Second Order Riemannian Mechanics

2015 ◽  
Vol 65 (6) ◽  
Author(s):  
Roman Matsyuk

AbstractThe technique of covariant differentiation is being implemented into the single integral variational calculus with second derivatives in Riemannian manifolds and into the Grässer-Rund-Weyssenhoff homogeneous generalized canonical framework for the parameter-invariant variational problem. As an example of physical nature we relate to each other the equations of motion with higher derivatives that describe the physical phenomena known as ‘Zitterbewegung’, the electromagnetic self-interaction, and spherical top. Although two of these phenomena, the quiver (the ‘Zitterbewegung’) and the radiation friction, have been known only within the framework of Special Relativity, our approach allows to generalize the corresponding equations of motion to the curved spacetime. From the geometrical point of view these generalizations depend neither on the signature of the Riemannian space nor on its dimension.

1954 ◽  
Vol 27 (3) ◽  
pp. 648-670
Author(s):  
Arturo Chiesa

Abstract The T-50 test, which has already been used for the last twenty years in many rubber factories has, up to the present time, been used solely as a means of checking vulcanization conditions, and so far no one has attempted to derive any mathematical relations from the results obtained. In the first part of this work it is shown that the T-50 test can be regarded as a useful and efficient means for studying much more complex and important problems, as, for example, the determination of the vulcanization characteristics of a rubber compound, both with respect to the ingredients and from the thermal point of view. In fact, this test makes it possible to obtain quantative data rapidly, which can be utilized to render any study easier and more conclusive than is possible with other tests commonly used, e.g., dynamometric parameters, aging tests, relaxation at elevated temperatures, etc. In making a study of the T-50 test, the approach was from the point of view of the chemical and physical nature of the phenomena involved in the test. It has been established with a considerable degree of exactitude that results obtained with the test conform to an energy law common to many chemical-physical phenomena, viz., the law of Arrhenius, which expresses the relation between the rate of a reaction and the temperature of a process by means of a parameter which depends on the activation energy of the process itself. In fact, analysis of experimental data shows clearly that, except in the case of mixtures having peculiar vulcanization characteristics, the activation energy of the process is practically independent of the composition of the mixture. One is led to believe, therefore, that such energy depends directly on the nature of the polymer itself and on its vulcanization reactions with sulfur, and that it is independent of the chemical and physical factors which control vulcanization reactions. On account of the nature of our work, we could not undertake a thorough scientific study of the aspect of this fact, considered from the point of view of the molecular structure of cross-linked high polymers. Instead, we have limited ourselves to developing an application of considerable practical utility, based on the fact that the activation energy is practically constant. In fact, a simple correlation diagram can be established which can be used for most types of vulcanizates, and by means of which it is possible to obtain directly the equivalent coefficients for passing from one vulcanization temperature to another with the longest temperature range which it was possible to use, i.e., from 78° to 151 ° C. It has thus been possible to interpret the results of the T-50 test for temperatures other than for the temperature at which the test was made, and thus to obtain a more complete and comprehensive picture directly related to the same chemico-physical phenomena. Passing on to the subject of dynamometric parameters, it is shown that, for these too, the above law applies with sufficient approximation. Again, since the order of magnitude of the activation energy is similar to that found by the T-50 test, the same correlation diagram applies, thus appreciably increasing its practical importance. Finally, the satisfactory accord between the activation energy values obtained by the T-50 test and those obtained by stress relaxation tests reveals an intimate relation between the two phenomena, which otherwise would appear to be completely different. Thus a further contribution is made to the complex study of relaxation.


