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19 pages, 313 KiB

Open AccessArticle

Non-Relativistic and Relativistic Lagrangian Pairing in Fluid Mechanics Inspired by Quantum Theory

by Sara Ismail-Sutton, Markus Scholle and Philip H. Gaskell

Symmetry 2025, 17(3), 315; https://doi.org/10.3390/sym17030315 - 20 Feb 2025

Viewed by 803

The pairing of non-relativistic and relativistic Lagrangians within the context of fluid mechanics, advancing methodologies for constructing Poincare-invariant Lagrangians, is explored. Through leveraging symmetries and Noether’s theorem in an inverse framework, three primary cases are investigated: potential flow, barotropic flow expressed in terms [...] Read more.

The pairing of non-relativistic and relativistic Lagrangians within the context of fluid mechanics, advancing methodologies for constructing Poincare-invariant Lagrangians, is explored. Through leveraging symmetries and Noether’s theorem in an inverse framework, three primary cases are investigated: potential flow, barotropic flow expressed in terms of Clebsch variables, and an extended Clebsch Lagrangian incorporating thermodynamic effects. To ensure physical correctness, the eigenvalue relation of the energy–momentum tensor, together with velocity normalisation, are applied as key criteria. The findings confirm that the relativistic Lagrangians successfully reduce to their non-relativistic counterparts in the limit

c \to \infty

. These results demonstrate a systematic approach that enhances the relationship between symmetries and variational formulations, providing the advantage of deriving Lagrangians that unify non-relativistic and relativistic theories. Full article

(This article belongs to the Section Physics)

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55 pages, 18955 KiB

Open AccessArticle

Structured Dynamics in the Algorithmic Agent

by Giulio Ruffini, Francesca Castaldo and Jakub Vohryzek

Entropy 2025, 27(1), 90; https://doi.org/10.3390/e27010090 - 19 Jan 2025

Cited by 1 | Viewed by 1327

Abstract

In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how [...] Read more.

In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how the requirement of tracking natural data drives the structural and dynamical properties of the agent. We first formalize the notion of a generative model using the language of symmetry from group theory, specifically employing Lie pseudogroups to describe the continuous transformations that characterize invariance in natural data. Then, adopting a generic neural network as a proxy for the agent dynamical system and drawing parallels to Noether’s theorem in physics, we demonstrate that data tracking forces the agent to mirror the symmetry properties of the generative world model. This dual constraint on the agent’s constitutive parameters and dynamical repertoire enforces a hierarchical organization consistent with the manifold hypothesis in the neural network. Our findings bridge perspectives from algorithmic information theory (Kolmogorov complexity, compressive modeling), symmetry (group theory), and dynamics (conservation laws, reduced manifolds), offering insights into the neural correlates of agenthood and structured experience in natural systems, as well as the design of artificial intelligence and computational models of the brain. Full article

(This article belongs to the Special Issue The Mathematics of Structured Experience: Exploring Dynamics, Topology, and Complexity in the Brain)

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15 pages, 373 KiB

Open AccessArticle

Conserved Vectors, Analytic Solutions and Numerical Simulation of Soliton Collisions of the Modified Gardner Equation

by Chaudry Masood Khalique, Carel Olivier and Boikanyo Pretty Sebogodi

AppliedMath 2024, 4(4), 1471-1485; https://doi.org/10.3390/appliedmath4040078 - 26 Nov 2024

Viewed by 737

Abstract

This paper aims to study the modified Gardner (mG) equation that was proposed in the literature a short while ago. We first construct conserved vectors of the mG equation by invoking three different techniques; namely the method of multiplier, Noether’s theorem, and the [...] Read more.

This paper aims to study the modified Gardner (mG) equation that was proposed in the literature a short while ago. We first construct conserved vectors of the mG equation by invoking three different techniques; namely the method of multiplier, Noether’s theorem, and the conservation theorem owing to Ibragimov. Thereafter, we present exact solutions to the mG equation by invoking a complete discrimination system for the fifth degree polynomial. Finally, we simulate collisions of solitons for the mG equation. Full article

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17 pages, 293 KiB

Open AccessArticle

Lie Symmetry Analysis, Closed-Form Solutions, and Conservation Laws for the Camassa–Holm Type Equation

by Jonathan Lebogang Bodibe and Chaudry Masood Khalique

Math. Comput. Appl. 2024, 29(5), 92; https://doi.org/10.3390/mca29050092 - 10 Oct 2024

Cited by 1 | Viewed by 1099

Abstract

In this paper, we study the Camassa–Holm type equation, which has applications in mathematical physics and engineering. Its applications extend across disciplines, contributing to our understanding of complex systems and helping to develop innovative solutions in diverse areas of research. Our main aim [...] Read more.

