1. Stability

2. Lagrangian Near Equilibium  A 1-dimensional Lagrangian can be expanded near equilibrium. Expand to second order

3. Second Derivative  The Lagrangian simplifies near equilibrium. Constant is arbitraryConstant is arbitrary Definition requires B = 0Definition requires B = 0  The equation of motion follows from the Lagrangian Depends only on D/FDepends only on D/F Rescale time coordinateRescale time coordinate  This gives two forms of an equivalent Lagrangian. stable unstable

4. Matrix Stability  A general set of coordinates gives rise to a matrix form of the Lagrangian. Normal modes for normal coordinates.Normal modes for normal coordinates.  The eigenfrequencies  2 determine stability. If stable, all positiveIf stable, all positive Diagonalization of VDiagonalization of V

5. 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

6. Dynamic Equilibrium  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.

7. 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

8. Lyapunov Stability  A Lyapunov function is defined on some region of a space X including 0. Continuous, real functionContinuous, real function  The derivative with respect to a map f is defined as a dot product.  If V exists such that V*  0, then the point 0 is stable.

9. Lyapunov Example  A 2D map f : R 2  R 2. (from Mathworld)  Define a Lyapunov function.  The derivative is negative so the origin is stable. next