Noether’s Theorem and Symmetry Edited by P.G.L. Leach and Andronikos Paliathanasis Printed Edition of the Special Issue Published in Symmetry www.mdpi.com/journal/symmetry Noether’s Theorem and Symmetry Noether’s Theorem and Symmetry Special Issue Editors P.G.L. Leach Andronikos Paliathanasis MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editors P.G.L. Leach Andronikos Paliathanasis University of Kwazulu Natal & Durban University of Technology Durban University of Technology Republic of South Arica Cyprus Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Symmetry (ISSN 2073-8994) from 2018 to 2019 (available at: https://www.mdpi.com/journal/symmetry/ special issues/Noethers Theorem Symmetry). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03928-234-0 (Pbk) ISBN 978-3-03928-235-7 (PDF) Cover image courtesy of Andronikos Paliathanasis. c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editors ..................................... vii Preface to ”Noether’s Theorem and Symmetry” ............................ ix Amlan K Halder, Andronikos Paliathanasis and P.G.L. Leach Noether’s Theorem and Symmetry Reprinted from: Symmetry 2018, 10, 744, doi:10.3390/sym10120744 ................. 1 Dmitry S. Kaparulin Conservation Laws and Stability of Field Theories of Derived Type Reprinted from: Symmetry 2019, 11, 642, doi:10.3390/sym11050642 ................. 22 Linyu Peng Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations Reprinted from: Symmetry 2019, 11, 884, doi:10.3390/sym11070884 ................. 42 V. Rosenhaus and Ravi Shankar Quasi-Noether Systems and Quasi-Lagrangians Reprinted from: Symmetry 2019, 11, 1008, doi:10.3390/sym11081008 ................. 53 M. Safdar, A. Qadir, M. Umar Farooq Comparison of Noether Symmetries and First Integrals of Two-Dimensional Systems of Second Order Ordinary Differential Equations by Real and Complex Methods Reprinted from: Symmetry 2019, 11, 1180, doi:10.3390/sym11091180 ................ 73 Marianna Ruggieri and Maria Paola Speciale Optimal System and New Approximate Solutions of a Generalized Ames’s Equation Reprinted from: Symmetry 2019, 11, 1230, doi:10.3390/sym11101230 ................. 85 Elena Recio, Tamara M. Garrido, Rafael de la Rosa and Mar´ıa S. Bruzon´ Reprinted from: Symmetry 2019, 11, 1031, doi:10.3390/sym11081031 ................. 96 Almudena P. M´arquez and Mar´ıa S. Bruz´on Symmetry Analysis and Conservation Laws of a Generalization of the Kelvin-Voigt Viscoelasticity Equation Reprinted from: Symmetry 2019, 11, 840, doi:10.3390/sym11070840 .................109 Hassan Azad, Khaleel Anaya, Ahmad Y. Al-Dweik and M. T. Mustafa Invariant Solutions of the WaveEquation on Static Spherically Symmetric Spacetimes AdmittingG7 Isometry Algebra Reprinted from: Symmetry 2018, 10, 665, doi:10.3390/sym10120665 .................118 Sumaira Saleem Akhtar, Tahir Hussain and Ashfaque H. Bokhari Positive Energy Condition and Conservation Laws in Kantowski-Sachs Spacetime via Noether Symmetries Reprinted from: Symmetry 2018, 10, 712, doi:10.3390/sym10120712 .................144 Ugur Camci F (R, G) Cosmology through Noether Symmetry Approach Reprinted from: Symmetry 2018, 10, 719, doi:10.3390/sym10120719 .................156 v About the Special Issue Editors P.G.L. Leach was born in Melbourne, Australia, just a few days before the bombing of Pearl Harbour. He matriculated from St Patrick’s College in 1958 and attended the University of Melbourne to obtain his BSc, BA, and Dip Ed. After teaching at High Schools for some years, he went to the Bendigo Institute of Technology (Victoria, Australia) and resumed his studies at La Trobe University in Melbourne. His supervisor was CJ Eliezer, a former student of PAM Dirac. He graduated with an MSc in 1977, Ph.D. in 1979, and, from the University of Natal, a DSc in 1996. In 1983, he went to the University of the Witwatersrand, Johannesburg, as a Professor of Applied Mathematics, and in 1990, to the University of Natal in Durban. Currently, he is Professor Emeritus at the University of KwaZulu Natal and Extraordinary Research Professor at the Durban University of Technology. Honors: Fellow of the University of Natal (1996); Elected Member of the Academy of Nonlinear Sciences (Russian Republic) (1997); Elected Fellow of the Royal Society of South Africa (1998); Gold Medalist of the South African Mathematical Society (2001); Distinguished Research Award of the South African Mathematical Society (2002). He has some 350 publications. Andronikos Paliathanasis received his BSc in 2008 from the Physics department of the National and Kapodistrian University of Athens. From the same department, he graduated with an MSc in 2010 and a Ph.D. in 2015. Since then, he has collaborated with the INFN Sez. Di Napoli, in Italy. In November 2015, he was awarded with a postdoctoral fellowship from CONICYT (Chile) in the Universidad Austral de Chile, where he was also an Adjunct Professor and gave various classes in the institute of physics and mathematics. Since June 2017, he has been affiliated with the Institute of System Science of Durban University of Technology, while since July 2019, he has also been a Research Fellow in the same institute. He has more than 100 publications in peer-reviewed journals on the subject of symmetries of differential equations as well as in gravitational physics and cosmology. vii Preface to ”Noether’s Theorem and Symmetry” This book deals with the two famous theorems of Emmy Noether, published a century ago. The work of Emmy Noether was pioneering for her period and changed the way that we approach physical theories and other applied mathematics theories today. There have been many different readings of her work and various attempts to extend and generalize the original results of 1918. In this book, in a series of articles, we summarize the above attempts to extend Noether’s work, and various applications are presented for the modern treatment of Noether’s theorems. P.G.L. Leach, Andronikos Paliathanasis Special Issue Editors ix S S symmetry Article Noether’s Theorem and Symmetry Amlan K. Halder 1, Andronikos Paliathanasis 2,3,*,† and Peter G.L. Leach 3,4 1 Department of Mathematics, Pondicherry University, Kalapet 605014, India; [email protected] 2 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia 5090000, Chile 3 Institute of Systems Science, Durban University of Technology, PO Box 1334, Durban 4000, South Africa; [email protected] 4 School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4000, South Africa * Correspondence: [email protected] † Current address: Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia 5090000, Chile. Received: 20 November 2018; Accepted: 9 December 2018; Published: 12 December 2018 Abstract: In Noether’s original presentation of her celebrated theorem of 1918, allowance was made for the dependence of the coefficient functions of the differential operator, which generated the infinitesimal transformation of the action integral upon the derivatives of the dependent variable(s), the so-called generalized, or dynamical, symmetries. A similar allowance is to be found in the variables of the boundary function, often termed a gauge function by those who have not read the original paper. This generality was lost after texts such as those of Courant and Hilbert or Lovelock and Rund confined attention to point transformations only. In recent decades, this diminution of the power of Noether’s theorem has been partly countered, in particular in the review of Sarlet and Cantrijn. In this Special Issue, we emphasize the generality of Noether’s theorem in its original form and explore the applicability of even more general coefficient functions by allowing for nonlocal terms. We also look for the application of these more general symmetries to problems in which parameters or parametric functions have a more general dependence on the independent variables. Keywords: Noether’s theorem; action integral; generalized symmetry; first integral; invariant; nonlocal transformation; boundary term; conservation laws; analytic mechanics 1. Introduction Noether’s theorem [1] treats the invariance of the functional of the calculus of variations—the action integral in mechanics—under an infinitesimal transformation. This transformation can be considered as being generated by a differential operator, which in this case is termed a Noether symmetry. The theorem was not developed ab initio by Noether. Not only is it steeped in the philosophy of Lie’s approach, but also, it is based on earlier work of more immediate relevance by a number of writers. Hamel [2,3] and Herglotz [4] had already applied the ideas developed in her paper to some specific finite groups. Fokker [5] did the same for specific