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300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-23 Temporary Title.md

@@ -50,3 +50,23 @@ These things relate directly to the eigenvectors and eigenvectors of the system.
 ## Eigenvalues and Eigenvectors
 $z = \beta^2 - 4 \omega^2$
 Solving for lambda ($\det[A - I\lambda]$) will lead us towards the same expression for z. This is what's under the square root.
+# Nonlinear Things (Finally!)
+## Nonlinear Pendulum
+### Undamped
+$$ \ddot{\theta} = -\frac{g}{l} \sin(\theta) $$
+$\dot \theta = \zeta = P(\theta, \zeta)$
+$$ {\bf J} = 
+\left[ \matrix{ \frac{\partial P}{\partial \theta} &  \frac{\partial P}{\partial \zeta} \\ \frac{\partial Q}{\partial \theta} & \frac{\partial Q}{\partial \zeta}} \right] = 
+
+\left[ \matrix{ 0 & 1\\ -\omega^2 \sin(\theta) & 0 } \right]  
+ $$
+ What ar the equilibrium points?
+$$\left[ \matrix{ \dot \theta \\ \dot \zeta} \right ] =  \left[ \matrix{ \zeta \\ -\omega^2 \sin(\theta) } \right] $$
+This system is in equilibrium when $\dot \theta = \dot \zeta = 0$, which is when $\zeta = \sin(\theta) = 0$ . 
+For $\bf J$:
+- $\tau = 0$
+- $\Delta = \omega^2 \cos(\theta)$
+Then:
+- $\theta$ is 0, $\Delta = \omega^2 >0$, center, marginally stable
+- $\theta = n \pi$, $\Delta = - \omega^2 <0$, saddle. Unstable
+What does the phase plane look like?