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Clausius/Cosserat/Maxwell/Weyl Equations: The Virial Theorem Revisited  [PDF]
Jean-Fran?ois Pommaret
Physics , 2015,
Abstract: In 1870, R. Clausius found the virial theorem which amounts to introduce the trace of the stress tensor when studying the foundations of thermodynamics, as a way to relate the absolute temperature of an ideal gas to the mean kinetic energy of its molecules. In 1901, H. Poincar{\'e} introduced a duality principle in analytical mechanics in order to study lagrangians invariant under the action of a Lie group of transformations. In 1909, the brothers E. and F. Cosserat discovered another approach for studying the same problem though using quite different equations. In 1916, H. Weyl considered again the same problem for the conformal group of transformations, obtaining at the same time the Maxwell equations and an additional specific equation also involving the trace of the impulsion-energy tensor. Finally, having in mind the space-time formulation of electromagnetism and the Maurer-Cartan equations for Lie groups, gauge theory has been created by C.N. Yang and R.L. Mills in 1954 as a way to introduce in physics the differential geometric methods available at that time, independently of any group action, contrary to all the previous approaches. The main purpose of this paper is to revisit the mathematical foundations of thermodynamics and gauge theory by using new differential geometric methods coming from the formal theory of systems of partial differential equations and Lie pseudogroups, mostly developped by D.C Spencer and coworkers around 1970. In particular, we justify and extend the virial theorem, showing that the Clausius/Cosserat/Maxwell/Weyl equations are nothing else but the formal adjoint of the Spencer operator appearing in the canonical Spencer sequence for the conformal group of space-time and are thus totally dependent on the group action. The duality principle also appeals to the formal adjoint of a linear differential operator used in differential geometry and to the extension modules used in homological algebra.
The stationary Weyl equation and Cosserat elasticity  [PDF]
Olga Chervova,Dmitri Vassiliev
Mathematics , 2010, DOI: 10.1088/1751-8113/43/33/335203
Abstract: The paper deals with the Weyl equation which is the massless Dirac equation. We study the Weyl equation in the stationary setting, i.e. when the spinor field oscillates harmonically in time. We suggest a new geometric interpretation of the stationary Weyl equation, one which does not require the use of spinors, Pauli matrices or covariant differentiation. We think of our 3-dimensional space as an elastic continuum and assume that material points of this continuum can experience no displacements, only rotations. This framework is a special case of the Cosserat theory of elasticity. Rotations of material points of the space continuum are described mathematically by attaching to each geometric point an orthonormal basis which gives a field of orthonormal bases called the coframe. As the dynamical variables (unknowns) of our theory we choose the coframe and a density. We choose a particular potential energy which is conformally invariant and then incorporate time into our action in the standard Newtonian way, by subtracting kinetic energy. The main result of our paper is the theorem stating that in the stationary setting our model is equivalent to a pair of Weyl equations. The crucial element of the proof is the observation that our Lagrangian admits a factorization.
Generalized Maxwell and Weyl Equations for Massless Particles  [PDF]
Valeri V. Dvoeglazov
Physics , 2001,
Abstract: I discuss generalized Maxwell and Weyl equations. They may lead to dynamics which are different from those accepted at the present time. For instance, the photon may have non-transverse components and the neutrino may be not in the chiral states.
Mechanical Weyl Modes in Topological Maxwell Lattices  [PDF]
D. Zeb Rocklin,Bryan Gin-ge Chen,Martin Falk,Vincenzo Vitelli,T. C. Lubensky
Physics , 2015,
Abstract: Topological mechanical structures exhibit robust properties protected by topological invariants. In this letter, we study a family of deformed square lattices that display topologically protected zero-energy bulk modes analogous to the massless fermion modes of Weyl semimetals. Our findings apply to sufficiently complex lattices satisfying the Maxwell criterion of equal numbers of constraints and degrees of freedom. We demonstrate that such systems exhibit pairs of oppositely charged Weyl points, corresponding to zero-frequency bulk modes, that can appear at the origin of the Brillouin zone and move away to the zone edge (or return to the origin) where they annihilate. We prove that the existence of these Weyl points leads to a wavenumber-dependent count of topological mechanical states at free surfaces and domain walls.
Cellular Automaton Approach to Pedestrian Dynamics - Applications  [PDF]
Carsten Burstedde,Ansgar Kirchner,Kai Klauck,Andreas Schadschneider,Johannes Zittartz
Physics , 2001,
Abstract: We present applications of a cellular automaton approach to pedestrian dynamics introduced in [1,2]. It is shown that the model is able to reproduce collective effects and self-organization phenomena encountered in pedestrian traffic, e.g. lane formation in counterflow through a large corridor and oscillations at doors. Furthermore we present simple examples where the model is applied to the simulation of evacuation processes.
