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A Unifying Action Principle for Classical Mechanical Systems 经典力学系统的统一作用原理
Pub Date : 2024-09-17 DOI: arxiv-2409.11063
A. Rothkopf, W. A. Horowitz
The modern theory of classical mechanics, developed by Lagrange, Hamilton andNoether, attempts to cast all of classical motion in the form of anoptimization problem, based on an energy functional called the classicalaction. The most important advantage of this formalism is the ability tomanifestly incorporate and exploit symmetries and conservation laws. Thisreformulation succeeded for unconstrained and holonomic systems that at mostobey position equality constraints. Non-holonomic systems, which obey velocitydependent constraints or position inequality constraints, are abundant innature and of central relevance for science, engineering and industry. Allattempts so far to solve non-holonomic dynamics as a classical actionoptimization problem have failed. Here we utilize the classical limit of aquantum field theory action principle to construct a novel classical action fornon-holonomic systems. We therefore put to rest the 190 year old question ofwhether classical mechanics is variational, answering in the affirmative. Weillustrate and validate our approach by solving three canonical model problemsby direct numerical optimization of our new action. The formalism developed inthis work significantly extends the reach of action principles to a large classof relevant mechanical systems, opening new avenues for their analysis andcontrol both analytically and numerically.
由拉格朗日、汉密尔顿和诺特发展起来的现代经典力学理论,试图将所有经典运动以优化问题的形式呈现出来,其基础是被称为经典作用的能量函数。这种形式主义最重要的优点是能够明确地纳入并利用对称性和守恒定律。这种重构方法适用于无约束系统和整体系统,这些系统最多服从位置相等约束。服从速度相关约束或位置不等式约束的非整体系统在自然界中比比皆是,而且与科学、工程和工业息息相关。迄今为止,将非全局动力学作为经典行动优化问题来解决的所有尝试都以失败告终。在这里,我们利用量子场论作用原理的经典极限,为非全局系统构建了一种新的经典作用。因此,我们平息了长达 190 年之久的经典力学是否具有变分性的问题,给出了肯定的答案。我们通过对新作用的直接数值优化解决了三个典型模型问题,从而证明并验证了我们的方法。这项工作中建立的形式主义极大地扩展了作用原理在一大类相关力学系统中的应用范围,为分析和数值控制这些系统开辟了新的途径。
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引用次数: 0
Crack Dynamics in Rotating, Initially Stressed Material Strips: A Mathematical Approach 旋转初始应力材料带的裂缝动力学:数学方法
Pub Date : 2024-09-15 DOI: arxiv-2409.11434
Soniya Chaudhary, Diksha, Pawan Kumar Sharma
The current study explores the analysis of crack in initially stressed,rotating material strips, drawing insights from singular integral equations. Inthis work, a self-reinforced material strip with finite thickness and infiniteextent, subjected to initial stress and rotational motion, has been consideredto examine the Griffith fracture. The edges of the strip are pushed by constantloads from punches moving alongside it. This study makes waves in the materialthat affect the fracture's movement. A distinct mathematical technique isutilized to streamline the resolution of a pair of singular integral equationsfeaturing First-order singularities. These obtained equations help usunderstand how the fracture behaves. The force acting at the fracture's edge ismodeled using the Dirac delta function. Then, the Hilbert transformation methodcalculates the stress intensity factor (SIF) at the fracture's edge.Additionally, the study explores various scenarios, including constantintensity force without punch pressure, rotation parameter, initial stress, andisotropy in the strip, deduced from the SIF expression. Numerical computationsand graphical analyses are conducted to assess the influence of various factorson SIF in the study. Finally, a comparison is made between the behavior offractures in the initially stressed and rotating reinforced material strip andthose in a standard material strip to identify any differences.
