An example in Hamiltonian dynamics

Henryk Żoła̧dek; Henryk Żoła̧dek

doi:10.3934/cam.2024020

Communications in Analysis and Mechanics

2024, Volume 16, Issue 2: 431-447. doi: 10.3934/cam.2024020

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Research article

An example in Hamiltonian dynamics

Henryk Żoła̧dek ^,

Institute of Mathematics, University of Warsaw, UL. Banacha 2, 02-097 Warsaw, Poland

Received: 11 November 2023 Revised: 03 April 2024 Accepted: 08 May 2024 Published: 11 June 2024
Primary 05C38, 15A15; Secondary 05A15, 15A18

We present an example of a three-degrees-of-freedom polynomial Hamilton function with a critical point characterized by indefinite quadratic part with a Morse index 2. This function generates a Hamiltonian system wherein all eigenvalues equal $ \pm \mathrm{i} $, but it lacks small-amplitude periodic solutions with a period $ \approx 2\pi. $
- Hamiltonian system,
- periodic solutions,
- Lyapunov theorem
Citation: Henryk Żoła̧dek. An example in Hamiltonian dynamics[J]. Communications in Analysis and Mechanics, 2024, 16(2): 431-447. doi: 10.3934/cam.2024020

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Abstract

We present an example of a three-degrees-of-freedom polynomial Hamilton function with a critical point characterized by indefinite quadratic part with a Morse index 2. This function generates a Hamiltonian system wherein all eigenvalues equal $ \pm \mathrm{i} $, but it lacks small-amplitude periodic solutions with a period $ \approx 2\pi. $

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