Classical mechanics: point particles and relativity
... In most textbooks on classical mechanics the topic of resisted motion is discussed using solely the first-or second-order term of the expansion, arguably for the sake of simplicity (e.g. see [14,36,37]). From the viewpoint of fluid mechanics, in turn, the well-known drag model is said to be pure quadratic. However, one should notice that the nondimensional drag coefficient involved is a function of, among others, the body Reynolds number: ...
... Usually the method of undetermined coefficients is employed to derive an appropriate finite difference formula. With such an approach, expansion (14) up to term of order n à is used for the n à -point stencil. This leads to a n à  n à Vandermonde system of linear equations that becomes poorly conditioned for large n à [21]. ...
In this work, planar free vibrations of a single physical pendulum are investigated both experimentally and numerically. The laboratory experiments are performed with pendula of different lengths, for a wide range of initial configurations, beyond the small angle regime. In order to approximate the air resistance, three models of damping are considered—involving the three components of the resistive force: linear (proportional to velocity), quadratic (velocity-squared) and acceleration-dependent (proportional to acceleration). A series of numerical experiments is discussed, in which the damping coefficients are estimated by means of several computational methods. Based on the observed efficiency, a gradient method for optimization is treated as the main tool for determination of a single set of damping parameters, independent of the system’s initial position. In the model of resistive force, the term proportional to acceleration, associated with the empirical Morison equation, seems to be indispensable for the successful approximation of the real pendulum motion.
... Certains auteurs l'utilisent pour indiquer la loi universelle de la gravitation, même si celle-ci n'est pas censée être une véritable loi. Nous faisons plutôt référence au principe de superposition [Gre04], qui indique que le système des forces est un système linéaire, les forces ...
... 4. Often one describes a fourth law as well, which represents the "principle of superposition" [Gre04]. It states that forces acting on an object add up like vectors, which is more of a supplement than an actual law, it arises as a necessity in modern classical mechanics. ...
This PhD thesis concerns the mathematical analysis of the interaction of an inviscid fluid with immersed structures. More precisely it revolves around two main problems: one of them is the asymptotic analysis of an infinitesimal immersed particle, the other one being the interaction of water waves with a submerged solid object. Concerning the first problem, we studied a system of second order non-linear ODEs, serving as a toy model for the motion of a rigid body immersed in a two-dimensional perfect fluid. The unknowns of the model describe the position of the object, that is the position of its center of mass and the angle of rotation; the equations arise from Newton’s second law with the consideration of a Kutta-Joukowski type lift force. It concerns the detailed analysis of the dynamic of this system when the solid inertia tends to 0. For the evolution of the position of the solid’s center of mass, the study highlights similarities with the motion of a charged particle in an electromagnetic field and the wellknown “guiding center approximation”; it turns out that the motion of the corresponding guiding center is given by a point-vortex equation. As for the angular equation, its evolution is given by a slowly-in-time modulated non-linear pendulum equation. Based on the initial values of the system one can distinguish qualitatively different regimes: for small angular velocities, by the Poincaré-Lindstedt method one observes a modulation in the fast time-scale oscillatory terms, for larger angular velocities however erratic rotational motion is observed, a consequence of Melnikov’s observations on the presence of a homoclinic tangle. About the other problem, the Cauchy problem for the water waves equations is considered in a fluid domain which has a free surface on the upper vertical limit and a flat bottom on which a solid object moves horizontally, its motion determined by the pressure forces exerted by the fluid. Two shallow water asymptotic regimes are detailed, well-posedness results are obtained for both the Saint-Venant and the Boussinesq system coupled with Newton’s equation characterizing the solid motion. Using the particular structure of the coupling terms one is able to go beyond the standard scale for the existence time of solutions to the Boussinesq system with a moving bottom. An extended numerical study has also been carried out for the latter system. A high order finite difference scheme is developed, extending the convergence ratio of previous, staggered grid based models. The discretized solid mechanics are adapted to represent important features of the original model, such as the dissipation due to the friction term. We observed qualitative differences for the transformation of a passing wave over a moving solid object as compared to an immobile one. The movement of the solid not only influences wave attenuation but it affects the shoaling process as well as the wave breaking. The importance of the coefficient of friction is also highlighted, influencing qualitative and quantitative properties of the coupled system. Furthermore, we showed the hydrodynamic damping effects of the waves on the solid motion, reminiscent of the so-called dead water phenomenon.
