6533b7cffe1ef96bd1259af9

RESEARCH PRODUCT

Time-Independent Canonical Perturbation Theory

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First we consider the perturbation calculation only to first order, limiting ourselves to only one degree of freedom. Furthermore, the system is to be conservative, ∂ H∕∂ t = 0, and periodic in both the unperturbed and perturbed case. In addition to periodicity, we shall require the Hamilton–Jacobi equation to be separable for the unperturbed situation. The unperturbed problem H0(J0) which is described by the action-angle variables J0 and w0 will be assumed to be solved. Thus we have, for the unperturbed frequency: $$\displaystyle{ \nu _{0} = \frac{\partial H_{0}} {\partial J_{0}} }$$ (10.1) and $$\displaystyle{ w_{0} =\nu _{0}t +\beta _{0}\;. }$$ (10.2) Then the new Hamiltonian reads, up to a perturbation term of first order: $$\displaystyle{ H = H_{0}{\bigl (J_{0}\bigr )} +\varepsilon H_{1}{\bigl (w_{0},J_{0}\bigr )}\;, }$$ (10.3) where \(\varepsilon\) is a small parameter.