Perturbation

Perturbed System  Simple system with added effect. Basic Lagrangian L 0Basic Lagrangian L 0 Perturbing term UPerturbing term U  Express as a perturbed Hamiltonian. Formed in the usual wayFormed in the usual way  Write as a first-order power series.  = 1 for perturbed motion

Stationary State  Time-independent systems can use J, w. Action-angle variablesAction-angle variables Unperturbed H 0 (J 0 ) onlyUnperturbed H 0 (J 0 ) only  Require a contact transformation for H(J). Identity for = 1Identity for = 1 Find the actionFind the action

Power Series  The Hamiltonian can be expressed in.

Periodic Variables  All dynamic variables are expressed as periodic functions of both old and new angle variables. Differ by a periodic functionDiffer by a periodic function Unit periodUnit period  Terms are also periodic in old angles. Choose to have mean = 0Choose to have mean = 0

Equating Terms  The mean value can be found for each term in the Hamiltonian Split V into average and oscillating termSplit V into average and oscillating term Can solve for S 1, S 2Can solve for S 1, S 2

Perturbed Charge  Charge under two forces Attractive Coulomb force Uniform magnetic field  Let the magnetic field be a perturbation. X Y Z

Perturbing Potential  The perturbing potential can be extracted from the Hamiltonian. Approximate A as smallApproximate A as small  Find the average value of V. Use angular momentum lUse angular momentum l Or use action variable JOr use action variable J

New Frequency  The perturbation is first order only. Alter the frequency accordingly.Alter the frequency accordingly. next