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Synchronization of perturbed non-linear Hamiltonians

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We propose a new method based on Lie transformations for simplifying perturbed Hamiltonians in one degree of freedom. The method is most useful when the unperturbed part has solutions in non-elementary functions. A non-canonical Lie transformation is used to eliminate terms from the perturbation that are not of the same form as those in the main part. The system is thus transformed into a modified version of the principal part. In conjunction with a time transformation, the procedure synchronizes the motions of the perturbed system onto those of the unperturbed part.

A specific algorithm is given for systems whose principal part consists of a kinetic energy plus an arbitrary potential which is polynomial in the coordinate; the perturbation applied to the principal part is a polynomial in the coordinate and possibly the momentum.

We demonstrate the strategy by applying it in detail to a perturbed Duffing system. Our procedure allow us to avoid treating the system as a perturbed harmonic oscillator. In contrast to a canonical simplification, our method involves only polynomial manipulations in two variables. Only after the change of time do we start manipulating elliptic functions in an exhaustive discussion of the flows.

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Miller, B.R., Coppola, V.T. Synchronization of perturbed non-linear Hamiltonians. Celestial Mech Dyn Astr 55, 331–350 (1993). https://doi.org/10.1007/BF00692993

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Key words

  • Duffing equation
  • Hamiltonian systems
  • Lie transformation
  • non-canonical transformations
  • perturbation theory
  • synchronization