... Where k : represents the interpolation number, T 6 : is an element of T 6 matrix expressed by the approximate value of the generalized coordinates. ...
... 1) The angular speed and acceleration of the trihedral: Speed: From the Frenet-Serret theorem [6] we obtain : ...
Actually, in the majority of mechanical systems such as industrial robots, mobile robots with embarked structure, etc… the end-effector mass widely exceeds that of the supported load, and lot of works do not take it into account in their algorithms. These systems must be designed based on dynamic model; moreover the synthesis of the algorithm of an autonomous manipulator must be accomplished by taking into account nonlinearity and complexity of its dynamic model, this is in order to increase the control effectiveness. Currently, tasks involving a high precision welding of complex forms with high characteristics are very required in industry in general. This paper introduces the modeling of the end-effector movement equations to allow a tracking of welding cords with complex forms in the acceptable working zone.
... Da mesma forma que as leis de Kepler (que são obtidas a partir das leis da Mecânica e da lei da Gravitação Universal) nos dão o entendimento dos movimentos dos planetas em nosso sistema solar[10,11], essas mesmas leis regem a astrodinâmica dos sistemas extra-solares. As três leis de Kepler para o movimento planetário podem ser descritas como: ...
... Mas a velocidade orbital do planetá e dada pela expressão da Mecânica Celeste[10,11,16], ...
This issue reports a brief review about transit method of detecting extrasolar planet (exoplanet). When a exoplanet crosses (transits) in front of its host star's disk along the line of sight, then the observed visual brightness of the star drops by a small amount. The transit method is responsible for 23% of some important new discoveries of exoplanets. In this sense, we explain didactically the conceptual and calculational details of transit method. As a result of application of the transit method, we applied it to exoplanet COROT-1b.
... Da mesma forma que as leis de Kepler (que são obtidas a partir das leis da Mecânica e da lei da Gravitação Universal) nos dão o entendimento dos movimentos dos planetas em nosso sistema solar [10, 11], essas mesmas leis regem a astrodinâmica dos sistemas extra-solares. As três leis de Kepler para o movimento planetário podem ser descritas como: ...
... Mas a velocidade orbital do planetá e dada pela expressão da Mecânica Celeste [10, 11, 16], ...
This issue reports a brief review about transit method of detecting extrasolar planet (exoplanet). When a exoplanet crosses (transits) in front of its host star's disk along the line of sight, then the observed visual brightness of the star drops by a small amount. The transit method is responsible for 23% of some important new discoveries of exoplanets. In this sense, we explain didactically the conceptual and calculational details of transit method. As a result of application of the transit method, we applied it to exoplanet COROT-1b.
... Gauss's law for gravity states that the gravitational flux through a closed surface is proportional to the enclosed mass [41]. In our analogy, each midge has an effective unit " mass, " and therefore Gauss's law for the force in the " pure gravity " regime ( ...
... Gauss's law for gravity states that the gravitational flux through a closed surface is proportional to the enclosed mass [41]. In our analogy, each midge has an effective unit " mass, " and therefore Gauss's law for the force in the " pure gravity " regime (Eq. ...
The collective motion of groups of animals emerges from the net effect of the
interactions between individual members of the group. In many cases, such as
birds, fish, or ungulates, these interactions are mediated by sensory stimuli
that predominantly arise from nearby neighbors. But not all stimuli in animal
groups are short range. Here, we consider mating swarms of midges, which
interact primarily via long-range acoustic stimuli. We exploit the similarity
in form between the decay of acoustic and gravitational sources to build a
model for swarm behavior. By accounting for the adaptive nature of the midges'
acoustic sensing, we show that our "adaptive gravity" model makes mean-field
predictions that agree well with experimental observations of laboratory
swarms. Our results highlight the role of sensory mechanisms and interaction
range in collective animal behavior. The adaptive interactions that we present
here open a new class of equations of motion, which may appear in other
biological contexts.