本研究从奇异积分方程中汲取灵感,探讨了初始受力旋转材料带的裂纹分析。在这项研究中,我们考虑了一个具有有限厚度和无限延伸的自加固带材,该带材受到初始应力和旋转运动的作用,以研究格里菲斯断裂。带材的边缘受到沿其运动的冲头的恒定载荷的推动。这项研究在材料中产生了影响断口运动的波浪。利用一种独特的数学技术简化了一对具有一阶奇异性的奇异积分方程的解析。这些求得的方程有助于我们理解断裂的行为。利用 Dirac delta 函数对作用在断裂边缘的力进行建模。此外,研究还探讨了各种情况,包括从 SIF 表达式推导出的无冲压的恒定强度力、旋转参数、初始应力和带状各向异性。通过数值计算和图形分析,评估了研究中各种因素对 SIF 的影响。最后,比较了初始应力和旋转增强材料带材的断裂行为与标准材料带材的断裂行为,以确定两者之间的差异。
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引用次数: 0
Effective Youngs Modulus of Two-Phase Elastic Composites by Repeated Isostrain and Isostress Constructions and Arithmetic-Geometric Mean 通过重复等应变和等应力构造以及算术几何平均数计算两相弹性复合材料的有效杨氏模量
Pub Date : 2024-09-15 DOI: arxiv-2409.09738
Jiashi Yang
A relationship is established between the effective Youngs modulus of atwo-phase elastic composite and a known mathematical mean value. Specifically,the effective Youngs modulus of a composite obtained from repeated parallel andserial constructions is equal to the arithmetic-geometric mean of the Youngsmoduli of the component materials. This result also applies to electriccircuits with resistors in repeated parallel and serial connections.
在两相弹性复合材料的有效杨氏模量与已知数学平均值之间建立了一种关系。具体来说,通过重复并联和串联结构得到的复合材料的有效杨氏模量等于各组成材料杨氏模量的算术几何平均数。这一结果同样适用于电阻器重复并联和串联的电路。
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引用次数: 0
The principle of minimum virtual work and its application in bridge engineering 最小虚功原理及其在桥梁工程中的应用
Pub Date : 2024-09-14 DOI: arxiv-2409.11431
Lukai Xiang
In mechanics, common energy principles are based on fixed boundaryconditions. However, in bridge engineering structures, it is usually necessaryto adjust the boundary conditions to make the structure's internal forcereasonable and save materials. However, there is currently little theoreticalresearch in this area. To solve this problem, this paper proposes the principleof minimum virtual work for movable boundaries in mechanics through theoreticalderivation such as variation method and tensor analysis. It reveals that theexact solution of the mechanical system minimizes the total virtual work of thesystem among all possible displacements, and the conclusion that the principleof minimum potential energy is a special case of this principle is obtained. Atthe same time, proposed virtual work boundaries and control conditions, whichadded to the fundamental equations of mechanics. The general formula ofmultidimensional variation method for movable boundaries is also proposed,which can be used to easily derive the basic control equations of themechanical system. The incremental method is used to prove the theory ofminimum value in multidimensional space, which extends the Pontryagin's minimumvalue principle. Multiple bridge examples were listed to demonstrate theextensive practical value of the theory presented in this article. The theoryproposed in this article enriches the energy principle and variation method,establishes fundamental equations of mechanics for the structural optimizationof movable boundary, and provides a path for active control of mechanicalstructures, which has important theoretical and engineering practicalsignificance.
在力学中,常见的能量原理都是基于固定的边界条件。然而,在桥梁工程结构中,通常需要对边界条件进行调整,以使结构的内力合理并节省材料。然而,目前这方面的理论研究还很少。为了解决这一问题,本文通过变分法和张量分析等理论推导,提出了力学中活动边界的最小虚功原理。它揭示了力学系统的精确解在所有可能的位移中使系统的总虚功最小,并得到了最小势能原理是该原理的特例的结论。同时,提出了虚功边界和控制条件,补充了力学基本方程。还提出了动边界多维变化法的一般公式,可用于方便地推导主题机械系统的基本控制方程。用增量法证明了多维空间最小值理论,扩展了庞特里亚金最小值原理。文章列举了多个桥梁实例,以证明本文提出的理论具有广泛的实用价值。本文提出的理论丰富了能量原理和变异法,建立了动边界结构优化的力学基本方程,为机械结构的主动控制提供了路径,具有重要的理论和工程实践意义。
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引用次数: 0
Observation of exceptional points in a spherical open elastic system 球形开放弹性系统中的异常点观测
Pub Date : 2024-09-13 DOI: arxiv-2409.08560
Hiroaki Deguchi, Kei Matsushima, Takayuki Yamada
Exceptional points (EPs) are spectral singularities in non-Hermitian systemswhere eigenvalues and their corresponding eigenstates coalesce simultaneously.In this study, we calculate scattering poles in an open spherical solid andpropose a depth-first search-based method to identify EPs. Using the proposedmethod, we numerically identify multiple EPs in a parameter space and confirmthe simultaneous degeneracy of scattering poles through numerical experiments.The proposed method and findings enable the exploration of applications inpractical three-dimension models.
