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Stability. Lagrangian Near Equilibium. A 1-dimensional Lagrangian can be expanded near equilibrium. Expand to second order. Second Derivative. The Lagrangian simplifies near equilibrium. Constant is arbitrary Definition requires B = 0 The equation of motion follows from the Lagrangian

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Presentation Transcript
lagrangian near equilibium
Lagrangian Near Equilibium
  • A 1-dimensional Lagrangian can be expanded near equilibrium.
    • Expand to second order
second derivative
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

matrix stability
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
orbital potentials
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

dynamic equilibrium
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
modified kepler
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

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

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