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Symmetry
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Symmetry in Hamiltonian Dynamical Systems
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Symmetry in Hamiltonian Dynamical Systems

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: 15 July 2024 | Viewed by 5935

Special Issue Editor

Prof. Dr. Fernando Haas

E-Mail Website
Guest Editor
Physics Institute, Federal University of Rio Grande do Sul, Av. Bento Gonçalves 9500, Porto Alegre 91501-970, RS, Brazil
Interests: lie symmetries; noether symmetries; integrable systems; generalized Hamiltonian systems; quantum plasmas; neutrino-plasma interactions; Bose-Einstein condensates

Special Issue Information

Dear Colleagues,

The search for Lie symmetry is a powerful method for the reduction in necessary variables and integration of dynamical systems in general. Opposite to chaotic systems, integrable systems have a sufficient degree of symmetry and exact constants of motion, or invariants. As a result, dynamical evolution in such systems is more regular and predictable. The quest for symmetry and integrability has many applications, such as in plasma physics, epidemics models, and climate prediction models, to name a few. On the other hand, Hamiltonian systems have a key role in the development of perturbation theory and quantum mechanics. The analysis of the geometric properties of Hamiltonian systems points to the relevance of Poisson structures, or non-canonical Hamiltonian systems and their diverse generalizations, such as Jacobi systems. Related to Hamiltonian systems, the deductive approach provided by Noether’s theorem has a central interest for problems admitting a variational description. In the case of continuous systems, completely integrable dynamical systems have an infinite number of conservation laws, together with the existence of soliton solutions. In plasma physics, special attention has been paid to electron hole structures and solitary waves derived by means of the Sagdeev potential method, with an underlying Hamiltonian structure.

We cordially and earnestly invite researchers to contribute their original and high-quality research papers which will inspire advances about symmetries and Hamiltonian systems and beyond. Potential topics include but are not limited to:

  • Lie symmetry
  • Noether symmetry
  • Dynamical algebra
  • Poisson mechanics
  • Perturbation theory
  • Jacobi systems
  • Integrable systems
  • Exact or approximate constants of motion
  • Finite dimensional dynamical systems
  • Solitons
  • Painlevé test
  • Ermakov systems
  • Extended Lie groups
  • Sagdeev potential
  • Reductive perturbation method
  • Nonlinear waves.

Prof. Dr. Fernando Haas
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Lie symmetry
  • Noether symmetry
  • generalized Hamiltonian systems
  • integrable dynamical systems
  • exact constants of motion
  • solitons
  • Painlevé analysis
  • nonlinear waves

Published Papers (6 papers)

