The modern theory of classical mechanics, developed by Lagrange, Hamilton and Noether, attempts to cast all of classical motion in the form of an optimization problem, based on an energy functional called the classical action. The most important advantage of this formalism is the ability to manifestly incorporate and exploit symmetries and conservation laws. This reformulation succeeded for unconstrained and holonomic systems that at most obey position equality constraints. Non-holonomic systems, which obey velocity dependent constraints or position inequality constraints, are abundant in nature and of central relevance for science, engineering and industry. All attempts so far to solve non-holonomic dynamics as a classical action optimization problem have failed. Here we utilize the classical limit of a quantum field theory action principle to construct a novel classical action for non-holonomic systems. We therefore put to rest the 190 year old question of whether classical mechanics is variational, answering in the affirmative. We illustrate and validate our approach by solving three canonical model problems by direct numerical optimization of our new action. The formalism developed in this work significantly extends the reach of action principles to a large class of relevant mechanical systems, opening new avenues for their analysis and control both analytically and numerically.
{"title":"A Unifying Action Principle for Classical Mechanical Systems","authors":"A. Rothkopf, W. A. Horowitz","doi":"arxiv-2409.11063","DOIUrl":"https://doi.org/arxiv-2409.11063","url":null,"abstract":"The modern theory of classical mechanics, developed by Lagrange, Hamilton and\u0000Noether, attempts to cast all of classical motion in the form of an\u0000optimization problem, based on an energy functional called the classical\u0000action. The most important advantage of this formalism is the ability to\u0000manifestly incorporate and exploit symmetries and conservation laws. This\u0000reformulation succeeded for unconstrained and holonomic systems that at most\u0000obey position equality constraints. Non-holonomic systems, which obey velocity\u0000dependent constraints or position inequality constraints, are abundant in\u0000nature and of central relevance for science, engineering and industry. All\u0000attempts so far to solve non-holonomic dynamics as a classical action\u0000optimization problem have failed. Here we utilize the classical limit of a\u0000quantum field theory action principle to construct a novel classical action for\u0000non-holonomic systems. We therefore put to rest the 190 year old question of\u0000whether classical mechanics is variational, answering in the affirmative. We\u0000illustrate and validate our approach by solving three canonical model problems\u0000by direct numerical optimization of our new action. The formalism developed in\u0000this work significantly extends the reach of action principles to a large class\u0000of relevant mechanical systems, opening new avenues for their analysis and\u0000control both analytically and numerically.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259815","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The current study explores the analysis of crack in initially stressed, rotating material strips, drawing insights from singular integral equations. In this work, a self-reinforced material strip with finite thickness and infinite extent, subjected to initial stress and rotational motion, has been considered to examine the Griffith fracture. The edges of the strip are pushed by constant loads from punches moving alongside it. This study makes waves in the material that affect the fracture's movement. A distinct mathematical technique is utilized to streamline the resolution of a pair of singular integral equations featuring First-order singularities. These obtained equations help us understand how the fracture behaves. The force acting at the fracture's edge is modeled using the Dirac delta function. Then, the Hilbert transformation method calculates the stress intensity factor (SIF) at the fracture's edge. Additionally, the study explores various scenarios, including constant intensity force without punch pressure, rotation parameter, initial stress, and isotropy in the strip, deduced from the SIF expression. Numerical computations and graphical analyses are conducted to assess the influence of various factors on SIF in the study. Finally, a comparison is made between the behavior of fractures in the initially stressed and rotating reinforced material strip and those in a standard material strip to identify any differences.
{"title":"Crack Dynamics in Rotating, Initially Stressed Material Strips: A Mathematical Approach","authors":"Soniya Chaudhary, Diksha, Pawan Kumar Sharma","doi":"arxiv-2409.11434","DOIUrl":"https://doi.org/arxiv-2409.11434","url":null,"abstract":"The current study explores the analysis of crack in initially stressed,\u0000rotating material strips, drawing insights from singular integral equations. In\u0000this work, a self-reinforced material strip with finite thickness and infinite\u0000extent, subjected to initial stress and rotational motion, has been considered\u0000to examine the Griffith fracture. The edges of the strip are pushed by constant\u0000loads from punches moving alongside it. This study makes waves in the material\u0000that affect the fracture's movement. A distinct mathematical technique is\u0000utilized to streamline the resolution of a pair of singular integral equations\u0000featuring First-order singularities. These obtained equations help us\u0000understand how the fracture behaves. The force acting at the fracture's edge is\u0000modeled using the Dirac delta function. Then, the Hilbert transformation method\u0000calculates the stress intensity factor (SIF) at the fracture's edge.\u0000Additionally, the study explores various scenarios, including constant\u0000intensity force without punch pressure, rotation parameter, initial stress, and\u0000isotropy in the strip, deduced from the SIF expression. Numerical computations\u0000and graphical analyses are conducted to assess the influence of various factors\u0000on SIF in the study. Finally, a comparison is made between the behavior of\u0000fractures in the initially stressed and rotating reinforced material strip and\u0000those in a standard material strip to identify any differences.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"210 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A relationship is established between the effective Youngs modulus of a two-phase elastic composite and a known mathematical mean value. Specifically, the effective Youngs modulus of a composite obtained from repeated parallel and serial constructions is equal to the arithmetic-geometric mean of the Youngs moduli of the component materials. This result also applies to electric circuits with resistors in repeated parallel and serial connections.
