Article
Optics
Andrea Aiello
Summary: Helicity, chirality, and spin angular momentum are three physical observables that play an important role in the study of optical fields. They can be expressed in terms of the electric and magnetic fields and exhibit gauge invariance and electric-magnetic democracy.
Article
Mechanics
W. Nakpim, S. Meleshko
Summary: Two-dimensional relativistic gas dynamics equations are studied in this paper, and a suitable Lagrangian is obtained by solving the Helmholtz problem to simplify the equations. The derived Lie algebra of these equations is used to apply Noether's theorem and obtain conservation laws in Lagrangian variables.
INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS
(2022)
Article
Physics, Applied
Long Ju, Yufeng Zhang, Faiza Afzal, Binlu Feng
Summary: The Zakharov-Kuznetsov (ZK) equation, a 2D generalization of the Korteweg-de Vries (KdV) equation, is extensively investigated. Conservation laws of the ZK equation are solved using three different methods: direct construction, strict self-adjoint property verification, and Noether's theorem. Exact solutions of the ZK equation are calculated using the extended Kudryashov method, and the conservation law analysis is carried out. The Hamiltonian structure, pre-symplectic mapping, and soliton solutions of the ZK equation are determined.
MODERN PHYSICS LETTERS B
(2023)
Article
Mathematics, Applied
E. I. Kaptsov, S. V. Meleshko
Summary: The paper analyzes a model of equations of magnetohydrodynamics (MHD) obtained from group classification. The model utilizes Lagrangian coordinates and includes four arbitrary functions. Conservation laws are obtained using Noether's theorem and exact solutions are obtained through explicit or numerical methods.
STUDIES IN APPLIED MATHEMATICS
(2023)
Article
Mathematics
Lili Xia, Xinsheng Ge
Summary: By applying the Lie symmetry method, group-invariant solutions are constructed for axially loaded Euler beams. The corresponding mathematical models of the beams are formulated. The determining equations of Lie symmetry are proposed via Lie point transformations, and the infinitesimal generators of symmetries of the systems are presented with Maple. Conserved vectors are derived by using two methods, namely, the multipliers method and Noether's theorem.
Article
Mathematics, Applied
Hemant Gandhi, Amit Tomar, Dimple Singh
Summary: The paper aims to achieve invariance analysis of the fractional-order HSC-KdV system of equations based on RL derivatives. The Lie symmetry analysis was used to obtain infinitesimal generators, and the system was reduced to nonlinear FODEs with the help of EK operators. The reduced system was solved using power series technique and its convergence was verified, while the conservation laws were constructed using Noether's theorem.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Physics, Multidisciplinary
Qi-Cheng Wu, Jun-Long Zhao, Yu-Liang Fang, Yu Zhang, Dong-Xu Chen, Chui-Ping Yang, Franco Nori
Summary: Noether's theorem is extended to a class of parity-time (PT)-symmetric systems with eigenvalue transitions from purely real to purely imaginary numbers. The generalized expectation value of an operator is found to be a constant of motion under certain symmetries in both the PT-symmetric unbroken and broken regimes. Experimental investigations using optical setups demonstrate the existence of the constant of motion and the potential application of PT-symmetric systems in quantum information theory and protocols.
SCIENCE CHINA-PHYSICS MECHANICS & ASTRONOMY
(2023)
Article
Materials Science, Multidisciplinary
Chaudry Masood Khalique, Anila Mehmood
Summary: This paper focuses on studying the Lie symmetries and symmetry reductions of the second extended Calogero-Bogoyavlenskii-Schiff (eCBS) equation, reducing it to a fourth-order ordinary differential equation using translation symmetries and solving it with three techniques to form closed-form solutions. The solutions are then portrayed graphically. Additionally, conserved vectors of the eCBS equation are computed through multiplier procedure and Noether's theorem.
RESULTS IN PHYSICS
(2021)
Article
Mathematics, Applied
Evgeniy I. Kaptsov, Vladimir A. Dorodnitsyn, Sergey V. Meleshko
Summary: The one-dimensional modified shallow water equations in Lagrangian coordinates are studied in this paper. The relations between symmetries and conservation laws in Lagrangian coordinates, mass Lagrangian variables, and Eulerian coordinates are discussed. Invariant finite-difference schemes are constructed for equations in Lagrangian coordinates, and these schemes have been tested numerically and compared with an ad hoc approximation-invariant scheme, demonstrating their effectiveness.
STUDIES IN APPLIED MATHEMATICS
(2022)
Article
Astronomy & Astrophysics
Carlos Heredia, Josep Llosa
Summary: This article aims to study nonlocal Lagrangians with an infinite number of degrees of freedom. The extension of Noether's theorem and Noether's identities for such Lagrangians are obtained. A Hamiltonian formalism is established for them. In addition, it is shown that n-order local Lagrangians can be treated as a particular case, and the standard results can be recovered. Finally, this formalism is applied to the case of p-adic open string field.
Article
Mathematics, Applied
M. Skopenkov
Summary: We propose a general algorithm for discretizing classical field theories from Lagrangians. Our work introduces a novel discrete Noether theorem that establishes a connection between symmetries and conservation laws, and presents an energy conservation theorem that does not rely on any symmetry. This allows for exact conservation laws in various theories, including lattice electrodynamics and gauge theory. Furthermore, we develop a conserved discrete energy-momentum tensor that approximates the continuum counterpart, particularly for free fields. The theory is formulated in topological terms, such as coboundary and products of cochains.
MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY
(2023)
Article
Physics, Multidisciplinary
Chaudry Masood Khalique, Mduduzi Yolane Thabo Lephoko
Summary: This article investigates the potential Kadomtsev-Petviashvili (pKP) equation and obtains exact solutions through symmetry reductions and the application of Kudryashov's method. The nature of each solution is further understood through 3D, 2D, and density plots, and a one-parameter group of transformations and conserved vectors for the pKP equation are also provided.
Article
Mathematical & Computational Biology
Zygmunt Pizlo, J. Acacio de Barros
Summary: Perceptual constancy refers to the fact that the perceived geometrical and physical characteristics of objects remain constant despite transformations of the objects such as rigid motion. This concept is essential in various aspects of our daily activities, and it is based on the geometric and physical permanence of objects.
FRONTIERS IN COMPUTATIONAL NEUROSCIENCE
(2021)
Article
Mathematics
Chaudry Masood Khalique, Karabo Plaatjie
Summary: In this article, a two-dimensional generalized shallow water wave equation is investigated by calculating Lie symmetries and performing symmetry reductions. An analytical solution is obtained by utilizing three translation symmetries of the equation, and more closed-form solutions are constructed using Kudryashov's approach. Furthermore, energy and linear momentum conservation laws are computed for the equation by engaging the multiplier approach and Noether's theorem.
Article
Mathematical & Computational Biology
Chaudry Masood Khalique, Kentse Maefo
Summary: This article investigates the Lie symmetries of a (2+1)-dimensional first extended Calogero-Bogoyavlenskii-Schiff equation and performs symmetry reductions, obtaining closed-form solutions with the aid of Kudryashov and (G'/G)-expansion techniques. The solutions are depicted visually and the conserved vectors of the equation are calculated using the multiplier approach and Noether's theorem.
MATHEMATICAL BIOSCIENCES AND ENGINEERING
(2021)