… constant forces  integrate EOM  parabolic trajectories. … linear restoring force  guess EOM solution  SHM … nonlinear restoring forces  ? linear.

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… 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

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)

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

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

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

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!)

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

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!

Derivatives:

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

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).