{"title":"Effective Youngs Modulus of Two-Phase Elastic Composites by Repeated Isostrain and Isostress Constructions and Arithmetic-Geometric Mean","authors":"Jiashi Yang","doi":"arxiv-2409.09738","DOIUrl":"https://doi.org/arxiv-2409.09738","url":null,"abstract":"A relationship is established between the effective Youngs modulus of a\u0000two-phase elastic composite and a known mathematical mean value. Specifically,\u0000the effective Youngs modulus of a composite obtained from repeated parallel and\u0000serial constructions is equal to the arithmetic-geometric mean of the Youngs\u0000moduli of the component materials. This result also applies to electric\u0000circuits with resistors in repeated parallel and serial connections.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In mechanics, common energy principles are based on fixed boundary conditions. However, in bridge engineering structures, it is usually necessary to adjust the boundary conditions to make the structure's internal force reasonable and save materials. However, there is currently little theoretical research in this area. To solve this problem, this paper proposes the principle of minimum virtual work for movable boundaries in mechanics through theoretical derivation such as variation method and tensor analysis. It reveals that the exact solution of the mechanical system minimizes the total virtual work of the system among all possible displacements, and the conclusion that the principle of minimum potential energy is a special case of this principle is obtained. At the same time, proposed virtual work boundaries and control conditions, which added to the fundamental equations of mechanics. The general formula of multidimensional variation method for movable boundaries is also proposed, which can be used to easily derive the basic control equations of the mechanical system. The incremental method is used to prove the theory of minimum value in multidimensional space, which extends the Pontryagin's minimum value principle. Multiple bridge examples were listed to demonstrate the extensive practical value of the theory presented in this article. The theory proposed in this article enriches the energy principle and variation method, establishes fundamental equations of mechanics for the structural optimization of movable boundary, and provides a path for active control of mechanical structures, which has important theoretical and engineering practical significance.
{"title":"The principle of minimum virtual work and its application in bridge engineering","authors":"Lukai Xiang","doi":"arxiv-2409.11431","DOIUrl":"https://doi.org/arxiv-2409.11431","url":null,"abstract":"In mechanics, common energy principles are based on fixed boundary\u0000conditions. However, in bridge engineering structures, it is usually necessary\u0000to adjust the boundary conditions to make the structure's internal force\u0000reasonable and save materials. However, there is currently little theoretical\u0000research in this area. To solve this problem, this paper proposes the principle\u0000of minimum virtual work for movable boundaries in mechanics through theoretical\u0000derivation such as variation method and tensor analysis. It reveals that the\u0000exact solution of the mechanical system minimizes the total virtual work of the\u0000system among all possible displacements, and the conclusion that the principle\u0000of minimum potential energy is a special case of this principle is obtained. At\u0000the same time, proposed virtual work boundaries and control conditions, which\u0000added to the fundamental equations of mechanics. The general formula of\u0000multidimensional variation method for movable boundaries is also proposed,\u0000which can be used to easily derive the basic control equations of the\u0000mechanical system. The incremental method is used to prove the theory of\u0000minimum value in multidimensional space, which extends the Pontryagin's minimum\u0000value principle. Multiple bridge examples were listed to demonstrate the\u0000extensive practical value of the theory presented in this article. The theory\u0000proposed in this article enriches the energy principle and variation method,\u0000establishes fundamental equations of mechanics for the structural optimization\u0000of movable boundary, and provides a path for active control of mechanical\u0000structures, which has important theoretical and engineering practical\u0000significance.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Exceptional points (EPs) are spectral singularities in non-Hermitian systems where eigenvalues and their corresponding eigenstates coalesce simultaneously. In this study, we calculate scattering poles in an open spherical solid and propose a depth-first search-based method to identify EPs. Using the proposed method, we numerically identify multiple EPs in a parameter space and confirm the simultaneous degeneracy of scattering poles through numerical experiments. The proposed method and findings enable the exploration of applications in practical three-dimension models.
