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Stability

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Stability

- A 1-dimensional Lagrangian can be expanded near equilibrium.
- Expand to second order

- 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

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

- Kepler orbits involve a moving system.
- Effective potential reduces to a single variable
- Second variable is cyclic

Veff

r0

r

r0

r

q

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.

- 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

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.

- A 2D map f: R2R2.
- (from Mathworld)

- Define a Lyapunov function.
- The derivative is negative so the origin is stable.

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