Stability
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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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Stability

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Stability

Stability


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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