{"title":"Observation of exceptional points in a spherical open elastic system","authors":"Hiroaki Deguchi, Kei Matsushima, Takayuki Yamada","doi":"arxiv-2409.08560","DOIUrl":"https://doi.org/arxiv-2409.08560","url":null,"abstract":"Exceptional points (EPs) are spectral singularities in non-Hermitian systems\u0000where eigenvalues and their corresponding eigenstates coalesce simultaneously.\u0000In this study, we calculate scattering poles in an open spherical solid and\u0000propose a depth-first search-based method to identify EPs. Using the proposed\u0000method, we numerically identify multiple EPs in a parameter space and confirm\u0000the simultaneous degeneracy of scattering poles through numerical experiments.\u0000The proposed method and findings enable the exploration of applications in\u0000practical three-dimension models.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Symmetries -- whether explicit, latent, or hidden -- are fundamental to understanding topological materials. This work introduces a prototypical spring-mass model that extends beyond established canonical models, revealing topological edge states with distinct profiles at opposite edges. These edge states originate from hidden symmetries that become apparent only in deformation coordinates, as opposed to the conventional displacement coordinates used for bulk-boundary correspondence. Our model realized through the intricate connectivity of a spinner chain, demonstrates experimentally distinct edge states at opposite ends. By extending this framework to two dimensions, we explore the conditions required for such edge waves and their hidden symmetry in deformation coordinates. We also show that these edge states are robust against disorders that respect the hidden symmetry. This research paves the way for advanced material designs with tailored boundary conditions and edge state profiles, offering potential applications in fields such as photonics, acoustics, and mechanical metamaterials.
{"title":"Edge States with Hidden Topology in Spinner Lattices","authors":"Udbhav Vishwakarma, Murthaza Irfan, Georgios Theocharis, Rajesh Chaunsali","doi":"arxiv-2409.07949","DOIUrl":"https://doi.org/arxiv-2409.07949","url":null,"abstract":"Symmetries -- whether explicit, latent, or hidden -- are fundamental to\u0000understanding topological materials. This work introduces a prototypical\u0000spring-mass model that extends beyond established canonical models, revealing\u0000topological edge states with distinct profiles at opposite edges. These edge\u0000states originate from hidden symmetries that become apparent only in\u0000deformation coordinates, as opposed to the conventional displacement\u0000coordinates used for bulk-boundary correspondence. Our model realized through\u0000the intricate connectivity of a spinner chain, demonstrates experimentally\u0000distinct edge states at opposite ends. By extending this framework to two\u0000dimensions, we explore the conditions required for such edge waves and their\u0000hidden symmetry in deformation coordinates. We also show that these edge states\u0000are robust against disorders that respect the hidden symmetry. This research\u0000paves the way for advanced material designs with tailored boundary conditions\u0000and edge state profiles, offering potential applications in fields such as\u0000photonics, acoustics, and mechanical metamaterials.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is shown that when the well-known minimal complementary energy variational principle in linear elastostatics is written in a different form with the strain tensor as an independent variable and the constitutive relation as one of the constraints, the removal of the constraints by Lagrange multipliers leads to a three-field variational principle with the displacement vector, stress field and strain field as independent variables. This three-field variational principle is without constrains and its variational functional is different from those of the existing three-field variational principles. The generalization is not unique. The procedure is mathematical and may be used in other branches of physics.
{"title":"On Generalizations of the Minimal Complementary Energy Variational Principle in Linear Elastostatics","authors":"Jiashi Yang","doi":"arxiv-2409.06875","DOIUrl":"https://doi.org/arxiv-2409.06875","url":null,"abstract":"It is shown that when the well-known minimal complementary energy variational\u0000principle in linear elastostatics is written in a different form with the\u0000strain tensor as an independent variable and the constitutive relation as one\u0000of the constraints, the removal of the constraints by Lagrange multipliers\u0000leads to a three-field variational principle with the displacement vector,\u0000stress field and strain field as independent variables. This three-field\u0000variational principle is without constrains and its variational functional is\u0000different from those of the existing three-field variational principles. The\u0000generalization is not unique. The procedure is mathematical and may be used in\u0000other branches of physics.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is generally assumed that the retarded Lienard-Wiechert electromagnetic field produced by a point particle depends on the acceleration of that source particle. This dependence is not real, it is an illusion. The true electromagnetic interaction is time symmetric (half retarded and half advanced) and depends only on the positions and velocities of the electrically charged particles. A different acceleration of the retarded source particle will result in a different position and velocity of the advanced source particle, changing in this way the Lorentz force felt by the test particle.