2015 ◽  
Vol 10 (18) ◽  
pp. 61
Author(s):  
Mauro Meireles

<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p>A eucaristia é um rito católico que visa reafirmar a fé cristã. O presente texto ocupa-se, portanto, da eucaristia enquanto ato instituinte do ser e busca pensar o referido rito a partir da perspectiva de Mikhail Bakhtin. Desta feita e a partir do enfoque da antropologia, se ancora em certas sínteses na medida em que, a validade daquilo que se pressupõe verdade, do ponto de vista da cognição, não depende do fato desta ser ou não ser conhecida por alguém. Pois, é o homem que une fé e rito, que une verdades da ciência e fenômenos físicos. Desta feita, tem-se então que, quando postulamos a existência de Deus a partir de certos escritos canônicos – e o reafirmamos no rito eucarístico – lhe conferimos existência e tangibilidade. Sobretudo, defende- se no decorrer do texto que a eucaristia, seja enquanto ato instituinte, seja enquanto rito, nada mais é do que uma enunciação de si sobre si que só tem significado, manifesto em seu conteúdo- sentido, se enunciado por aquele que experimenta e executa o ato.</p></div></div><div class="layoutArea"><div class="column"><p><span>P</span><span>ALAVRAS</span><span>-C</span><span>HAVE</span><span>: </span><span>Eucaristia. Rito Eucarístico. Mikhail Bakhtin. Rito e Fé. Antropologia da Religião. </span></p><p><span>A</span><span>BSTRACT </span></p><p><span>Eucharist is a Catholic rite that aims to reaffirm the Christian faith. This paper analyzes the Eucharist rite as instituted act of being and aims think that rite in the Mikhail Bakhtin ́s perspective. So, from an anthropology's approach, it is based in some perspectives that recognize validity in of what is assumed true, in a cognition point of view, depends on whether this is or is not known to anyone. That is because the man is the one who unites faith and rite, truths of science and physical phenomena. Therefore, when we postulate the existence of God from certain canonical writings - and reaffirm the Eucharistic rite, we give him existence and tangibility. Most of all, this text argued that the Eucharist is as instituted act or a rite, is nothing more than a statement of itself that only has meaning, manifest in their content-sense, if enunciated by one who experience and performs the act. </span></p><p><span>K</span><span>EYWORDS</span><span>: </span><span>Eucharist. Eucharistic rite. Mikhail Bakhtin. Ritual and faith. Anthropology of Religion </span></p></div></div></div>


1987 ◽  
Vol 02 (05) ◽  
pp. 1591-1615 ◽  
Author(s):  
V.A. BEREZIN

A method for the phenomenological description of particle production is proposed. Correspondingly modified equations of motion and energy-momentum tensor are obtained. In order to illustrate this method we reconsider from the new point of view of (i) the C-field Hoyle-Narlikar cosmology, (ii) the influence of the particle production process on metric inside the event horizon of a charged black hole and (iii) a nonsingular cosmological model.


Author(s):  
Nicolas Delcey ◽  
Philippe Baucour ◽  
Didier Chamagne ◽  
Geneviève Wimmer ◽  
Odile Bouger ◽  
...  

The pantograph strip interface involves many physical phenomena. Temperature evolution is one of them. This problem includes various thermal flux and sources. More specifically, due to the train motion, a moving zigzag heat source occurs. This paper deals with a thermal 2D Alternating Direction Implicit (ADI) numerical method for temperature estimations in the train pantograph carbon strip, the aims being a better wear problems anticipation and the creation of a preventive maintenance. For that, an electrical model is coupled to the thermal one to take into account all Joule effects. The ADI strategy enables a significant computation time reduction against most classical resolution methods. Besides, the model involves two mathematical processes: the first one is an appropriate variable transform which induces a fixed surface heat production, while the second is based on locally refined meshes. Various numerical tests are presented and discussed in order to show the accuracy of the scheme. From a physical point of view, the results are much interesting. Further investigations, depending on the different parameters, should lead us to predict the strip critical thermal phases.


2019 ◽  
Vol 34 (02) ◽  
pp. 1950001 ◽  
Author(s):  
Pavel A. Bolokhov

We argue that quaternions form a natural language for the description of quantum-mechanical wave functions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper, we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the nonrelativistic limit to find the Schrödinger’s equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the noninteracting nonrelativistic limit, the quaternionic description effectively reduces to the regular complex wave function language. We provide an easy-to-use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell’s equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.


2012 ◽  
Vol 22 (10) ◽  
pp. 1230033 ◽  
Author(s):  
L. A. DARRIBA ◽  
N. P. MAFFIONE ◽  
P. M. CINCOTTA ◽  
C. M. GIORDANO

