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Mazrooei-Sebdani R, Toghraei O. Properties of Orbits Space of Indefinite Hamiltonian Resonances. mmr 2025; 11 (1) :19-34
URL: http://mmr.khu.ac.ir/article-1-3373-en.html
URL: http://mmr.khu.ac.ir/article-1-3373-en.html
1- Isfahan University of Technology , mazrooei@iut.ac.ir
2- Isfahan University of Technology
2- Isfahan University of Technology
Abstract: (87 Views)
ABSTRACT
The study of Hamiltonian systems around the elliptic equilibrium points, which is a non-generic subject in the study of Hamiltonian systems, has received attention in recent decades. Such systems appear in many applied models, including molecular physics, galactic dynamics, and mechanics.
Consider a Hamiltonian ofn 
degrees of freedom, whose quadratic part is as follows,
The vector ω ω 1 , ω 2 , …, ω n 
is called frequency vector (related to 
) and its components are called frequency. If all frequencies are non-zero, ω 
is called non-degenerate, and if at least one of the frequencies is zero, we say ω
Definition. Any integer-valued vector perpendicular to the frequency vectorω k n 
is a resonant vector for the frequency vector ω if
k ω j n k j ω j 
If there is such an annihilator vector forω ω ω
Material and methods
In this paper, we focus on indefinite Hamiltonian resonances, so that by introducing some important physical models, we review their occurrence in different parts of science including the satellite problem, plasma interactions, Pais-Ollenbeck oscillators (PU) and a problem of cosmology. Then, by using the normal form of indefinite Hamiltonian resonances, we will discuss the space structure of the corresponding vector field.
Results and discussion
As we mentioned there are many models passing indefinite Hamiltonian resonances. We clear the space of orbits of indefinite Hamiltonian resonances topologically. Specifically, we will see it can be unbounded in the comparison to space of orbits of other Hamiltonian resonances.
The study of Hamiltonian systems around the elliptic equilibrium points, which is a non-generic subject in the study of Hamiltonian systems, has received attention in recent decades. Such systems appear in many applied models, including molecular physics, galactic dynamics, and mechanics.
Consider a Hamiltonian of
degrees of freedom, whose quadratic part is as follows,
| (1) |  |
is called frequency vector (related to ) and its components are called frequency. If all frequencies are non-zero, is called non-degenerate, and if at least one of the frequencies is zero, we say is degenerate.Definition. Any integer-valued vector perpendicular to the frequency vector
is called a resonance or annihilator vector. In other words, is a resonant vector for the frequency vector if If there is such an annihilator vector for
, we call the resonance frequency vector. If at least one of the components of the non-degenerate resonance frequency vector is negative, the Hamiltonian resonance is called indefinite.Material and methods
In this paper, we focus on indefinite Hamiltonian resonances, so that by introducing some important physical models, we review their occurrence in different parts of science including the satellite problem, plasma interactions, Pais-Ollenbeck oscillators (PU) and a problem of cosmology. Then, by using the normal form of indefinite Hamiltonian resonances, we will discuss the space structure of the corresponding vector field.
Results and discussion
As we mentioned there are many models passing indefinite Hamiltonian resonances. We clear the space of orbits of indefinite Hamiltonian resonances topologically. Specifically, we will see it can be unbounded in the comparison to space of orbits of other Hamiltonian resonances.
Keywords: Hamiltonian resonance, Indefinite resonance, Normal form, Action-angle coordinates, Space of orbits
Type of Study: Research Paper |
Subject:
Geometry of dynamical systems
Received: 2024/02/19 | Revised: 2025/10/6 | Accepted: 2025/06/30 | Published: 2025/05/31 | ePublished: 2025/05/31
Received: 2024/02/19 | Revised: 2025/10/6 | Accepted: 2025/06/30 | Published: 2025/05/31 | ePublished: 2025/05/31
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