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====== Math 445 lab 7: differential equations ======

**Problem 1:** In class we developed a linear system of differential equations of the form $dx/dt = f(x) = Ax$ for the plane pendulum, where

<latex>
x = \left(\begin{array}{l} x_1\\ x_2\end{array} \right) = \left(\begin{array}{l} \theta \\ v_{\theta} \end{array} \right)
</latex>

We used the approximation $\sin \theta \approx \theta$ for small $\theta$, i.e. small amplitude oscillations.  A chief result of this derivation was that the frequency of oscillation of the pendulum is given by $\omega = \sqrt{g/l}$, where $g$ is the acceleration due to gravity and $l$ is the length of the pendulum.

However if $\theta$ gets large the approximation $\sin \theta \approx \theta$ is no longer valid. This lab problem is an investigation of how the frequency of oscillation of the pendulum changes when the amplitude of the oscillation is large.

**(a)** Revise the derivation from class to develop a nonlinear system of differential equations $dx/dt= f(x)$ that is valid for large $\theta$. 

**(b)** Set up Matlab code to integrate this system of equations numerically, using Matlab's ''ode45'' function.

**%%(c)%%** Determine the frequency of oscillation of the nonlinear pendulum for 
for the constants $g=9.8 \; m/s^2$ and $l=1.0 \; m$ and a variety of oscillation amplitudes.
The oscillation amplitude is given by the angle $\theta_0$ at which the pendulum is released at time $t=0$, with no initial velocity. Determine the frequency $\omega$ for 
$\theta_0 = 0.05, 0.10, ..., 0.30$ and plot $\omega$ versus $\theta_0$. 

**(d)** For moderate amplitudes, the frequency $\omega$ of the nonlinear pendulum should vary with $\theta_0$ as

<latex>
\omega = \sqrt{g/l} + c \; \theta_0^2
</latex>

for some value of $c$.  Determine what the value of $c$ is from your graph in %%(c)%%.

----

**Problem 2:** For the linear pendulum, we produced in lecture //time series// plots of position $\theta$ and (angular) velocity $d\theta/dt$ as a function of time, and a //phase portrait// of $\theta$ versus $d\theta/dt$.

**With no damping** from air resistance, the equations of motion of the linear pendulum are 

\begin{eqnarray*}
\frac{d}{dt} \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) = \left[ \begin{array}{cc} 0 & 1 \\ -g/l & 0 \end{array} \right] \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right)
\end{eqnarray*}

and the time series and phase portrait (for $g=9.8$ and $l=1$) look like 

{{:gibson:teaching:fall-2014:math445:timeseries_linear_nodamp.png?nolink&400}}  {{:gibson:teaching:fall-2014:math445:phaseportrait_linear_nodamp.png?nolink&400}}

**If we include the damping of air resistance**, the equations of motion of the linear pendulum are 

\begin{eqnarray*}
\frac{d}{dt} \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) = \left[ \begin{array}{cc} 0 & 1 \\ -g/l & -\alpha/m \end{array} \right] \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right)
\end{eqnarray*}

and the time series and phase portrait for (for $g=9.8$, $l=1$, and $\alpha=1$) look like

{{:gibson:teaching:fall-2014:math445:timeseries_linear_damp.png?nolink&400}}  {{:gibson:teaching:fall-2014:math445:phaseportrait_linear_damp.png?nolink&400}}
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