在本研究中,我们计算了开放球形固体中的散射极点,并提出了一种基于深度优先搜索的方法来识别EPs。利用所提出的方法,我们在参数空间中数值识别了多个 EPs,并通过数值实验证实了散射极点的同时退化性。
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引用次数: 0
Edge States with Hidden Topology in Spinner Lattices 旋转晶格中具有隐藏拓扑结构的边缘状态
Pub Date : 2024-09-12 DOI: arxiv-2409.07949
Udbhav Vishwakarma, Murthaza Irfan, Georgios Theocharis, Rajesh Chaunsali
Symmetries -- whether explicit, latent, or hidden -- are fundamental tounderstanding topological materials. This work introduces a prototypicalspring-mass model that extends beyond established canonical models, revealingtopological edge states with distinct profiles at opposite edges. These edgestates originate from hidden symmetries that become apparent only indeformation coordinates, as opposed to the conventional displacementcoordinates used for bulk-boundary correspondence. Our model realized throughthe intricate connectivity of a spinner chain, demonstrates experimentallydistinct edge states at opposite ends. By extending this framework to twodimensions, we explore the conditions required for such edge waves and theirhidden symmetry in deformation coordinates. We also show that these edge statesare robust against disorders that respect the hidden symmetry. This researchpaves the way for advanced material designs with tailored boundary conditionsand edge state profiles, offering potential applications in fields such asphotonics, acoustics, and mechanical metamaterials.
对称性--无论是显性的、潜在的还是隐性的--都是了解拓扑材料的基础。这项研究引入了一种原型弹簧质量模型,它超越了既有的典型模型,揭示了拓扑边缘态,在相对边缘具有独特的轮廓。这些边缘态源于隐藏的对称性,只有在形变坐标(而不是用于体界对应的传统位移坐标)中才会显现出来。我们的模型通过纺锤链错综复杂的连通性来实现,在实验中展示了相对两端不同的边缘状态。通过将这一框架扩展到二维,我们探索了这种边缘波所需的条件及其在形变坐标中隐藏的对称性。我们还证明,这些边缘状态对尊重隐藏对称性的紊乱具有鲁棒性。这项研究为具有定制边界条件和边缘状态剖面的先进材料设计铺平了道路,为光子学、声学和机械超材料等领域提供了潜在应用。
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引用次数: 0
On Generalizations of the Minimal Complementary Energy Variational Principle in Linear Elastostatics 论线性弹性力学中最小互补能量变分原理的一般化
Pub Date : 2024-09-10 DOI: arxiv-2409.06875
Jiashi Yang
It is shown that when the well-known minimal complementary energy variationalprinciple in linear elastostatics is written in a different form with thestrain tensor as an independent variable and the constitutive relation as oneof the constraints, the removal of the constraints by Lagrange multipliersleads to a three-field variational principle with the displacement vector,stress field and strain field as independent variables. This three-fieldvariational principle is without constrains and its variational functional isdifferent from those of the existing three-field variational principles. Thegeneralization is not unique. The procedure is mathematical and may be used inother branches of physics.
研究表明,当把线性弹性力学中著名的最小互补能量变分原理换一种形式写成以应变张量为自变量、以构成关系为约束条件之一时,通过拉格朗日乘法器去除约束条件,就会得到以位移矢量、应力场和应变场为自变量的三场变分原理。这种三场变分原理没有约束条件,其变分函数与现有的三场变分原理不同。这种概括并不是唯一的。该过程是数学化的,可用于物理学的其他分支。
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引用次数: 0
The illusion of acceleration in the retarded Lienard-Wiechert electromagnetic field 延缓的李纳德-维切特电磁场中的加速幻觉
Pub Date : 2024-09-09 DOI: arxiv-2409.05338
Calin Galeriu
It is generally assumed that the retarded Lienard-Wiechert electromagneticfield produced by a point particle depends on the acceleration of that sourceparticle. This dependence is not real, it is an illusion. The trueelectromagnetic interaction is time symmetric (half retarded and half advanced)and depends only on the positions and velocities of the electrically chargedparticles. A different acceleration of the retarded source particle will resultin a different position and velocity of the advanced source particle, changingin this way the Lorentz force felt by the test particle.
