Hamiltonian perturbation and resonance

Write H as an integrable part depending only on actions plus a small angle-dependent correction. The unperturbed frequencies are derivatives of the integrable Hamiltonian. Expand a periodic correction in Fourier modes of the angles. A nonresonant oscillatory mode can often be removed to first order by a near-identity canonical transformation, leaving its angle average as the leading change in the effective Hamiltonian. The transformation must remain small throughout the domain where it is used.

A numerical coefficient called ε is not by itself a sufficient smallness test. For a quartic correction to a harmonic oscillator, its energy relative to the quadratic energy grows with amplitude. The same ε can be negligible for one orbit and dominant for another. Always identify the dimensionless combination involving the action or amplitude that controls the approximation. An expansion that is excellent for energy can also accumulate a large phase error after many cycles if its frequency correction is ignored.

Use normalized mass and harmonic angular frequency equal to one, with a positive weak quartic term εq⁴/4. The unperturbed substitution q equal to the square root of twice J times sine of θ converts that term into εJ² times sine to the fourth power. Its angle average is three eighths of εJ². Differentiating the averaged Hamiltonian with respect to J predicts a frequency that increases with action for a positive quartic correction.

This result describes the leading anharmonic frequency shift, not an exact sinusoidal solution. The true orbit contains waveform corrections as well. In the averaged Hamiltonian, J denotes the canonical action after the near-identity averaging and agrees with harmonic action at zeroth order. Substituting zeroth-order harmonic action into the leading frequency correction changes only higher-order terms, but the linear energy term requires the first-order action conversion for first-order energy accuracy. Using initial amplitude A in the frequency shift therefore permits J approximately A squared over two. The expansion requires εA squared to be small.

A Fourier term with angle combination k dotted into θ oscillates at k dotted into the unperturbed frequency vector. Removing it perturbatively involves division by that combination. Near zero, the proposed transformation is no longer small even if ε is small. With periodic external forcing, the relevant combinations also include integer multiples of the drive frequency. The correct response is to retain the slowly varying resonant angle and derive a reduced resonant Hamiltonian rather than force a divergent nonresonant formula.

Resonance can exchange actions between modes, create islands, or change long-term stability, depending on the Hamiltonian. It is not identical to constructive interference of two prescribed waves. Equal-frequency waves of fixed amplitude add coherently but do not by themselves model energy supplied by a drive or transferred between dynamical oscillators. A lesson diagram should expose that distinction: phase alignment illustrates a slowly varying relative phase, while actual resonance dynamics needs coupled evolution equations and an energy-accounting model.