{"title":"The illusion of acceleration in the retarded Lienard-Wiechert electromagnetic field","authors":"Calin Galeriu","doi":"arxiv-2409.05338","DOIUrl":"https://doi.org/arxiv-2409.05338","url":null,"abstract":"It is generally assumed that the retarded Lienard-Wiechert electromagnetic\u0000field produced by a point particle depends on the acceleration of that source\u0000particle. This dependence is not real, it is an illusion. The true\u0000electromagnetic interaction is time symmetric (half retarded and half advanced)\u0000and depends only on the positions and velocities of the electrically charged\u0000particles. A different acceleration of the retarded source particle will result\u0000in a different position and velocity of the advanced source particle, changing\u0000in this way the Lorentz force felt by the test particle.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ansgar SiemensZentrum für Optische Quantentechnologien, Fachbereich Physik, Universität Hamburg, Peter SchmelcherZentrum für Optische Quantentechnologien, Fachbereich Physik, Universität HamburgHamburg Center for Ultrafast Imaging, Universität Hamburg
We explore the scattering dynamics of classical Coulomb-interacting clusters of ions confined to a helical geometry. Ion clusters of equally charged particles constrained to a helix can form many-body bound states, i.e. they exhibit stable motion of Coulomb-interacting identical ions. We analyze the scattering and fragmentation behavior of two ion clusters, thereby understanding the rich phenomenology of their dynamics. The scattering dynamics is complex in the sense that it exhibits cascades of decay processes involving strongly varying cluster sizes. These processes are governed by the internal energy flow and the underlying oscillatory many-body potential. We specifically focus on the impact of the collision energy on the dynamics of individual ions during and immediately after the collision of two clusters, and on the internal dynamics of ion clusters that are excited during a cluster collision.
{"title":"Classical scattering and fragmentation of clusters of ions in helical confinement","authors":"Ansgar SiemensZentrum für Optische Quantentechnologien, Fachbereich Physik, Universität Hamburg, Peter SchmelcherZentrum für Optische Quantentechnologien, Fachbereich Physik, Universität HamburgHamburg Center for Ultrafast Imaging, Universität Hamburg","doi":"arxiv-2409.04852","DOIUrl":"https://doi.org/arxiv-2409.04852","url":null,"abstract":"We explore the scattering dynamics of classical Coulomb-interacting clusters\u0000of ions confined to a helical geometry. Ion clusters of equally charged\u0000particles constrained to a helix can form many-body bound states, i.e. they\u0000exhibit stable motion of Coulomb-interacting identical ions. We analyze the\u0000scattering and fragmentation behavior of two ion clusters, thereby\u0000understanding the rich phenomenology of their dynamics. The scattering dynamics\u0000is complex in the sense that it exhibits cascades of decay processes involving\u0000strongly varying cluster sizes. These processes are governed by the internal\u0000energy flow and the underlying oscillatory many-body potential. We specifically\u0000focus on the impact of the collision energy on the dynamics of individual ions\u0000during and immediately after the collision of two clusters, and on the internal\u0000dynamics of ion clusters that are excited during a cluster collision.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"58 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
David Cichra, Vít Průša, K. R. Rajagopal, Casey Rodriguez, Martin Vejvoda
The concept of "effective mass" is frequently used for the simplification of complex lumped parameter systems (discrete dynamical systems) as well as materials that have complicated microstructural features. From the perspective of wave propagation, it is claimed that for some bodies described as metamaterials, the corresponding "effective mass" can be frequency dependent, negative or it may not even be a scalar quantity. The procedure has even led some authors to suggest that Newton's second law needs to be modified within the context of classical continuum mechanics. Such absurd physical conclusions are a consequence of appealing to the notion of "effective mass" with a preconception for the constitutive structure of the metamaterial and using a correct mathematical procedure. We show that such unreasonable physical conclusions would not arise if we were to use the appropriate "effective constitutive relation" for the metamaterial, rather than use the concept of "effective mass" with an incorrect predetermined constitutive relation.
{"title":"The conclusion that metamaterials could have negative mass is a consequence of improper constitutive characterisation","authors":"David Cichra, Vít Průša, K. R. Rajagopal, Casey Rodriguez, Martin Vejvoda","doi":"arxiv-2409.05906","DOIUrl":"https://doi.org/arxiv-2409.05906","url":null,"abstract":"The concept of \"effective mass\" is frequently used for the simplification of\u0000complex lumped parameter systems (discrete dynamical systems) as well as\u0000materials that have complicated microstructural features. From the perspective\u0000of wave propagation, it is claimed that for some bodies described as\u0000metamaterials, the corresponding \"effective mass\" can be frequency dependent,\u0000negative or it may not even be a scalar quantity. The procedure has even led\u0000some authors to suggest that Newton's second law needs to be modified within\u0000the context of classical continuum mechanics. Such absurd physical conclusions\u0000are a consequence of appealing to the notion of \"effective mass\" with a\u0000preconception for the constitutive structure of the metamaterial and using a\u0000correct mathematical procedure. We show that such unreasonable physical\u0000conclusions would not arise if we were to use the appropriate \"effective\u0000constitutive relation\" for the metamaterial, rather than use the concept of\u0000\"effective mass\" with an incorrect predetermined constitutive relation.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"2022 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142186292","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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