一般认为,点粒子产生的迟滞李纳-维切特电磁场取决于该源粒子的加速度。这种依赖关系并不真实,只是一种假象。真正的电磁相互作用是时间对称的(一半迟滞,一半提前),只取决于带电粒子的位置和速度。延迟源粒子的加速度不同,会导致提前源粒子的位置和速度不同,从而改变测试粒子感受到的洛伦兹力。
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引用次数: 0
Classical scattering and fragmentation of clusters of ions in helical confinement 螺旋约束下离子簇的经典散射和碎裂
Pub Date : 2024-09-07 DOI: arxiv-2409.04852
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 clustersof ions confined to a helical geometry. Ion clusters of equally chargedparticles constrained to a helix can form many-body bound states, i.e. theyexhibit stable motion of Coulomb-interacting identical ions. We analyze thescattering and fragmentation behavior of two ion clusters, therebyunderstanding the rich phenomenology of their dynamics. The scattering dynamicsis complex in the sense that it exhibits cascades of decay processes involvingstrongly varying cluster sizes. These processes are governed by the internalenergy flow and the underlying oscillatory many-body potential. We specificallyfocus on the impact of the collision energy on the dynamics of individual ionsduring and immediately after the collision of two clusters, and on the internaldynamics of ion clusters that are excited during a cluster collision.
我们探索了限制在螺旋几何中的经典库仑相互作用离子簇的散射动力学。受限于螺旋形的带相同电荷粒子的离子群可以形成多体束缚态,即它们表现出库仑相互作用的相同离子的稳定运动。我们分析了两个离子团簇的散射和碎裂行为,从而理解了它们丰富的动力学现象。散射动力学是复杂的,因为它表现出一连串的衰变过程,涉及强烈变化的离子簇大小。这些过程受内部能量流和底层振荡多体势能的支配。我们特别关注碰撞能量对单个离子在两个团簇碰撞期间和碰撞后的动力学的影响,以及在团簇碰撞期间被激发的离子团簇的内部动力学。
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引用次数: 0
The conclusion that metamaterials could have negative mass is a consequence of improper constitutive characterisation 超材料可能具有负质量的结论是不恰当的构成特征描述的结果
Pub Date : 2024-09-06 DOI: arxiv-2409.05906
David Cichra, Vít Průša, K. R. Rajagopal, Casey Rodriguez, Martin Vejvoda
The concept of "effective mass" is frequently used for the simplification ofcomplex lumped parameter systems (discrete dynamical systems) as well asmaterials that have complicated microstructural features. From the perspectiveof wave propagation, it is claimed that for some bodies described asmetamaterials, the corresponding "effective mass" can be frequency dependent,negative or it may not even be a scalar quantity. The procedure has even ledsome authors to suggest that Newton's second law needs to be modified withinthe context of classical continuum mechanics. Such absurd physical conclusionsare a consequence of appealing to the notion of "effective mass" with apreconception for the constitutive structure of the metamaterial and using acorrect mathematical procedure. We show that such unreasonable physicalconclusions would not arise if we were to use the appropriate "effectiveconstitutive relation" for the metamaterial, rather than use the concept of"effective mass" with an incorrect predetermined constitutive relation.
有效质量 "的概念常用于简化复杂的整块参数系统(离散动力系统)以及具有复杂微观结构特征的材料。从波传播的角度来看,对于某些被描述为超材料的物体,相应的 "有效质量 "可能与频率有关,也可能是负值,甚至可能不是标量。有些学者甚至认为,牛顿第二定律需要在经典连续介质力学中加以修正。这种荒谬的物理结论是在对超材料的构成结构缺乏正确理解的情况下诉诸 "有效质量 "概念并使用正确数学方法的结果。我们证明,如果我们对超材料使用适当的 "有效构成关系",而不是使用带有不正确的预定构成关系的 "有效质量 "概念,就不会出现这种不合理的物理结论。
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引用次数: 0
期刊
arXiv - PHYS - Classical Physics
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