PerturbationPerturbed SystemStationary StatePower SeriesPeriodic VariablesEquating TermsPerturbed ChargePerturbing PotentialNew FrequencyPerturbationPerturbationPerturbed SystemPerturbed SystemSimple system with added Simple system with added effect.effect.•Basic Lagrangian Basic Lagrangian LL00•Perturbing term Perturbing term UUExpress as a perturbed Express as a perturbed Hamiltonian.Hamiltonian.•Formed in the usual wayFormed in the usual wayWrite as a first-order power Write as a first-order power series.series. = 1 for perturbed motion= 1 for perturbed motion),,(),,(),,(0tqqUtqqLtqqLjjjjjppqUqLp0ULL0ULqptpqHjj0),,(),,(),,(),,(0tpqVtpqHtpqH VHH0Stationary StateStationary StateTime-independent systems Time-independent systems can use can use J, wJ, w..•Action-angle variablesAction-angle variables•Unperturbed Unperturbed HH00((JJ00)) only onlyRequire a contact Require a contact transformation for transformation for HH((JJ)) . .•Identity for Identity for  = 1 = 1•Find the actionFind the action),(),(),(0220100JwSJwSJwJwSkk),()(),(000000wJVJHwJH kkkkkwSwSJwSJ0220100Power SeriesPower SeriesThe Hamiltonian can be expressed in The Hamiltonian can be expressed in ..kkkklklkkkwSJVwSJHwSwSJJHVwSJHJHJH0102001010220100)()( )()()()(2210JHJHJHJHPeriodic VariablesPeriodic VariablesAll dynamic variables are All dynamic variables are expressed as periodic expressed as periodic functions of both old and functions of both old and new angle variables.new angle variables.•Differ by a periodic functionDiffer by a periodic function•Unit periodUnit periodTerms are also periodic in Terms are also periodic in old angles.old angles.•Choose to have mean = 0Choose to have mean = 0kkkkJSJSww2210)()(0201kkkkwJSwJSEquating TermsEquating TermsThe mean value can be The mean value can be found for each term in the found for each term in the HamiltonianHamiltonian•Split Split VV into average and into average and oscillating termoscillating term•Can solve for Can solve for SS11, , SS22),()(00101wJVwSJHJHkk),()(01wJVJH kklklkwSJWwSwSJJHJH010101022)(001020010102osckkkkosclklkwSJVwSvwSwSJJHWVV 0),(0010wJWwSvkkPerturbed ChargePerturbed ChargeCharge under two forcesCharge under two forces•Attractive Coulomb forceAttractive Coulomb force•Uniform magnetic fieldUniform magnetic fieldLet the magnetic field be a Let the magnetic field be a perturbation.perturbation.rZeAcepmH2221XYZ rZepmH22021ByAx21BxAy210zAPerturbing PotentialPerturbing PotentialThe perturbing potential can The perturbing potential can be extracted from the be extracted from the Hamiltonian.Hamiltonian.•Approximate Approximate AA as small as smallFind the average value of Find the average value of VV..•Use angular momentum Use angular momentum ll•Or use action variable Or use action variable JJ   2222AmceApmceV ApmceV)(2xyypxpmceBV JmceBlmceBVz42New FrequencyNew FrequencyThe perturbation is first order only.The perturbation is first order only.•Alter the frequency accordingly.Alter the frequency