My question is regarding applying averaging theory to a perturbed Hamiltonian. Now, my Hamiltonian is of the form $$H=H_0 + R(q_i,p_i)$$ Where R is the disturbing potential which is a function of the coordinates ($q_i$) and momenta ($p_i$). Now, I would like to average this R over one of the coordinates say, $q_1$. What I do here is: $$<H> = H_0 + <R> = H_0 + \frac{1}{2 \pi} \int_0^{2 \pi} R dq_1$$

Now, this is where my problem begins. My averaged R has the form $$<R> = f_1 + f_2 + f_3 + ...$$ What I observe is that many of the terms in $<R>$ have a division by zero suggesting resonance. My question is, what do I do with these terms when writing hamiltons equations of motion from $<H>$?



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