Stability

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Stability

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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 arbitrary • Definition requires B = 0 • The equation of motion follows from the Lagrangian • Depends only on D/F • Rescale time coordinate • This gives two forms of an equivalent Lagrangian. stable unstable

4. A general set of coordinates gives rise to a matrix form of the Lagrangian. Normal modes for normal coordinates. The eigenfrequencies w2 determine stability. If stable, all positive Diagonalization of V Matrix Stability

5. Orbital Potentials • Kepler orbits involve a moving system. • Effective potential reduces to a single variable • Second variable is cyclic Veff r0 r r0 r q

6. A perturbed orbit varies slightly from equilibrium. Perturbed velocity Track the difference from the equation of motion Apply a Taylor expansion. Keep first order Small perturbations are stable with same frequency. Dynamic Equilibrium

7. Modified Kepler • Kepler orbits can have a perturbed potential. • Not small at small r • Two equilibrium points • Test with second derivative • Test with dr Veff r0 r rA stable unstable

8. A Lyapunov function is defined on some region of a space X including 0. Continuous, 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. Lyapunov Stability

9. Lyapunov Example • A 2D map f: R2R2. • (from Mathworld) • Define a Lyapunov function. • The derivative is negative so the origin is stable. next