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Symmetry
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Symmetry and Applications of Differential Geometry to the Differential Equations...
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Symmetry and Applications of Differential Geometry to the Differential Equations of Mathematical Physics

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

Deadline for manuscript submissions: 31 August 2024 | Viewed by 5489

Special Issue Editor

Dr. Jean-Pierre Magnot

E-Mail Website
Guest Editor
Lyc’ee Jeanne d’Arc, Avenue de Grande Bretagne, F-63000 Clermont-Ferrand, France
Interests: mathematics; mathematical physics; differential geometry; integrable systems; operational research

Special Issue Information

Dear Colleagues,

Geometric methods in the theory of differential equations in mathematical physics have a very long history. Over more than one century, the technical gaps that a direct (numerical) analysis can meet in solving and understanding differential equations have met a complementary approach in geometry, in order, for example, to describe no-go directions of investigations or to produce new solutions or invariants. Under the term "differential equation", I can embrace ordinaly differential equations, partial differential equations, integro-differential equations, equations in higher structures and quantum field theories, and solutions can also be understood in a very significant way: strong or weak solutions, under constraints or not, approximate solutions in a way to determine, or even formal solutions.

The symmetry properties of equations can be studied in order to determine solutions with particular properties or to distinguish equations that do not have the same intrinsic properties. Studies on symmetry properties associated with the concept of quantum calculus or stochastic analysis may also be investigated.

Therefore, it seems exciting to propose a Special Issue that intends to gather all these interrelated topics. These ideas and methods have a significant effect on everyday life, as new tools are developed and achieve revolutionary research results, bringing scientists even closer to exact sciences, encouraging the emergence of new approaches, techniques, and perspectives in geometric approaches of differential equations. Please note that all submitted papers should be within the scope of the journal.

Dr. Jean-Pierre Magnot
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

  • Gelfand's formal geometry
  • diffeologies
  • symmetries
  • first integrals
  • Hamiltonian geometry
  • h-principle
  • homotopy methods
  • approximation methods
  • approximate symmetries
  • symmetries in SPDE
  • higher homotopy
  • gauge fields
  • non-geometric fluxes
  • quantum field theory

Published Papers (5 papers)

