Synchronization of perturbed non-linear Hamiltonians |
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| Authors: | Bruce R Miller Vincent T Coppola |
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| Institution: | (1) National Institute of Standards and Technology, 20899 Gaithersburg, MD, USA;(2) Department of Aerospace Engineering, University of Michigan, 48109-2109 Ann Arbor, MI, USA |
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| Abstract: | 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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| Keywords: | Duffing equation Hamiltonian systems Lie transformation non-canonical transformations perturbation theory synchronization |
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