... Robert Hakl hakl@ipm.cz on singular differential equations where the nonlinear term presents some types of discontinuity (see Refs. [1,6,12,17,18,21,22]). ...
... Remark 1 Note that from the proof of the uniqueness, it follows that (5) has at most one solution provided δ ∈ [0, 1), λ > 0, and g, h ∈ L ω fulfill (6). That means that if there exists an ω-periodic solution to (5), then it is unique. ...
We study the existence of periodic solutions to differential equations with a singularity at the state variable, extending the recently found existing relation to differential equations with attractive singularity, where the order of the singularity and the regularity of the input functions are closely connected. The results related to the classic open problems in the field of singular equations. The question about uniqueness is also discussed. © 2015 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg
... Para hacer los cálculos, utilice del Teorema del Trabajo y la Energía para hallar la Velocidad del Niño Esquimal, así como la Fuerza de Reacción Normal entre el Niño y el Montículo Esférico, mientras se desliza sobre su Iglú. ¿Se recuperan los casos conocidos cuando µ k → 0 del Modelo 3? (De Lande, O. L. y Pierrus, J. 2010), (Greiner, W., 1989). ...
En la enseñanza de la Mecánica, es siempre recomendable poder discutir en clase Problemas Integradores y Holísticos. En ese sentido proponemos analizar el clásico problema del Niño Esquimal que resbala por su Iglú desde diferentes puntos de vista, y que sirva como Experiencia de Aprendizaje Mediado (EAM) usando para ello la Estrategia de Modelización Progresiva (EMP). Dicho problema, aparece en la mayoría de los textos de Física General Universitarios y es abordado únicamente con argumentos de Conservación de Energía. En este trabajo lo planteamos y resolvemos usando la “Dinámica Traslacional con y sin Fricción”, “Conservación de Energía”, “Teorema Trabajo-Energía con y sin Fricción”, “Dinámica Rotacional con y sin Fricción” y usando la “Dinámica Lagrangiana con y sin Fricción”, con la intención de mostrar como con un mismo problema se puede resolver de muy diferentes maneras, ayudando al estudiante a integrar sus conocimientos de Mecánica de una forma general.
... current-voltage, motion). Traditionally, governing equations are derived from principles that have been formalized from general empirical observations consistent with certain hypotheses, for instance Newton's laws are based on the constant-mass hypothesis (Greiner, 2006). However, in practice it is hard to derive governing equations from the existing rules, because either the practitioners do not have sufficient knowledge or they do not have sufficient time. ...
In physical sciences, dynamic systems are modeled using their parameters within governing equations that often form a system of ordinary differential equations (SODE). This system consists of multiple equations, each of which relates the time derivative of a single parameter to several parameters. A parameter can appear in multiple equations, and this parameter potentially links the equations to each other. Although in certain cases the SODE can be written by domain experts, it is often unknown. With advances in sensor technology, large quantities of data can be sampled from dynamic systems, thus enabling the data-driven discovery of closed-form SODEs. State-of-the-art approaches are based on sparse single-task learning, which means that each equation from the SODE is learned independently. Omitting the coupling features of equations leads to SODEs that weakly identify the dynamic system. Furthermore, the convexity of the sparse penalty included in the learning criterion gives an SODE that is biased with respect to the true SODE. To reduce such a bias, we propose a multitask learning (MTL) based penalty which can learn the closed-form SODE with unbiasedness. The purpose of each task is to discover a single equation. But discovering an SODE is nontrivial, as dynamic systems are often nonlinear and the available data are noisy. Our proposal improves SODE identification by harnessing a nonconvex sparse matrix-structured penalty which takes into account the coupling feature as well as addresses the bias issue. Experimental results, based on noisy data simulated from known SODEs, confirm that, compared to single-task learning, MTL is more effective for recovering the closed-form SODE, and the proposed nonconvexity ensures that it can be estimated with unbiasedness. We also show the benefits of our approach on a real-world public dataset sampled from a laboratory-based ecological experiment.
... The Grand Unified Theories of today suggest . . . W. Greiner [93] ...