The Role of the Pauli-Lubanski Vector for the Dirac, Weyl, Proca, Maxwell, and Fierz-Pauli Equations  [PDF]
Sergey I. Kryuchkov,Nathan A. Lanfear,Sergei K. Suslov
Physics , 2015,
Abstract: We analyze basic relativistic wave equations for the classical fields, such as Dirac's equation, Weyl's two-component equation for massless neutrinos, and the Proca, Maxwell, and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubanski vector and the Casimir operators of the Poincare group. In general, in this group-theoretical approach, the above wave equations arise in certain overdetermined forms, which can be reduced to the conventional ones by a Gaussian elimination. A connection between the spin of a particle/field and consistency of the corresponding overdetermined system is emphasized in the massless case.
Three Dimensional Nonlinear Dynamics of Slender Structures: Cosserat Rod Element Approach  [PDF]
D. Q. Cao,Dongsheng Liu,Charles H. -T. Wang
Mathematics , 2004,
Abstract: In this paper, the modelling strategy of a Cosserat rod element (CRE) is addressed systematically for 3-dimensional dynamical analysis of slender structures. We employ the exact nonlinear kinematic relationships in the sense of Cosserat theory, and adopt the Bernoulli hypothesis. For the sake of simplicity, the Kirchoff constitutive relations are adopted to provide an adequate description of elastic properties in terms of a few elastic moduli. A deformed configuration of the rod is described by the displacement vector of the deformed centroid curves and an orthonormal moving frame, rigidly attached to the cross-section of the rod. The position of the moving frame relative to the inertial frame is specified by the rotation matrix, parametrized by a rotational vector. The approximate solutions of the nonlinear partial differential equations of motion in quasi-static sense are chosen as the shape functions with up to third order nonlinear terms of generic nodal displacements. Based on the Lagrangian constructed by the Cosserat kinetic energy and strain energy expressions, the principle of virtual work is employed to derive the ordinary differential equations of motion with third order nonlinear generic nodal displacements. A simple example is presented to illustrate the use of the formulation developed here to obtain the lower order nonlinear ordinary differential equations of motion of a given structure. The corresponding nonlinear dynamical responses of the structures have been presented through numerical simulations by Matlab software.
Defective beams in MEMS: a model of non-ideal rods using a Cosserat approach for component level modelling  [PDF]
Tim Gould,Charles Wang
Physics , 2004, DOI: 10.1088/0960-1317/15/1/012
Abstract: We present and derive a technique for the introduction of defects into a beam model based on the Cosserat theory of rods. The technique is designed for the derivation of component models of non-ideal rods for use in MEMS devices. We also present a worked through example of blob/nick defects (where the rod has an area with an excess/lack of material) and a guide for a model with random pits and blobs along the length of the beam. Finally we present a component level model of a beam with a defect and compare it to results from a Finite Element Analysis simulation. We test the Cosserat model for two cases without any defect and four with a defect. Results are in good agreement with a maximum 0.5% difference for the ideal case and under 1% differences for all but one of the defective cases, the exception being a 2% error in an extreme case for which the model is expected to break down. Overall, the Cosserat model with and without defects provides an accurate way of modelling long slender beams. In addition, simulation times are greatly reduced through this approach and further development for both component level models as well as as FEA components is important for practical yet accurate modelling of MEMS both for prediction and comparison.
A Pedestrian Approach to the Measurement Problem in Quantum Mechanics  [PDF]
Stephen Boughn,Marcel Reginatto
Physics , 2013, DOI: 10.1140/epjh/e2013-40004-9
Abstract: The quantum theory of measurement has been a matter of debate for over eighty years. Most of the discussion has focused on theoretical issues with the consequence that operational prescriptions, which are integral to experimental physics, have been largely ignored. This has undoubtedly exacerbated attempts to find a solution to the "measurement problem". In this paper, we fully embrace the ensemble interpretation of quantum mechanics that obviates the need to entertain reduction of the state vector, one of the primary dilemmas of the measurement problem. The other major aspect of the measurement problem, the necessity of describing measurements in terms of classical concepts, remains. However, we argue that the ultimate interface with experiments is described by operational prescriptions and not in terms of the concepts of classical theory. The pedestrian approach presented here suggests that the measurement problem is, in some sense, ill-posed and might never be resolved. This state of affairs is, in part, the result of searching for a theoretical answer to what is fundamentally an experimental question. This point of view is tenable so long as one is willing to view physical theories as providing models of nature rather than complete descriptions of reality. Among other things, these considerations lead us to suggest that the Copenhagen interpretation's insistence on the classicality of the measurement apparatus should be replaced by the requirement that a measurement, which is specified operationally, should simply be of sufficient precision.
Berry phase and anomalous velocity of Weyl fermions and Maxwell photons  [PDF]
Michael Stone
Physics , 2015,
Abstract: We consider two systems of wave equations whose wave-packet solutions have trajectories that are altered by the "anomalous velocity" effect of a Berry curvature. The first is the matrix Weyl equation describing cyclotron motion of a charged massless fermion. The second is Maxwell equations for the whispering-gallery modes of light in a cylindrical waveguide. In the case of the massless fermion, the anomalous velocity is obscured by the contribution from the magnetic moment. In the whispering gallery modes the anomalous velocity causes the circumferential light ray to creep up the cylinder at the rate of one wavelength per orbit, and can be identified as a continuous version of the Imbert-Federov effect.
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