In this paper, we study the Camassa–Holm type equation, which has applications in mathematical physics and engineering. Its applications extend across disciplines, contributing to our understanding of complex systems and helping to develop innovative solutions in diverse areas of research. Our main aim is to construct closed-form solutions of the equation using a powerful technique, namely the Lie group analysis method. Firstly, we derive the Lie point symmetries of the equation. Thereafter, the equation is reduced to non-linear ordinary differential equations using symmetry reductions. Furthermore, the solutions of the equation are derived using the extended Jacobi elliptic function technique, the simplest equation method, and the power series method. In conclusion, we construct conservation laws for the equation using Noether’s theorem and the multiplier approach, which plays a crucial role in understanding the behavior of non-linear equations, especially in physics and engineering, and these laws are derived from fundamental principles such as the conservation of mass, energy, momentum, and angular momentum. Full article

(This article belongs to the Special Issue Symmetry Methods for Solving Differential Equations)

9 pages, 250 KiB

Open AccessArticle

On the Damped Pinney Equation from Noether Symmetry Principles

by Fernando Haas

Symmetry 2024, 16(10), 1310; https://doi.org/10.3390/sym16101310 - 4 Oct 2024

Viewed by 863

Abstract

There are several versions of the damped form of the celebrated Pinney equation, which is the natural partner of the undamped time-dependent harmonic oscillator. In this work, these dissipative versions of the Pinney equation are briefly reviewed. We show that Noether’s theorem for [...] Read more.

There are several versions of the damped form of the celebrated Pinney equation, which is the natural partner of the undamped time-dependent harmonic oscillator. In this work, these dissipative versions of the Pinney equation are briefly reviewed. We show that Noether’s theorem for the usual time-dependent harmonic oscillator as a guiding principle for derivation of the Pinney equation also works in the damped case, selecting a Noether symmetry-based damped Pinney equation. The results are extended to general nonlinear damped Ermakov systems. A certain time-rescaling always allows to remove the damping from the final equations. Full article

(This article belongs to the Special Issue Symmetry in Hamiltonian Dynamical Systems)

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18 pages, 1023 KiB

Open AccessEditor’s ChoiceReview

Nuclear Symmetry Energy in Strongly Interacting Matter: Past, Present and Future

by Jirina R. Stone

Symmetry 2024, 16(8), 1038; https://doi.org/10.3390/sym16081038 - 13 Aug 2024

Cited by 1 | Viewed by 1958

Abstract

The concept of symmetry under various transformations of quantities describing basic natural phenomena is one of the fundamental principles in the mathematical formulation of physical laws. Starting with Noether’s theorems, we highlight some well–known examples of global symmetries and symmetry breaking on the [...] Read more.

The concept of symmetry under various transformations of quantities describing basic natural phenomena is one of the fundamental principles in the mathematical formulation of physical laws. Starting with Noether’s theorems, we highlight some well–known examples of global symmetries and symmetry breaking on the particle level, such as the separation of strong and electroweak interactions and the Higgs mechanism, which gives mass to leptons and quarks. The relation between symmetry energy and charge symmetry breaking at both the nuclear level (under the interchange of protons and neutrons) and the particle level (under the interchange of u and d quarks) forms the main subject of this work. We trace the concept of symmetry energy from its introduction in the simple semi-empirical mass formula and liquid drop models to the most sophisticated non-relativistic, relativistic, and ab initio models. Methods used to extract symmetry energy attributes, utilizing the most significant combined terrestrial and astrophysical data and theoretical predictions, are reviewed. This includes properties of finite nuclei, heavy-ion collisions, neutron stars, gravitational waves, and parity–violating electron scattering experiments such as CREX and PREX, for which selected examples are provided. Finally, future approaches to investigation of the symmetry energy and its properties are discussed. Full article

