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Symmetry
Special Issues
Geometrization of PDEs and Their Solution by Means of Symmetries
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Geometrization of PDEs and Their Solution by Means of Symmetries

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

Deadline for manuscript submissions: closed (31 October 2020) | Viewed by 1880

Special Issue Editors

Prof. Dr. Michael Tsamparlis

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Guest Editor
Faculty of Physics, Department of Astronomy, Astrophysics and Mechanics, University of Athens, Panepistemiopolis, 157 83 Athens, Greece
Interests: mathematical physics; cosmology; general relativity; conservation laws; lie algebra; fundamental symmetry
Special Issues, Collections and Topics in MDPI journals
Dr. Andronikos Paliathanasis

E-Mail Website
Guest Editor
Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa
Interests: symmetries; integrability; collineations; gravitational physics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In recent years, a large number of studies have related PDEs with geometric objects, mainly with a metric tensor. For each PDE, a metric tensor can be defined, i.e., Riemannian or non-Riemannian space, which acts as the “phase space” of the PDE. The Lie symmetries defined by the geometric elements defining the geometry of that space are related with the first integrals and the invariants of the PDE. The last elements can be used to facilitate the solution of the PDE and, if there are enough of them, even to solve the PDE. This Special Issue offers an opportunity to collect a large amount of work and the obtained results on this topic existing so far in the literature and, at the same time, present new avenues and ideas for future steps in this topic.

Prof. Dr. Michael Tsamparlis
Dr. Andronikos Paliathanasis
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

  • first integrals of PDEs
  • invariants of PDEs
  • symmetries
  • geometry of a PDE

Published Papers (1 paper)

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Research

23 pages, 386 KiB  
Article
Integrable and Superintegrable Potentials of 2d Autonomous Conservative Dynamical Systems
by Antonios Mitsopoulos, Michael Tsamparlis and Andronikos Paliathanasis
Symmetry 2020, 12(10), 1655; https://doi.org/10.3390/sym12101655 - 10 Oct 2020
Cited by 10 | Viewed by 1536
Abstract
We consider the generic quadratic first integral (QFI) of the form [...] Read more.
We consider the generic quadratic first integral (QFI) of the form I=Kab(t,q)q˙aq˙b+Ka(t,q)q˙a+K(t,q) and require the condition dI/dt=0. The latter results in a system of partial differential equations which involve the tensors Kab(t,q), Ka(t,q), K(t,q) and the dynamical quantities of the dynamical equations. These equations divide in two sets. The first set involves only geometric quantities of the configuration space and the second set contains the interaction of these quantities with the dynamical fields. A theorem is presented which provides a systematic solution of the system of equations in terms of the collineations of the kinetic metric in the configuration space. This solution being geometric and covariant, applies to higher dimensions and curved spaces. The results are applied to the simple but interesting case of two-dimensional (2d) autonomous conservative Newtonian potentials. It is found that there are two classes of 2d integrable potentials and that superintegrable potentials exist in both classes. We recover most main previous results, which have been obtained by various methods, in a single and systematic way. Full article
(This article belongs to the Special Issue Geometrization of PDEs and Their Solution by Means of Symmetries)
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