Simple system with added effect. Basic Lagrangian L0 Perturbing term U Express as a perturbed Hamiltonian. Formed in the usual way Write as a first-order power series. l = 1 for perturbed motion Perturbed System
Time-independent systems can use J, w. Action-angle variables Unperturbed H0(J0) only Require a contact transformation for H(J) . Identity for l = 1 Find the action Stationary State
Power Series • The Hamiltonian can be expressed in l.
All dynamic variables are expressed as periodic functions of both old and new angle variables. Differ by a periodic function Unit period Terms are also periodic in old angles. Choose to have mean = 0 Periodic Variables
The mean value can be found for each term in the Hamiltonian Split V into average and oscillating term Can solve for S1, S2 Equating Terms
Perturbed Charge • Charge under two forces • Attractive Coulomb force • Uniform magnetic field • Let the magnetic field be a perturbation. Z Y X
The perturbing potential can be extracted from the Hamiltonian. Approximate A as small Find the average value of V. Use angular momentum l Or use action variable J Perturbing Potential
New Frequency • The perturbation is first order only. • Alter the frequency accordingly. next