Dynamic Equilibrium

Orbital Potentials  Kepler orbits involve a moving system. Effective potential reduces to a single variable Second variable is cyclic r V eff r0r0 r  r0r0

Radial Perturbation  A perturbed orbit varies slightly from equilibrium. Perturbed velocityPerturbed velocity Track the difference from the equation of motionTrack the difference from the equation of motion  Apply a Taylor expansion. Keep first orderKeep first order  Small perturbations are stable with same frequency.

Modified Kepler  Kepler orbits can have a perturbed potential. Not small at small r Two equilibrium points Test with second derivative Test with  r r V eff r0r0 rArA stable unstable

Inverted Pendulum  An inverted pendulum may have an oscillating support. Driving frequency  Moment of inertia I  The apparent acceleration of gravity is adjusted by the oscillation. l m

Mathieu’s Equation  Substitute variables to get a standard form. a compares natural frequency to driving frequencya compares natural frequency to driving frequency q is relates to the amplitude of oscillation  0 /  2q is relates to the amplitude of oscillation  0 /  2  is a dimensionless time variable  is a dimensionless time variable

Infinite Series  The Mathieu equation is soluble as an infinite series. Infinite Fourier seriesInfinite Fourier series  Solutions are unstable for real or complex . Builds up exponentiallyBuilds up exponentially  Purely imaginary m has stable motion. Dominant term n = 0Dominant term n = 0 Fundamental frequency  ’Fundamental frequency  ’

Stability Regions  There are an infinite set of stable regions. Mirror behavior in negative q  There is well defined for q = 0. Stable for a > 0 (normal pendulum) Unstable for a < 0 (inverted pendulum) March and Hughes, Quadrupole Storage Mass Spectrometry next