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Research

14 pages, 309 KiB  
Article
On Equation Manifolds, the Vinogradov Spectral Sequence, and Related Diffeological Structures
by Jean-Pierre Magnot and Enrique G. Reyes
Symmetry 2024, 16(2), 192; https://doi.org/10.3390/sym16020192 - 06 Feb 2024
Viewed by 487
Abstract
We consider basic diffeological structures that can be highlighted naturally within the theory of the Vinogradov spectral sequence and equation manifolds. These interrelated features are presented in a rigorous and accurate way, that complements some heuristic formulations appearing in very recent literature. We [...] Read more.
We consider basic diffeological structures that can be highlighted naturally within the theory of the Vinogradov spectral sequence and equation manifolds. These interrelated features are presented in a rigorous and accurate way, that complements some heuristic formulations appearing in very recent literature. We also propose a refined definition of the Vinogradov spectral sequence using diffeologies. Full article
(This article belongs to the Special Issue Symmetry and Applications of Differential Geometry to the Differential Equations of Mathematical Physics)
9 pages, 1284 KiB  
Article
Surfaces Family with Bertrand Curves as Common Asymptotic Curves in Euclidean 3–Space E3
by Maryam T. Aldossary and Rashad A. Abdel-Baky
Symmetry 2023, 15(7), 1440; https://doi.org/10.3390/sym15071440 - 18 Jul 2023
Cited by 1 | Viewed by 881
Abstract
The main result of this paper is constructing a surfaces family with the similarity of Bertrand curves in Euclidean 3–space E3. Then, by utilizing the Serret–Frenet frame, we conclude the sufficient and necessary conditions of surfaces family interpolating Bertrand curves as [...] Read more.
The main result of this paper is constructing a surfaces family with the similarity of Bertrand curves in Euclidean 3–space E3. Then, by utilizing the Serret–Frenet frame, we conclude the sufficient and necessary conditions of surfaces family interpolating Bertrand curves as common asymptotic curves. Consequently, the expansion to the ruled surfaces family is also depicted. As implementations of our main results, we demonstrate some examples to confirm the method. Full article
(This article belongs to the Special Issue Symmetry and Applications of Differential Geometry to the Differential Equations of Mathematical Physics)
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30 pages, 495 KiB  
Article
Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control
by Luis Blanco Díaz, Cristina Sardón, Fernando Jiménez Alburquerque and Javier de Lucas
Symmetry 2023, 15(6), 1285; https://doi.org/10.3390/sym15061285 - 19 Jun 2023
Viewed by 1606
Abstract
A Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. [...] Read more.
A Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. This superposition rule can be obtained using the geometric features of the Lie system, its symmetries, and the symmetric properties of certain morphisms involved. Even if a superposition rule for a Lie system is known, the explicit analytic expression of its solutions frequently is not. This is why this article focuses on a novel geometric attempt to integrate Lie systems analytically and numerically. We focus on two families of methods based on Magnus expansions and on Runge–Kutta–Munthe–Kaas methods, which are here adapted, in a geometric manner, to Lie systems. To illustrate the accuracy of our techniques we analyze Lie systems related to Lie groups of the form SL(n,R), which play a very relevant role in mechanics. In particular, we depict an optimal control problem for a vehicle with quadratic cost function. Particular numerical solutions of the studied examples are given. Full article
(This article belongs to the Special Issue Symmetry and Applications of Differential Geometry to the Differential Equations of Mathematical Physics)
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16 pages, 713 KiB  
Article
Timelike Ruled Surfaces with Stationary Disteli-Axis
by Areej A. Almoneef and Rashad A. Abdel-Baky
Symmetry 2023, 15(5), 998; https://doi.org/10.3390/sym15050998 - 28 Apr 2023
Cited by 1 | Viewed by 753
Abstract
This paper derives the declarations for timelike ruled surfaces with stationary timelike Disteli-axis by the E. Study map. This prepares the ability to determine a set of Lorentzian invariants which explain the local shape of timelike ruled surfaces. As a result, the Hamilton [...] Read more.
This paper derives the declarations for timelike ruled surfaces with stationary timelike Disteli-axis by the E. Study map. This prepares the ability to determine a set of Lorentzian invariants which explain the local shape of timelike ruled surfaces. As a result, the Hamilton and Mannhiem formulae of surfaces theory are attained at Lorentzian line space and their geometrical explanations are examined. Then, we define and explicate the kinematic geometry of a timelike Plűcker conoid created by the timelike Disteli-axis. Additionally, we provide the relationships through timelike ruled surface and the order of contact with its timelike Disteli-axis. Full article
(This article belongs to the Special Issue Symmetry and Applications of Differential Geometry to the Differential Equations of Mathematical Physics)
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23 pages, 418 KiB  
Article
Decomposing Euler–Poincaré Flow on the Space of Hamiltonian Vector Fields
by Oğul Esen, Javier De Lucas, Cristina Sardon Muñoz and Marcin Zając
Symmetry 2023, 15(1), 23; https://doi.org/10.3390/sym15010023 - 22 Dec 2022
Viewed by 1042
Abstract
The main result of this paper is a matched-pair decomposition of the space of symmetric contravariant tensors TQ. From this procedure two complementary Lie subalgebras of TQ under mutual interaction arise. Introducing a lift operator, the matched pair decomposition of [...] Read more.
The main result of this paper is a matched-pair decomposition of the space of symmetric contravariant tensors TQ. From this procedure two complementary Lie subalgebras of TQ under mutual interaction arise. Introducing a lift operator, the matched pair decomposition of the space of Hamiltonian vector fields is determined. According to this realization, the Euler–Poincaré flows on such spaces are decomposed into two subdynamics: one is the Euler–Poincaré formulation of isentropic fluid flows, and the other one corresponds with Euler–Poincaré equations on contravariant tensors of order n⩾2. Full article
(This article belongs to the Special Issue Symmetry and Applications of Differential Geometry to the Differential Equations of Mathematical Physics)
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