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Hamiltonian Formulation

Reformulating the system dynamics using a Hamiltonian H(q,p,t)=H0+ϵH1H(q, p, t) = H_0 + \epsilon H_1. The perturbation ϵH1\epsilon H_1 is treated as a small deviation from the integrable, unperturbed Hamiltonian H0H_0.
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The statement of the theorem

Define the total Hamiltonian HH for the system as H=H0+ϵH1H = H_0 + \epsilon H_1, where H0H_0 is the unperturbed, integrable Hamiltonian and ϵH1\epsilon H_1 is the perturbation. The dynamics are governed by Hamilton's canonical equations:\ndqidt=∂H∂pi=∂H0∂pi+ϵ∂H1∂pianddpidt=−∂H∂qi=−∂H0∂qi−ϵ∂H1∂qi\frac{d q_i}{dt} = \frac{\partial H}{\partial p_i} = \frac{\partial H_0}{\partial p_i} + \epsilon \frac{\partial H_1}{\partial p_i} \quad \text{and} \quad \frac{d p_i}{dt} = -\frac{\partial H}{\partial q_i} = -\frac{\partial H_0}{\partial q_i} - \epsilon \frac{\partial H_1}{\partial q_i} \nFor the Kepler problem, H0=−μ/(2a)H_0 = -\mu / (2a). The perturbation ϵH1\epsilon H_1 is typically expanded in terms of the small parameter ϵ\epsilon and solved iteratively.