This article, the last in a series of three articles, attempts to unravel the underlying physics of recent experiments regarding the contradictory properties of the neutron lifetime that has been a complete riddle for quite some time. So far, none of the advanced theories beyond the Standard Models (SMs) of particle physics and cosmology have shown sufficient potential to resolve this mystery. We also try to explain the blatant contradiction between the predictions of particle physics and experiments concerning the nature and properties of the (so far undetected) dark matter and dark energy particles. To this end the novel concepts of both negative and hypercomplex matter (giving rise to the concept of matter flavor) are introduced, replacing the field of real numbers by hypercomplex numbers. This extension of the number system in physics leads to both novel internal symmetries requiring new elementary particles – as outlined in Part I and II, and to novel types of matter. Hypercomplex numbers are employed in place of the widely accepted (but never observed) concept of extra space dimensions – and, hence, also to question the corresponding concept of supersymmetry. To corroborate this claim, we report on the latest experimental searches for novel and supersymmetric elementary particles by direct searches at the Large Hadron Collider (LHC) and other colliders as well as numerous other dedicated experiments that all have come up empty handed. The same holds true for the dark matter search at European Council for Nuclear Research (CERN) [CERN Courier Team, “Funky physics at KIT,” in CERN Courier, 2020, p. 11]. In addition, new experiments looking for dark or hidden photons (e.g., FUNK at Karlsruhe Institute of Technology, CAST at CERN, and ALPS at Desy, Hamburg) are discussed that all produced negative results for the existence of the hitherto unseen but nevertheless gravitationally noticeably dark matter. In view of this contradicting outcome, we suggest a four-dimensional Minkowski spacetime, assumed to be a quasi de Sitter space, dS 1,3 , complemented by a dual spacetime , denoted by DdS 1,3 , in which the dark matter particles that are supposed to be of negative mass reside. This space is endowed with an imaginary time coordinate, −i t and an imaginary speed of light, i c . This means that time is considered a complex quantity , but energy m (i c ) ² > 0. With this construction visible and dark matter both represent positive energies, and hence gravitation makes no distinction between these two types of matter. As dark matter is supposed to reside in dual space DdS 1,3 , it is principally undetectable in our spacetime. That this is evident has been confirmed by numerous astrophysical observations. As the concept of matter flavor may possibly resolve the contradictory experimental results concerning the lifetime of the neutron [J. T. Wilson, “Space based measurement of the neutron lifetime using data from the neutron spectrometer on NASA’s messenger mission,” Phys. Rev. Res., vol. 2, p. 023216, 2020] this fact could be considered as a first experimental hint for the actual existence of hypercomplex matter. In canonical gravity the conversion of electromagnetic into gravity-like fields (as surmised by Faraday and Einstein) should be possible, but not in cosmological gravity (hence these attempts did not succeed), and thus these conversion fields are outside general relativity. In addition, the concept of hypercomplex mass in conjunction with magnetic monopoles emerging from spin ice materials is discussed that may provide the enabling technology for long sought propellantless space propulsion.
... O estudo do movimento dos corpos tanto nos últimos anos do ensino médio quanto nos períodos iniciais dos cursos superiores de Ciências, Licenciaturas e Engenharias está baseado em modelos simplificados. Os corpos são abstraídos e tratados como partículas pontuais [Greiner 2004] que nada mais são do que entidades matemáticas sem dimensão. Além do mais, forças resistivas são desprezadas, como é o caso das forças de interação com o ar e outras deixadas para um estudo posterior, como é o caso da força de atrito de superfície. ...
