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
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
A perturbed orbit varies slightly from equilibrium.
Track the difference from the equation of motion
Apply a Taylor expansion.
Keep first order
Small perturbations are stable with same frequency.
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.