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

@@ -76,4 +76,16 @@ How do we know which way the saddle points will kick us? The eigenvalues. The ce
 ### Damped
 What about when we have damping?
 ![[Pasted image 20240923133900.png]]
-Now we have stable spirals!
+Now we have stable spirals!
+$$ {\bf J} = 
+\left[ \matrix{ \frac{\partial P}{\partial \theta} &  \frac{\partial P}{\partial \zeta} \\ \frac{\partial Q}{\partial \theta} & \frac{\partial Q}{\partial \zeta}} \right] = 
+\left[ \matrix{ 0 & 1\\ -\omega^2 \sin(\theta) & -\beta } \right]  
+ $$
+  What are the equilibrium points?
+$$\left[ \matrix{ \dot \theta \\ \dot \zeta} \right ] =  \left[ \matrix{ \zeta \\ -\beta\zeta-\omega^2 \sin(\theta) } \right] $$
+For $\bf J$:
+- $\tau = -\beta$
+- $\Delta = \pm\omega^2$
+Then:
+- $\theta$ is 0, $\Delta = \omega^2 >0$, spiral. Stable
+- $\theta = n \pi$, $\Delta = - \omega^2 <0$, saddle. Unstable