O estudo do lançamento de projéteis em diferentes níveis de ensino está baseado numa modelagem teórica simplificada, onde o projétil é abstraído em uma partícula movendo-se em duas dimensões e, normalmente, não sujeito ao efeito resistivo do ar. Neste trabalho, é apresentada uma proposta experimental simples, envolvendo a utilização de brinquedos de baixo custo, com o objetivo de implantar atividades investigativas em sala de aula, despertando uma postura crítica acerca dos modelos teóricos estudados. Como forma de demostrar o relacionamento entre a modelagem teórica-matemática e a experimentação, é realizada uma revisão detalhada de dois dos modelos teóricos existentes para o lançamento oblíquo de uma partícula: o modelo padrão sem resistência do ar e o modelo sujeito a uma força de arrasto linearmente proporcional à velocidade da partícula. No primeiro modelo é utilizada a estratégia usual, utilizando-se a cinemática vetorial, no segundo, são desenvolvidas as equações do movimento a partir das leis de Newton, as quais são resolvidas exatamente. Em ambos os casos, são obtidas expressões analíticas para o alcance dos projéteis em função do ângulo de lançamento -- e do coeficiente de resistência do ar no segundo modelo. Além da comparação e discussão entre os resultados dos modelos matemáticos, é proposta uma abordagem experimental investigativa de baixo custo para o lançamento de projéteis reais, utilizando brinquedos. Nessa oportunidade, estudantes são desafiados a levantar concepções espontâneas acerca da validade do modelo matemático sem resistência do ar, ao qual foram expostos previamente em sala de aula. Finalmente, os estudantes são apresentados aos resultados do modelo com resistência da ar, sem serem mostrados necessariamente os detalhes do desenvolvimento matemático, sendo feitas considerações a respeitos de alguns dos melhoramentos. Por fim, são apresentadas evidências acerca da eficiência da atividade investigativa na compreensão sobre o que seja um modelo matemático e sua relação com a realidade.
... Moreover, and more precisely, the first law defines the force qualitatively, the second law measures the force quantitively. The third law states that an isolated single force does not exist [18][19][20][21]. Throughout nearly three centuries, these three laws have been stated in many different ways and we will summarize them as follows: ...
The concept of mathematical probability was established in 1933 by Andrey Nikolaevich Kolmogorov by defining a system of five axioms. This system can be enhanced to encompass the imaginary numbers set after the addition of three novel axioms. As a result, any random experiment can be executed in the complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. We aim here to incorporate supplementary imaginary dimensions to the random experiment occurring in the “real” laboratory in R and therefore to compute all the probabilities in the sets R, M, and C. Accordingly, the probability in the whole set C = R + M is constantly equivalent to one independently of the distribution of the input random variable in R, and subsequently the output of the stochastic experiment in R can be determined absolutely in C. This is the consequence of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. We will apply this innovative paradigm to Isaac Newton’s classical mechanics and to prove as well in an original way an important property at the foundation of statistical physics.
... Eq. (7) describes damped harmonic oscillation, well-known from introductory classical mechanics (Landau and Lifshitz, 1982;Greiner, 2003;Thornton and Marion, 2004). The solution has the form ...
We consider gradient descent with `momentum', a widely used method for loss function minimization in machine learning. This method is often used with `Nesterov acceleration', meaning that the gradient is evaluated not at the current position in parameter space, but at the estimated position after one step. In this work, we show that the algorithm can be improved by extending this `acceleration' --- by using the gradient at an estimated position several steps ahead rather than just one step ahead. How far one looks ahead in this `super-acceleration' algorithm is determined by a new hyperparameter. Considering a one-parameter quadratic loss function, the optimal value of the super-acceleration can be exactly calculated and analytically estimated. We show explicitly that super-accelerating the momentum algorithm is beneficial, not only for this idealized problem, but also for several synthetic loss landscapes and for the MNIST classification task with neural networks. Super-acceleration is also easy to incorporate into adaptive algorithms like RMSProp or Adam, and is shown to improve these algorithms.
... This can be modeled like a spring-mass system with masses connected in series such as the next figure. In a spring series system, the spring constant equivalent (keq) can be found knowing the spring constant values of each spring in the system [8]: ...
Using the physics of sound waves as a foundation, subatomic particles and their behaviors are modeled with classical mechanics to calculate the Planck energy, the electron's energy and the energy levels of the first two atoms: hydrogen and helium. Five different methods are used to calculate energies, including spring-mass systems and wave systems, and all five are found to be equal in their calculations.
... A solução da equação diferencial acima pode ser vista em detalhes nas referências[7,8], fazendo-se ...
Doppler method, also called radial velocities method, is a technique used in detecting of exoplanets. About 70% of known exoplanets were discovered using this technique. The Doppler method consist in determine spectral shifts of host star of exoplanet, which enable measurement of the variation of radial velocity of host star towards of Earth or away from Earth. This movement of host star around center of mass occurs due to the existence of exoplanet. In this work we present a pedagogical review of the Doppler method applied to detection of exoplanets. In this sense, we analyze theoretical elements of this technique and their application.
