1. … constant forces  integrate EOM  parabolic trajectories. … linear restoring force  guess EOM solution  SHM … nonlinear restoring forces  ? linear spring x F nonlinear spring? x F Real Oscillators

2. The spring of air : P, V m A P atm +x 0 use Ideal Gas Law: P V=NRT chamber volume: V =Ax Stable Equilibrium at x eq = NRT / (mg + AP atm ) 0 0 EOM WTF! (whoa there, fella)

3. Taylor Series Expansions: Turns a function into a polynomial near x = a Example:

4. Expand around x = -3: 0 th order1 st order2 nd order

5. Expand around x = 2: 0 th order 1 st order 2 nd order

6. Expand N RT/x around x eq : Is it safe to linearize it? Better check a unitless ratio. How about: (Yes, excellent choice Dr. Hafner!)

7. Displacement 5% of x eq : 0.05.0025 …. SHM with Perhaps you would prefer…...

8. Simple Pendulum: Length: L Mass: m  m g c o s  T mg cos  sin  mg mg Stable Equilibrium: Displace by   mg cos  -x EOM: Expand it!

9. Derivatives:

10. Now express as a unitless ratio of the dependent variable and some parameter of the system: SHM with Displacement 5% of length: 0.05 0.0000625 …

11. The world is not linear. However, one can use a Taylor expansion to linearize an EOM by assuming only small perturbations around a point of stable equilibrium (which may not be the origin).