Normal form for lower dimensional elliptic tori in Hamiltonian systems

Chiara Caracciolo; Chiara Caracciolo

doi:10.3934/mine.2022051

Mathematics in Engineering

2022, Volume 4, Issue 6: 1-40. doi: 10.3934/mine.2022051

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Normal form for lower dimensional elliptic tori in Hamiltonian systems

Chiara Caracciolo ^,

Department of Mathematics "F. Enriques", Milano University, via Saldini 50, 20133 - Milano, Italy

Received: 04 March 2021 Accepted: 16 October 2021 Published: 23 November 2021

We give a proof of the convergence of an algorithm for the construction of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. The existence of such invariant tori is proved by leading the Hamiltonian to a suitable normal form. In particular, we adapt the procedure described in a previous work by Giorgilli and co-workers, where the construction was made so as to be used in the context of the planetary problem. We extend the proof of the convergence to the cases in which the two sets of frequencies, describing the motion along the torus and the transverse oscillations, have the same order of magnitude.
- lower dimensional invariant tori,
- KAM theory,
- normal form methods,
- perturbation theory for Hamiltonian systems
Citation: Chiara Caracciolo. Normal form for lower dimensional elliptic tori in Hamiltonian systems[J]. Mathematics in Engineering, 2022, 4(6): 1-40. doi: 10.3934/mine.2022051

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Abstract

We give a proof of the convergence of an algorithm for the construction of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. The existence of such invariant tori is proved by leading the Hamiltonian to a suitable normal form. In particular, we adapt the procedure described in a previous work by Giorgilli and co-workers, where the construction was made so as to be used in the context of the planetary problem. We extend the proof of the convergence to the cases in which the two sets of frequencies, describing the motion along the torus and the transverse oscillations, have the same order of magnitude.

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