(This article belongs to the Special Issue Strongly Interacting Matter at Extreme Conditions—the Role of Symmetry Energy in Nuclear Physics and Astrophysics)

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14 pages, 3344 KiB

Open AccessArticle

Anisotropic Hyperelastic Strain Energy Function for Carbon Fiber Woven Fabrics

by Renye Cai, Heng Zhang, Chenxiang Lai, Zexin Yu, Xiangkun Zeng, Min Wu, Yankun Wang, Qisen Huang, Yiwei Zhu and Chunyu Kong

Materials 2024, 17(10), 2456; https://doi.org/10.3390/ma17102456 - 20 May 2024

Viewed by 1662

Abstract

The present paper introduces an innovative strain energy function (SEF) for incompressible anisotropic fiber-reinforced materials. This SEF is specifically designed to understand the mechanical behavior of carbon fiber-woven fabric. The considered model combines polyconvex invariants forming an integrity basisin polynomial form, which is [...] Read more.

The present paper introduces an innovative strain energy function (SEF) for incompressible anisotropic fiber-reinforced materials. This SEF is specifically designed to understand the mechanical behavior of carbon fiber-woven fabric. The considered model combines polyconvex invariants forming an integrity basisin polynomial form, which is inspired by the application of Noether’s theorem. A single solution can be obtained during the identification because of the relationship between the SEF we have constructed and the material parameters, which are linearly dependent. The six material parameters were precisely determined through a comparison between the closed-form solutions from our model and the corresponding tensile experimental data with different stretching ratios, with determination coefficients consistently reaching a remarkable value of 0.99. When considering only uniaxial tensile tests, our model can be simplified from a quadratic polynomial to a linear polynomial, thereby reducing the number of material parameters required from six to four, while the fidelity of the model’s predictive accuracy remains unaltered. The comparison between the results of numerical calculations and experiments proves the efficiency and accuracy of the method. Full article

(This article belongs to the Section Carbon Materials)

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12 pages, 293 KiB

Open AccessArticle

Noether’s Theorem of Herglotz Type for Fractional Lagrange System with Nonholonomic Constraints

by Yuanyuan Deng and Yi Zhang

Fractal Fract. 2024, 8(5), 296; https://doi.org/10.3390/fractalfract8050296 - 18 May 2024

Cited by 3 | Viewed by 886

Abstract

This research aims to investigate the Noether symmetry and conserved quantity for the fractional Lagrange system with nonholonomic constraints, which are based on the Herglotz principle. Firstly, the fractional-order Herglotz principle is given, and the Herglotz-type fractional-order differential equations of motion for the [...] Read more.

This research aims to investigate the Noether symmetry and conserved quantity for the fractional Lagrange system with nonholonomic constraints, which are based on the Herglotz principle. Firstly, the fractional-order Herglotz principle is given, and the Herglotz-type fractional-order differential equations of motion for the fractional Lagrange system with nonholonomic constraints are derived. Secondly, by introducing infinitesimal generating functions of space and time, the Noether symmetry of the Herglotz type is defined, along with its criteria, and the conserved quantity of the Herglotz type is given. Finally, to demonstrate how to use this method, two examples are provided. Full article

27 pages, 401 KiB

Open AccessReview

A Geometric Approach to the Sundman Transformation and Its Applications to Integrability

by José F. Cariñena

Symmetry 2024, 16(5), 568; https://doi.org/10.3390/sym16050568 - 6 May 2024

Cited by 2 | Viewed by 2108

Abstract

A geometric approach to the integrability and reduction of dynamical systems, both when dealing with systems of differential equations and in classical physics, is developed from a modern perspective. The main ingredients of this analysis are infinitesimal symmetries and tensor fields that are [...] Read more.