... (1-a) and (1-b) into Eq. (75), neglecting the effect of the term 1/ω p τ 2 in the denominator of Eq. (75), and recalling the relation of angle aberration where sin 2 [32], the single−pass gain of the scattered radiation power in the laboratory frame is given by ...
A theoretical analysis is proposed to investigate the Compton scattering as well as the inverse Compton scattering. In this analysis, the incident and scattered electromagnetic (EM) waves are classically described using Maxwell's equations, while the interacting electron is represented quantum mechanically by a wavepacket with a finite spreading length. The density matrix formalism is introduced to analyze the interaction between the electron wave and the EM field in a quantum mechanical manner. The interaction mechanism is caused by the effective coupling between EM and electron waves satisfying the energy and momentum conservation rules. The effects of the electron relaxation time due to Coulomb collisions among electrons, as well as the finite spreading size of an electron, are newly considered. Expressions for the radiated power of the scattered wave are derived for Compton and inverse Compton scattering.
... Los comportamientos oscilatorios son ubicuos en la naturaleza en sistemas de todaíndole, entre ellos, en cuerpos astronómicos tales como asteroides, satélites, planetas y estrellas; en sistemas biológicos, tanto a nivel orgánico [1] como a nivel bioquímico [2]; en diferentes reacciones químicas [3,4]; en sistemas mecánicos; en circuitos electrónicos, etc. Asimismo, los objetos con comportamientos oscilatorios pueden acoplarse a otros similares lo que da lugar en muchos casos a otro fenómeno muy común que es la sincronización y caracterizada por la la constancia de las diferencias de fase y la razón entre los períodos [5,6] Uno de los comportamientos más sencillos y comunes de describir es el del péndulo simple que se lo describe en textos de física básica, de mecánica clásica [7][8][9][10], de métodos matemáticos de la física [11,12] ya sean analíticos o basados enálgebra computacional [13] y en textos especializados en oscilaciones [14,15] Por otra parte, dado el carácter no lineal que puede presentar el péndulo simple en su forma más general, es también objeto de análisis en libros y artículos de dinámica no lineal [16][17][18][19][20][21]. Los sistemas acoplados se presentan en muchas ramas de la física, ofreciendo una gran gama de características que dependen tanto de las propiedades de los sistemas independientes, como de las propias características de acoplamiento. ...
Based on a discrete model of a damped pendulum, we study the behavior of two coupled pendula, applying a simple connection factor between them. We find that the results of this model fit with the real data. The behavior of this system is studied as a function of its different parameters in which regions of resonance are observed and probable conditions are set for synchronization and chaos.
En la Enseñanza de la Dinámica Newtoniana a nivel universitario solemos construir Modelos Mecánicos para la comprensión de la situación Física estudiada. Por lo regular construimos primeramente modelos simples (toy models) que posteriormente iremos sofisticando con la intención de crear un modelo más complejo y por tanto más realista al final. Aseguramos con ello que el modelo sofisticado es mejor que el modelo simple (Estrategia de Modelización Progresiva, EMP). ¿Esto es siempre así?. Presentamos aquí la construcción de Dos Modelos Mecánicos (uno Sofisticado y otro Simple) para estudiar el Movimiento de una Cuerda Masiva que Resbala sobre una Mesa con y sin Fricción Seca y mostramos que los principales resultados obtenidos en el modelo simple son tan buenos y generales como los obtenidos del modelo sofisticado. Lo anterior nos muestra como poder estudiar el problema de la Cuerda que Desliza por una Mesa en diferentes estadios de la Educación Universitaria sin detrimento de la compresión completa del problema planteado. Como complemento a lo anterior discutimos también la versión Lagrangiana del Problema, así como una posible generalización al problema planteado.