The reader can find in the literature a lot of different techniques to study the dynamics of a given system and also, many suitable numerical integrators to compute them. Notwithstanding the recent work of [Maffione et al., 2011b] for mappings, a detailed comparison among the widespread indicators of chaos in a general system is still lacking. Such a comparison could lead to select the most efficient algorithms given a certain dynamical problem. Furthermore, in order to choose the appropriate numerical integrators to compute them, more comparative studies among numerical integrators are also needed. This work deals with both problems. We first extend the work of [Maffione et al., 2011b] for mappings to the 2D [Hénon & Heiles, 1964] potential, and compare several variational indicators of chaos: the Lyapunov Indicator (LI); the Mean Exponential Growth Factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI) and its generalized version, the Generalized Alignment Index (GALI); the Fast Lyapunov Indicator (FLI) and its variant, the Orthogonal Fast Lyapunov Indicator (OFLI); the Spectral Distance (D) and the Dynamical Spectra of Stretching Numbers (SSNs). We also include in the record the Relative Lyapunov Indicator (RLI), which is not a variational indicator as the others. Then, we test a numerical technique to integrate Ordinary Differential Equations (ODEs) based on the Taylor method implemented by [Jorba & Zou, 2005] (called taylor), and we compare its performance with other two well-known efficient integrators: the [Prince & Dormand, 1981] implementation of a Runge–Kutta of order 7–8 (DOPRI8) and a Bulirsch–Stöer implementation. These tests are run under two very different systems from the complexity of their equations point of view: a triaxial galactic potential model and a perturbed 3D quartic oscillator. We first show that a combination of the FLI/OFLI, the MEGNO and the GALI 2N succeeds in describing in detail most of the dynamical characteristics of a general Hamiltonian system. In the second part, we show that the precision of taylor is better than that of the other integrators tested, but it is not well suited to integrate systems of equations which include the variational ones, like in the computing of almost all the preceeding indicators of chaos. The result which induces us to draw this conclusion is that the computing times spent by taylor are far greater than the times consumed by the DOPRI8 and the Bulirsch–Stöer integrators in such cases. On the other hand, the package is very efficient when we only need to integrate the equations of motion (both in precision and speed), for instance to study the chaotic diffusion. We also notice that taylor attains a greater precision on the coordinates than either the DOPRI8 or the Bulirsch–Stöer.


Author(s):  
Elias Paraskevopoulos ◽  
Sotirios Natsiavas

An investigation is carried out for deriving conditions on the correct application of Newton’s law of motion to mechanical systems subjected to constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and anholonomic constraints. The focus is on establishment of conditions, so that the form of Newton’s law remains invariant when imposing an additional set of motion constraints on a system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold. The latter is weaker than a similar condition employed frequently in the literature, but holding on Riemannian manifolds only. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space (and not on the dual space) of a manifold. Finally, the Euler-Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated.


2014 ◽  
Vol 11 (04) ◽  
pp. 1450034 ◽  
Author(s):  
Leonardo Colombo ◽  
Pedro Daniel Prieto-Martínez

In this paper, we consider an intrinsic point of view to describe the equations of motion for higher-order variational problems with constraints on higher-order trivial principal bundles. Our techniques are an adaptation of the classical Skinner–Rusk approach for the case of Lagrangian dynamics with higher-order constraints. We study a regular case where it is possible to establish a symplectic framework and, as a consequence, to obtain a unique vector field determining the dynamics. As an interesting application we deduce the equations of motion for optimal control of underactuated mechanical systems defined on principal bundles.


Author(s):  
J. Gratus ◽  
T. Banaszek

Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat space–time due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordinate transformation and an integral. Transformations involving integrals have not been seen before. This is significantly different from the familiar transformation rules for a dipole, where the components transform as tensors. It enables quadrupoles to be correctly defined in general relativity and to prescribe the equations of motion for a quadrupole in a coordinate system adapted to its motion and then transform them to the laboratory coordinates. An example is given of another unusual feature: a quadrupole which is free of dipole terms in polar coordinates has dipole terms in Cartesian coordinates. It is shown that dipoles, electric dipoles, quadrupoles and electric quadrupoles can be defined without reference to a metric and in a coordinates-free manner. This is particularly useful given their complicated coordinate transformation.


1969 ◽  
Vol 91 (4) ◽  
pp. 931-938 ◽  
Author(s):  
David L. Sikarskie ◽  
Burton Paul

The dynamics of a widely used class of hammer impact machines are investigated on the basis of a two-degree-of-freedom idealization. The difficulty in the problem is due to the repetitive impact which introduces a nonlinearity in the system. It is the purpose of the analysis to develop a solution for the steady-state behavior of the system. There are several ways this can be done. One of the most efficient ways, from the point of view of ease of parametric studies of the system, is to convert the problem to a “boundary” value problem. With this technique, the system is governed by the equations of motion between impacts, and further satisfies additional conditions at the beginning and end of each impact cycle. Since the solution is obtained in only one cycle, it thus represents a straightforward method of studying the effect of various system parameters. A fundamental assumption in the analysis is that the steady-state response of the system has a period equal to the forcing period. This is verified for one set of parameters through the use of high-speed movies of an actual machine. There are several other interesting features in the analysis, including multivaluedness of the solution, allowable solution domain, and stability of solution, which have not been completely resolved to date.


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