A geometric approach to the integrability and reduction of dynamical systems, both when dealing with systems of differential equations and in classical physics, is developed from a modern perspective. The main ingredients of this analysis are infinitesimal symmetries and tensor fields that are invariant under the given dynamics. A particular emphasis is placed on the existence of alternative invariant volume forms and the associated Jacobi multiplier theory, and then the Hojman symmetry theory is developed as a complement to the Noether theorem and non-Noether constants of motion. We also recall the geometric approach to Sundman infinitesimal time-reparametrisation for autonomous systems of first-order differential equations and some of its applications to integrability, and an analysis of how to define Sundman transformations for autonomous systems of second-order differential equations is proposed, which shows the necessity of considering alternative tangent bundle structures. A short description of alternative tangent structures is provided, and an application to integrability, namely, the linearisability of scalar second-order differential equations under generalised Sundman transformations, is developed. Full article

(This article belongs to the Special Issue Selected Papers Symmetry 2023—The Fourth Edition of the International Conference on Symmetry)

14 pages, 283 KiB

Open AccessArticle

Noether Symmetry of Multi-Time-Delay Non-Conservative Mechanical System and Its Conserved Quantity

by Xingyu Ji, Zhengwei Yang and Xianghua Zhai

Symmetry 2024, 16(4), 475; https://doi.org/10.3390/sym16040475 - 14 Apr 2024

Cited by 1 | Viewed by 1113

Abstract

The study of multi-time-delay dynamical systems has highlighted many challenges, especially regarding the solution and analysis of multi-time-delay equations. The symmetry and conserved quantity are two important and effective essential properties for understanding complex dynamical behavior. In this study, a multi-time-delay non-conservative mechanical [...] Read more.

The study of multi-time-delay dynamical systems has highlighted many challenges, especially regarding the solution and analysis of multi-time-delay equations. The symmetry and conserved quantity are two important and effective essential properties for understanding complex dynamical behavior. In this study, a multi-time-delay non-conservative mechanical system is investigated. Firstly, the multi-time-delay Hamilton principle is proposed. Then, multi-time-delay non-conservative dynamical equations are deduced. Secondly, depending on the infinitesimal group transformations, the invariance of the multi-time-delay Hamilton action is studied, and Noether symmetry, Noether quasi-symmetry, and generalized Noether quasi-symmetry are discussed. Finally, Noether-type conserved quantities for a multi-time-delay Lagrangian system and a multi-time-delay non-conservative mechanical system are obtained. Two examples in terms of a multi-time-delay non-conservative mechanical system and a multi-time-delay Lagrangian system are given. Full article

(This article belongs to the Section Mathematics)

13 pages, 266 KiB

Open AccessArticle

Symmetry Analysis of the Two-Dimensional Stationary Gas Dynamics Equations in Lagrangian Coordinates

by Sergey V. Meleshko and Evgeniy I. Kaptsov

Mathematics 2024, 12(6), 879; https://doi.org/10.3390/math12060879 - 16 Mar 2024

Cited by 5 | Viewed by 927

Abstract

This article analyzes the symmetry of two-dimensional stationary gas dynamics equations in Lagrangian coordinates, including the search for equivalence transformations, the group classification of equations, the derivation of group foliations, and the construction of conservation laws. The consideration of equations in Lagrangian coordinates [...] Read more.

This article analyzes the symmetry of two-dimensional stationary gas dynamics equations in Lagrangian coordinates, including the search for equivalence transformations, the group classification of equations, the derivation of group foliations, and the construction of conservation laws. The consideration of equations in Lagrangian coordinates significantly simplifies the procedure for obtaining conservation laws, which are derived using the Noether theorem. The final part of the work is devoted to group foliations of the gas dynamics equations, including for the nonstationary isentropic case. The group foliations approach is usually employed for equations that admit infinite-dimensional groups of transformations (which is exactly the case for the gas dynamics equations in Lagrangian coordinates) and may make it possible to simplify their further analysis. The results obtained in this regard generalize previously known results for the two-dimensional shallow water equations in Lagrangian coordinates. Full article

(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)

57 pages, 754 KiB

Open AccessFeature PaperReview

Roadmap of the Multiplier Method for Partial Differential Equations

by Juan Arturo Alvarez-Valdez and Guillermo Fernandez-Anaya

Mathematics 2023, 11(22), 4572; https://doi.org/10.3390/math11224572 - 7 Nov 2023

Cited by 1 | Viewed by 2741

Abstract

This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking [...] Read more.