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Research

16 pages, 1031 KiB  
Article
Symmetry Breaking and Dynamic Transition in the Negative Mass Term Klein–Gordon Equations
by Ferenc Márkus and Katalin Gambár
Symmetry 2024, 16(2), 144; https://doi.org/10.3390/sym16020144 - 26 Jan 2024
Cited by 1 | Viewed by 585
Abstract
Through the discussion of three physical processes, we show that the Klein–Gordon equations with a negative mass term describe special dynamics. In the case of two classical disciplines—mechanics and thermodynamics—the Lagrangian-based mathematical description is the same, even though the nature of the investigated [...] Read more.
Through the discussion of three physical processes, we show that the Klein–Gordon equations with a negative mass term describe special dynamics. In the case of two classical disciplines—mechanics and thermodynamics—the Lagrangian-based mathematical description is the same, even though the nature of the investigated processes seems completely different. The unique feature of this type of equation is that it contains wave propagation and dissipative behavior in one framework. The dissipative behavior appears through a repulsive potential. The transition between the two types of dynamics can be specified precisely, and its physical meaning is clear. The success of the two descriptions inspires extension to the case of electrodynamics. We reverse the suggestion here. We create a Klein–Gordon equation with a negative mass term, but first, we modify Maxwell’s equations. The repulsive interaction that appears here results in a charge spike. However, the Coulomb interaction limits this. The charge separation is also associated with the high-speed movement of the charged particle localized in a small space domain. As a result, we arrive at a picture of a fast vibrating phenomenon with an electromagnetism-related Klein–Gordon equation with a negative mass term. The calculated maximal frequency value ω=1.74×1021 1/s. Full article
(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)
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16 pages, 672 KiB  
Article
Detecting Phase Transitions through Non-Equilibrium Work Fluctuations
by Matteo Colangeli, Antonio Di Francesco and Lamberto Rondoni
Symmetry 2024, 16(1), 125; https://doi.org/10.3390/sym16010125 - 20 Jan 2024
Viewed by 775
Abstract
We show how averages of exponential functions of path-dependent quantities, such as those of Work Fluctuation Theorems, detect phase transitions in deterministic and stochastic systems. State space truncation—the restriction of the observations to a subset of state space with prescribed probability—is introduced to [...] Read more.
We show how averages of exponential functions of path-dependent quantities, such as those of Work Fluctuation Theorems, detect phase transitions in deterministic and stochastic systems. State space truncation—the restriction of the observations to a subset of state space with prescribed probability—is introduced to obtain that result. Two stochastic processes undergoing first-order phase transitions are analyzed both analytically and numerically: a variant of the Ehrenfest urn model and the 2D Ising model subject to a magnetic field. In the presence of phase transitions, we prove that even minimal state space truncation makes averages of exponentials of path-dependent variables sensibly deviate from full state space values. Specifically, in the case of discontinuous phase transitions, this approach is strikingly effective in locating the transition value of the control parameter. As this approach works even with variables different from those of fluctuation theorems, it provides a new recipe to identify order parameters in the study of non-equilibrium phase transitions, profiting from the often incomplete statistics that are available. Full article
(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)
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12 pages, 710 KiB  
Article
Adiabatic Manipulation of a System Interacting with a Spin Bath
by Benedetto Militello and Anna Napoli
Symmetry 2023, 15(11), 2028; https://doi.org/10.3390/sym15112028 - 08 Nov 2023
Cited by 1 | Viewed by 716
Abstract
The Stimulated Raman Adiabatic Passage, a very efficient technique for manipulating a quantum system based on the adiabatic theorem, is analyzed in the case where the manipulated physical system is interacting with a spin bath. The exploitation of the rotating wave approximation allows [...] Read more.
The Stimulated Raman Adiabatic Passage, a very efficient technique for manipulating a quantum system based on the adiabatic theorem, is analyzed in the case where the manipulated physical system is interacting with a spin bath. The exploitation of the rotating wave approximation allows for the identification of a constant of motion, which simplifies both the analytical and the numerical treatment, which allows for evaluating the total unitary evolution of the system and bath. The efficiency of the population transfer process is investigated in several regimes, including the weak and strong coupling with the environment and the off-resonance. The formation of appropriate Zeno subspaces explains the lowering of the efficiency in the strong damping regime. Full article
(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)
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17 pages, 319 KiB  
Article
Complex Quantum Hydrodynamics in Momentum Space with Broken Time-Reversal Symmetry
by Dieter Schuch and Moise Bonilla-Licea
Symmetry 2023, 15(7), 1347; https://doi.org/10.3390/sym15071347 - 01 Jul 2023
Viewed by 795
Abstract
Shortly after Schrödinger’s wave mechanics in terms of complex wave functions was published, Madelung formulated this theory in terms of two real hydrodynamic-like equations. This version is also the formal basis of Bohmian mechanics, albeit with a different ontological interpretation. A point of [...] Read more.
Shortly after Schrödinger’s wave mechanics in terms of complex wave functions was published, Madelung formulated this theory in terms of two real hydrodynamic-like equations. This version is also the formal basis of Bohmian mechanics, albeit with a different ontological interpretation. A point of criticism raised by Pauli against Bohmian mechanics is its missing symmetry between position and momentum that is present in classical phase space as well as in the quantum mechanical position and momentum representations. Both Madelung’s quantum hydrodynamics formulation and Bohmian mechanics are usually expressed only in position space. Recently, with the use of complex quantities, we were able to provide a hydrodynamic formulation also in momentum space. In this paper, we extend this formalism to include dissipative systems with broken time-reversal symmetry. In classical Hamiltonian mechanics and conventional quantum mechanics, closed systems with reversible time-evolution are usually considered. Extending the discussion to include open systems with dissipation, another form of symmetry is broken, that under time-reversal. There are different ways of describing such systems; for instance, Langevin and Fokker–Planck-type equations are commonly used in classical physics. We now investigate how these aspects can be incorporated into our complex hydrodynamic description and what modifications occur in the corresponding equations, not only in position, but particularly in momentum space. Full article
(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)
19 pages, 857 KiB  
Article
Finite Reservoirs Corrections to Hamiltonian Systems Statistics and Time Symmetry Breaking
by Matteo Colangeli, Antonio Di Francesco and Lamberto Rondoni
Symmetry 2023, 15(6), 1268; https://doi.org/10.3390/sym15061268 - 15 Jun 2023
Cited by 1 | Viewed by 891
Abstract
We consider several Hamiltonian systems perturbed by external agents that preserve their Hamiltonian structure. We investigate the corrections to the canonical statistics resulting from coupling such systems with possibly large but finite reservoirs and from the onset of processes breaking the time-reversal symmetry. [...] Read more.
We consider several Hamiltonian systems perturbed by external agents that preserve their Hamiltonian structure. We investigate the corrections to the canonical statistics resulting from coupling such systems with possibly large but finite reservoirs and from the onset of processes breaking the time-reversal symmetry. We analyze exactly solvable oscillator systems and perform simulations of relatively more complex ones. This indicates that the standard statistical mechanical formalism needs to be adjusted in the ever more investigated nano-scale science and technology. In particular, the hypothesis that heat reservoirs be considered infinite and be described by the classical ensembles is found to be critical when exponential quantities are considered since the large size limit may not coincide with the infinite size canonical result. Furthermore, process-dependent emergent irreversibility affects ensemble averages, effectively frustrating, on a statistical level, the time reversal invariance of Hamiltonian dynamics that are used to obtain numerous results. Full article
(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)
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36 pages, 506 KiB  
Article
On Rational Solutions of Dressing Chains of Even Periodicity
by Henrik Aratyn, José Francisco Gomes, Gabriel Vieira Lobo and Abraham Hirsz Zimerman
Symmetry 2023, 15(1), 249; https://doi.org/10.3390/sym15010249 - 16 Jan 2023
Cited by 1 | Viewed by 1127
Abstract
We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalent Painlevé equations invariant under AN−1(1) symmetry. This formalism identifies rational solutions (as well [...] Read more.
We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalent Painlevé equations invariant under AN−1(1) symmetry. This formalism identifies rational solutions (as well as special function solutions) with points on orbits of fundamental shift operators of AN−1(1) affine Weyl groups acting on seed configurations defined as first-order polynomial solutions of the underlying dressing chains. This approach clarifies the structure of rational solutions and establishes an explicit and systematic method towards their construction. For the special case of the N=4 dressing chain equations, the method yields all the known rational (and special function) solutions of the Painlevé V equation. The formalism naturally extends to N=6 and beyond as shown in the paper. Full article
(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)
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