Stability

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# Stability - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Stability' - beauregard

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Presentation Transcript

### 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

Matrix Stability
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.

Dynamic Equilibrium
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

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
• 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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