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Symmetry
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Symmetry in Geometric Mechanics and Mathematical Physics
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Symmetry in Geometric Mechanics and Mathematical Physics

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

Deadline for manuscript submissions: 15 April 2025 | Viewed by 5723

Special Issue Editors

Prof. Dr. Manuel de León

E-Mail Website
Guest Editor
Campus Cantoblanco Consejo Superior de Investigaciones Científicas C/Nicolás Cabrera, Instituto de Ciencias Matematicas, 13–15, 28049 Madrid, Spain
Interests: differential geometry; symplectic geometry; poisson manifolds; cosserat media; nonholonomic mechanics; geometric integrators; optimal control theory
Prof. Dr. Narciso Román-Roy

E-Mail Website
Guest Editor
Department of Mathematics, Campus Nord, Universitat Politècnica de Catalunya, Ed. C3, 08034 Barcelona, Spain
Interests: Mathematical Physics; Applied Differential Geometry

Special Issue Information

Symmetries are present in natural phenomena, especially in physics. Since the celebrated Noether's Theorem that associates with each symmetry a conserved quantity, the study of symmetries and their implications in the equations of mechanics and mathematical physics has been a constantly growing topic. Since the pioneering work of Evariste Galois, the concept of symmetry has been closely linked to that of the group, and in our field, which requires differentiability, to that of the Lie group. One of the major achievements is the so-called Marsden–Weinstein Theorem of symplectic reduction, which allows us to reduce the dynamical system to another phase space with less dimension. The use of Lie groups and their infinitesimal approximations, Lie algebras, is essential. The moment map becomes a bridge connecting dynamical systems to more manageable algebraic versions. Furthermore, the gauge theories or the construction of the standard model itself are based on the properties of Lie groups.

The concept of the group has been extended to that of the groupoid, which has begun to be used both in particle mechanics (to develop new geometric integrators) and continuum mechanics (for the study of uniformity, evolution, growth, and aging). Symmetries are also extremely useful in the fields of engineering (robotics, for example) and in the study of control systems and economics.

The aim of this Special Issue is to address different aspects of the importance of symmetries in all these areas through the inclusion of quality articles by renowned authors.

Prof. Dr. Manuel de León
Prof. Dr. Narciso Román-Roy
Guest Editors

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

  • particle mechanics
  • control systems
  • continuum mechanics
  • Lie groups
  • mathematical physics
  • Lie algebras

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (4 papers)

