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## Perturbation

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**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