HAT: Hamiltonian Systems—Applications and Theory

Topic Information

Dear Colleagues,

Hamiltonian systems have been one of the most influential ideas in the theory of dynamical systems and in the formulation of physical theories ever since their discovery by J. L. Lagrange and W. R. Hamilton. Due to their appealing theoretical properties, they are a central node at the intersection of topology, geometry and dynamical systems. Moreover, they are being used to lay the foundations of an ever-growing number of applications that span from classical and quantum mechanics to statistical physics and general relativity, from chemical reactions to thermodynamics, and from evolutionary game theory to models of biological evolution and data science. The flourishing of so many different fields of application has, in turn, increased the number of relevant questions that must be addressed about these systems. The result is that there is currently so much investigation going on concerning Hamiltonian systems and their Lagrangian counterparts that it is hard—if not impossible—to keep track of all the new questions and results appearing in the literature, and there is a pressing need for synthesis of all these directions. The aim of this Topic is to provide an opportunity to anyone interested in Hamiltonian systems, from very different perspectives, to join their works together as part of a collection, which will take a broad as well as unified view of the state of the art regarding the knowledge on Hamiltonian systems with regard to both theory and applications.

Dr. Alessandro Bravetti
Prof. Dr. Manuel De León
Dr. Ángel Alejandro García-Chung
Dr. Marcello Seri
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Entropy entropy	2.1	4.9	1999	22.3 Days	CHF 2600
Fractal and Fractional fractalfract	3.6	4.6	2017	23.7 Days	CHF 2700
Mathematical and Computational Applications mca	1.9	-	1996	25.4 Days	CHF 1400
Mathematics mathematics	2.3	4.0	2013	18.3 Days	CHF 2600
Symmetry symmetry	2.2	5.4	2009	17.3 Days	CHF 2400

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Published Papers (2 papers)

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All Entropy Fractal Fract MCA Mathematics Symmetry

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10 pages, 5266 KiB  

Open AccessReview

Two Quantum Triatomic Hamiltonians: Applications to Non-Adiabatic Effects in NO₂ Spectroscopy and in Kr + OH(A²Σ⁺) Electronic Quenching

by Carlo Petrongolo

Symmetry 2025, 17(3), 346; https://doi.org/10.3390/sym17030346 - 25 Feb 2025

Viewed by 345

Abstract

This review discusses two triatomic Hamiltonians and their applications to some non-adiabatic spectroscopic and collision problems. Carter and Handy in 1984 presented the first Hamiltonian in bond lengths–bond angle coordinates, that is here applied for studying the NO₂ spectroscopy: vibronic states, internal [...] Read more.

This review discusses two triatomic Hamiltonians and their applications to some non-adiabatic spectroscopic and collision problems. Carter and Handy in 1984 presented the first Hamiltonian in bond lengths–bond angle coordinates, that is here applied for studying the NO₂ spectroscopy: vibronic states, internal dynamics, and interaction with the radiation due to the $\tilde{\mathrm{X}}$²A′(A₁)−$\tilde{\mathrm{A}}$²A′(B₂) conical intersection. The second Hamiltonian was reported by Tennyson and Sutcliffe in 1983 in Jacobi coordinates and is here employed in the study of the Kr + OH(A²Σ⁺) electronic quenching due to conical intersection and Renner–Teller interactions among the 1²A′, 2²A′, and 1²A″ electronic species. Within the non-relativistic approximation and the expansion method in diabatic electronic representations, the formalism is exact and allows a unified study of various non-adiabatic interactions between electronic states. The rotation, inversion, and nuclear permutation symmetries are considered for defining rovibronic representations, which are symmetry adapted for ABC and AB₂ molecules, and the matrix elements of the Hamiltonians are then computed. Full article

(This article belongs to the Topic HAT: Hamiltonian Systems—Applications and Theory)

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14 pages, 307 KiB  

Open AccessArticle

The Number of Limit Cycles Bifurcating from an Elementary Centre of Hamiltonian Differential Systems

by Lijun Wei, Yun Tian and Yancong Xu

Mathematics 2022, 10(9), 1483; https://doi.org/10.3390/math10091483 - 29 Apr 2022

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Abstract

This paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function [...] Read more.

This paper studies the number of small limit cycles produced around an elementary center for Hamiltonian differential systems with the elliptic Hamiltonian function $H=\frac{1}{2}{y}^2+\frac{1}{2}{x}^2-\frac{2}{3}{x}^3+\frac{a}{4}{x}^4(a\ne 0)$ under two types of polynomial perturbations of degree m, respectively. It is proved that the Hamiltonian system perturbed in Liénard systems can have at least $\left[\frac{3m-1}{4}\right]$ small limit cycles near the center, where $m\le 101$, and that the related near-Hamiltonian system with general polynomial perturbations can have at least $m+\left[\frac{m+1}{2}\right]-2$ small-amplitude limit cycles, where $m\le 16$. Furthermore, in any of the cases, the bounds for limit cycles can be reached by studying the isolated zeros of the corresponding first order Melnikov functions and with the help of Maple programs. Here, $[·]$ represents the integer function. Full article

(This article belongs to the Topic HAT: Hamiltonian Systems—Applications and Theory)

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