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Research

26 pages, 406 KiB  
Article
On Matrices of Generalized Octonions (Cayley Numbers)
by Seda Yamaç Akbıyık
Symmetry 2024, 16(12), 1567; https://doi.org/10.3390/sym16121567 - 22 Nov 2024
Abstract
This article focuses on generalized octonions which include real octonions, split octonions, semi octonions, split semi octonions, quasi octonions, split quasi octonions and para octonions in special cases. We make a classification according to the inner product and vector parts and give the [...] Read more.
This article focuses on generalized octonions which include real octonions, split octonions, semi octonions, split semi octonions, quasi octonions, split quasi octonions and para octonions in special cases. We make a classification according to the inner product and vector parts and give the polar forms for lightlike generalized octonions. Furthermore, the matrix representations of the generalized octonions are given and some properties of these representations are achieved. Also, powers and roots of the matrix representations are presented. All calculations in the article are achieved by using MATLAB R2023a and these codes are presented with an illustrative example. Full article
(This article belongs to the Special Issue Symmetry in Geometric Mechanics and Mathematical Physics)
14 pages, 2601 KiB  
Article
Sedenion Algebra Model as an Extension of the Standard Model and Its Link to SU(5)
by Qiang Tang and Jau Tang
Symmetry 2024, 16(5), 626; https://doi.org/10.3390/sym16050626 - 17 May 2024
Viewed by 887
Abstract
In the Standard Model, ad hoc hypotheses assume the existence of three generations of point-like leptons and quarks, which possess a point-like structure and follow the Dirac equation involving four anti-commutative matrices. In this work, we consider the sedenion hypercomplex algebra as an [...] Read more.
In the Standard Model, ad hoc hypotheses assume the existence of three generations of point-like leptons and quarks, which possess a point-like structure and follow the Dirac equation involving four anti-commutative matrices. In this work, we consider the sedenion hypercomplex algebra as an extension of the Standard Model and show its close link to SU(5), which is the underlying symmetry group for the grand unification theory (GUT). We first consider the direct-product quaternion model and the eight-element octonion algebra model. We show that neither the associative quaternion model nor the non-associative octonion model could generate three fermion generations. Instead, we show that the sedenion model, which contains three octonion sub-algebras, leads naturally to precisely three fermion generations. Moreover, we demonstrate the use of basis sedenion operators to construct twenty-four 5 × 5 generalized lambda matrices representing SU(5) generators, in analogy to the use of octonion basis operators to generate Gell-Mann’s eight 3 × 3 lambda-matrix generators for SU(3). Thus, we provide a link between the sedenion algebra and Georgi and Glashow’s SU(5) GUT model that unifies the electroweak and strong interactions for the Standard Model’s elementary particles, which obey SU(3)⊗SU(2)⊗U(1) symmetry. Full article
(This article belongs to the Special Issue Symmetry in Geometric Mechanics and Mathematical Physics)
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35 pages, 542 KiB  
Article
More Insights into Symmetries in Multisymplectic Field Theories
by Arnoldo Guerra IV and Narciso Román-Roy
Symmetry 2023, 15(2), 390; https://doi.org/10.3390/sym15020390 - 1 Feb 2023
Cited by 6 | Viewed by 1348
Abstract
This work provides a general overview for the treatment of symmetries in classical field theories and (pre)multisymplectic geometry. The geometric characteristics of the relation between how symmetries are interpreted in theoretical physics and in the geometric formulation of these theories are clarified. Finally, [...] Read more.
This work provides a general overview for the treatment of symmetries in classical field theories and (pre)multisymplectic geometry. The geometric characteristics of the relation between how symmetries are interpreted in theoretical physics and in the geometric formulation of these theories are clarified. Finally, a general discussion is given on the structure of symmetries in the presence of constraints appearing in singular field theories. Symmetries of some typical theories in theoretical physics are analyzed through the construction of the relevant multimomentum maps which are the conserved quantities (by Noether’s theorem) on the (pre)multisymplectic phase spaces. Full article
(This article belongs to the Special Issue Symmetry in Geometric Mechanics and Mathematical Physics)
39 pages, 498 KiB  
Article
Geodesic and Newtonian Vector Fields and Symmetries of Mechanical Systems
by José F. Cariñena and Miguel-C. Muñoz-Lecanda
Symmetry 2023, 15(1), 181; https://doi.org/10.3390/sym15010181 - 7 Jan 2023
Cited by 3 | Viewed by 2059
Abstract
Geodesic vector fields and other distinguished vector fields on a Riemann manifold were used in the study of free motions on such a manifold, and we applied the geometric Hamilton–Jacobi theory for the search of geodesic vector fields from Hamilton–Jacobi vector fields and [...] Read more.
Geodesic vector fields and other distinguished vector fields on a Riemann manifold were used in the study of free motions on such a manifold, and we applied the geometric Hamilton–Jacobi theory for the search of geodesic vector fields from Hamilton–Jacobi vector fields and the same for closed vector fields. These properties were appropriately extended to the framework of Newtonian and generalised Newtonian systems, in particular systems defined by Lagrangians of the mechanical type and velocity-dependent forces. Conserved quantities and a generalised concept of symmetry were developed, particularly for Killing vector fields. Nonholonomic constrained Newtonian systems were also analysed from this perspective, as well as the relation among Newtonian vector fields and Hamilton–Jacobi equations for conformally related metrics. Full article
(This article belongs to the Special Issue Symmetry in Geometric Mechanics and Mathematical Physics)
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