This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking at the history of the method and synthesizing the newest developments, we hope to give it the attention that it deserves to help develop the vast amount of work still needed to understand it and make the best use of it. It is also an interesting and a relevant method in itself that could possibly give interesting results in areas of mathematics such as modern algebra, group theory, topology, etc. The paper will be structured in such a manner that the last review known for this method will be presented to understand the theoretical framework of the method and then later work done will be presented. The information of four recent papers further developing the method will be synthesized and presented in such a manner that anyone interested in learning this method will have the most relevant information available and have all details cited for checking. Full article

(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications, 2nd Edition)

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13 pages, 281 KiB

Open AccessArticle

Diffeomorphism Covariance of the Canonical Barbero–Immirzi–Holst Triad Theory

by Donald Salisbury

Universe 2023, 9(11), 458; https://doi.org/10.3390/universe9110458 - 25 Oct 2023

Viewed by 1520

Abstract

The vanishing phase space generator of the full four-dimensional diffeomorphism-related symmetry group in the context of the Barbero–Immirz–Holst Lagrangian is derived directly, for the first time, from Noether’s second theorem. Its applicability in the construction of classical diffeomorphism invariants is reviewed. Full article

(This article belongs to the Special Issue Recent Advances in Gravity: A Themed Issue in Honor of Prof. Jorge Pullin on His 60th Anniversary)

35 pages, 482 KiB

Open AccessArticle

General Fractional Noether Theorem and Non-Holonomic Action Principle

by Vasily E. Tarasov

Mathematics 2023, 11(20), 4400; https://doi.org/10.3390/math11204400 - 23 Oct 2023

Cited by 5 | Viewed by 1763

Abstract

Using general fractional calculus (GFC) of the Luchko form and non-holonomic variational equations of Sedov type, generalizations of the standard action principle and first Noether theorem are proposed and proved for non-local (general fractional) non-Lagrangian field theory. The use of the GFC allows [...] Read more.

Using general fractional calculus (GFC) of the Luchko form and non-holonomic variational equations of Sedov type, generalizations of the standard action principle and first Noether theorem are proposed and proved for non-local (general fractional) non-Lagrangian field theory. The use of the GFC allows us to take into account a wide class of nonlocalities in space and time compared to the usual fractional calculus. The use of non-holonomic variation equations allows us to consider field equations and equations of motion for a wide class of irreversible processes, dissipative and open systems, non-Lagrangian and non-Hamiltonian field theories and systems. In addition, the proposed GF action principle and the GF Noether theorem are generalized to equations containing general fractional integrals (GFI) in addition to general fractional derivatives (GFD). Examples of field equations with GFDs and GFIs are suggested. The energy–momentum tensor, orbital angular-momentum tensor and spin angular-momentum tensor are given for general fractional non-Lagrangian field theories. Examples of application of generalized first Noether’s theorem are suggested for scalar end vector fields of non-Lagrangian field theory. Full article

(This article belongs to the Section E4: Mathematical Physics)

13 pages, 309 KiB

Open AccessArticle

A Relationship between the Schrödinger and Klein–Gordon Theories and Continuity Conditions for Scattering Problems

by Markus Scholle and Marcel Mellmann

Symmetry 2023, 15(9), 1667; https://doi.org/10.3390/sym15091667 - 29 Aug 2023

Cited by 1 | Viewed by 1264

Abstract

A rigorous analysis is undertaken based on the analysis of both Galilean and Lorentz (Poincaré) invariance in field theories in general in order to (i) identify a unique analytical scheme for canonical pairs of Lagrangians, one of them having Galilean, the other one [...] Read more.

A rigorous analysis is undertaken based on the analysis of both Galilean and Lorentz (Poincaré) invariance in field theories in general in order to (i) identify a unique analytical scheme for canonical pairs of Lagrangians, one of them having Galilean, the other one Poincaré invariance; and (ii) to obtain transition conditions for the state function purely from Hamilton’s principle and extended Noether’s theorem applied to the aforementioned symmetries. The general analysis is applied on Schrödinger and Klein–Gordon theory, identifying them as a canonical pair in the sense of (i) and exemplified for the scattering problem for both theories for a particle beam against a potential step in order to show that the transition conditions that result according to (ii) in a ‘natural’ way reproduce the well-known ‘methodical’ continuity conditions commonly found in the literature, at least in relevant cases, closing a relevant argumentation gap in quantum mechanical scattering problems. Full article

(This article belongs to the Special Issue Symmetry and Asymmetry in Quantum Mechanics)

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