Abstract
The word "average" and its variations became popular in the sixties and implicitly carried the idea that "averaging" methods lead to "average" Hamiltonians. However, given the Hamiltonian H = H0(J) + ∈R(θ, J), (∈ < < 1), the problem of transforming it into a new Hamiltonian H* (J*) (dependent only on the new actions J*), through a canonical transformation given by zero-average trigonometrical series has no general solution at orders higher than the first.
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Ferraz-Mello, S. Do Average Hamiltonians Exist?. Celestial Mechanics and Dynamical Astronomy 73, 243–248 (1999). https://doi.org/10.1023/A:1008363517421
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DOI: https://doi.org/10.1023/A:1008363517421