# Stability - PowerPoint PPT Presentation

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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

### 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

• 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

### 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

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.

### 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

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 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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