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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Bradbury, T.C.: 1966,Theoretical Mechanics, John Wiley & Sons, New York.
Byrd, P.F. and Friedman, M.D.: 1954,Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, Berlin.
Coppola, V.T.: 1989,Ph.D. thesis, Cornell University.
Deprit, A.: 1969,Celestial Mechanics 1, 12.
Deprit, A. and Rom, A.: 1969,Celestial Mechanics 2, 166.
Henrard, J.: 1970,Celestial Mechanics 3, 107.
Henrard, J. and Wauthier, P.: 1988,Celestial Mechanics 44, 227.
Howland, R.A.: 1988a,Celestial Mechanics 44, 209.
Howland, R.A.: 1988b,Celestial Mechanics 45, 407.
Meyer, K.R.: 1991, in:Computer Aided Proofs in Analysis, pp. 190–210, Springer-Verlag.
Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T.: 1988,Numerical Recipes in C, Cambridge University Press.
Sanders, J.A. and Verhulst, F.: 1985,Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag.
Symbolics: 1988,Macsyma Reference Manual, Symbolics Inc., Burlington, MA.
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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
- Duffing equation
- Hamiltonian systems
- Lie transformation
- non-canonical transformations
- perturbation theory