Abstract
In the previous chapter we have studied integrable hyperhamiltonian systems. We want now to study hyperhamiltonian perturbations of integrable systems. By this, we mean in general hyperhamiltonian perturbations of hyperhamiltonian integrable systems; however, as we have seen, standard Hamiltonian systems are a special case of hyperhamiltonian systems, and the same holds for integrable systems.
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Notes
- 1.
The notion of “simpler” is ambiguous, and actually what in the spirit of Poincaré theory is a simpler form can—and in general will—contain infinitely many terms, albeit one started from a very simple nonlinear expression.
- 2.
The theory was developed by Poincaré for non-resonant systems—in which case one can always arrive at least formally at a linear system—and by his pupil Dulac in the resonant case.
- 3.
Albeit one should more precisely say it is the action by conjugation of the one parameter group generated by H.
- 4.
By Darboux theorem [10, 12, 30, 82] this can always be reduced locally to the standard form; in particular, it can be brought to be constant in convenient (Darboux) coordinates.
- 5.
As in the general case, here \(H_m\) should be meant as the expression taken by \(H_m\) after the previous normalization steps have been performed. That is, at each step after computing the \(\widetilde{H}_k\) we will rename them as \(H_k\).
- 6.
Quite similarly to the Poincaré-Dulac case, here Birkhoff worked out the case of nonresonant (Hamiltonian) systems, and Gustavsson extended the result to the resonant case. Note that here by “non resonant” it is meant there are no resonances beside the trivial ones (corresponding to eigenvalues coming in pairs of complex conjugate ones, see above).
- 7.
Note also that with such a notation we can discuss at the same time also the problem of Lie transforms acting on a Dirac vector field, simply extending the range of the summation index \(\alpha \) relative to the f; see below for this extension.
- 8.
The reason for the seemingly odd notation \(\widetilde{\mathcal{H}}_{(0)}^\beta \) (note here the subscript “(0)” does not refer to homogeneity degree) will be clear in the following.
- 9.
In general non constant; see also Remark 6.6 below.
- 10.
We are not aware of any attempt to identify an equivalent of the Bargmann metric in the quaternionic case; in particular, we ignore if there is any real obstacle to its identification. Solving this problem would be a necessary step towards the full development of a quaternionic perturbation theory.
- 11.
Recall that these are the terms obtained after the previous steps in the normalization procedure.
- 12.
We stress that here we are not (yet) putting the F in standard Hamiltonian form, nor considering the case of quadratic \(\mathcal{F}^\alpha \).
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Gaeta, G., Rodríguez, M.A. (2017). Hyperhamiltonian Dynamics and Perturbation Theory. In: Lectures on Hyperhamiltonian Dynamics and Physical Applications. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-54358-1_6
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DOI: https://doi.org/10.1007/978-3-319-54358-1_6
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