theta = -(g/\ell) x_1$. Putting all this together, we can write |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
gibson/teaching/spring-2016/math445/lecture/pendulum.1461788634.txt.gz · Last modified: 2016/04/27 13:23 by gibson
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