En este artículo se aborda un modo no habitual (poco usado en la bibliografía sobre Relatividad Restringida) de llegar a las conocidas transformaciones de Lorentz a partir de consideraciones provenientes del espacio bidimensional de Minkowski. Su característica fundamental, proveniente de la forma no euclidiana del elemento de longitud, es que para pasar de un sistema de referencia a otro se usan rotaciones hiperbólicas; esto supone dejar los círculos (habituales en las rotaciones euclidianas) y pasar a las hipérbolas. Usando esta geometría hiperbólica y las relaciones para ángulos hiperbólicos (equivalentes y muy parecidas a las de ángulos circulares) se generan fácilmente las transformaciones de Lorentz. Para obtener la dilatación temporal, la contracción de longitud y la suma relativista de velocidades se utilizan el espacio minkowskiano y geometría sencilla. Por otro lado, como se sabe, el conocido enfoque de simetrías (vía grupo de Lorentz o Poincaré), es más potente con el análisis tensorial y muy físico. Si bien este enfoque geométrico no es tan poderoso, es muy sencillo y sin duda presenta un carácter pedagógico especialmente para pregrado. Y, por supuesto, si quisiéramos pasar del espacio bidimensional al cuatridimensional, esta vía geométrica se complicaría mucho.
We provide a Maupertuis-type principle for the following system of ODE, of interest in special relativity: where and is a function of class . As an application, we prove the existence of multiple periodic solutions with prescribed energy for a relativistic N-centre type problem in the plane.
In applied mathematics, the name Monte Carlo is given to the method of solving problems by means of experiments with random numbers. This name, after the casino at Monaco, was first applied around 1944 to the method of solving deterministic problems by reformulating them in terms of a problem with random elements which could then be solved by large-scale sampling. But, by extension, the term has come to mean any simulation that uses random numbers. In the twentieth century and present time, Monte Carlo methods have become among the fundamental techniques of simulation in modern science. This was accomplished after a long history of efforts done by prominent and distinguished mathematicians and scientists. This book is an illustration of the use of Monte Carlo methods when applied to solve specific problems in mathematics, engineering, physics, statistics, or science in general.
The existence and multiplicity of T -periodic solutions to a class of differential equations with attractive singularities at the origin are investigated in the paper. The approach is based on a new method of construction of strict upper and lower functions. The multiplicity results of Ambrosetti–Prodi type are established using a priori estimates and certain properties of topological degree.
A force field in is called central or radially symmetric if its magnitude only depends on the distance to the origin and its direction is proportional to the vector position, that is, it always points to the origin. The study of a radially symmetric system is reduced to a second-order scalar equation with a repulsive singularity. We review some of the most interesting models with radial symmetry.
We have constrained possible new interactions which produce nonrelativistic potentials between polarized neutrons and unpolarized matter proportional to ασ[over →]·v[over →] where σ[over →] is the neutron spin and v[over →] is the relative velocity. We use existing data from laboratory measurements on the very long T_{1} and T_{2} spin relaxation times of polarized ^{3}He gas in glass cells. Using the best available measured T_{2} of polarized ^{3}He gas atoms as the polarized source and the Earth as an unpolarized source, we obtain constraints on two new interactions. We present a new experimental upper bound on possible vector-axial-vector (V_{VA}) type interactions for ranges between 1 and 10^{8} m. In combination with previous results, we set the most stringent experiment limits on g_{V}g_{A} ranging from ∼μm to ∼10^{8} m. We also report what is to our knowledge the first experimental upper limit on the possible torsion fields induced by the Earth on its surface. Dedicated experiments could further improve these bounds by a factor of ∼100. Our method of analysis also makes it possible to probe many velocity dependent interactions which depend on the spins of both neutrons and other particles which have never been searched for before experimentally.
In 2012, Barry Cox and James Hill published a new and elegant way to deduce the formulae of extended relativity, developed by Erasmo Recami and others during the 1960s and 1970s. Extended relativity applies to superluminal or FTL (faster than light in the vacuum) particles, called tachyons by the late American physicist Gerald Feinberg. However, tachyons generate causality paradoxes in special relativity when used for signalling. These paradoxes are safely removed only in a space-time containing a preferred inertial reference frame. Moreover, quantum properties should be taken into account. They cause further trouble for tachyons because, for instance, spin-zero FTL particles satisfying the Klein-Gordon equation cannot be localized as pointlike particles, contrary to all other known fundamental constituents of matter. Perhaps the most interesting suggestion in Cox and Hill's paper is that a certain freedom is allowed in the theory of superluminal particles as no such particle has been reliably detected until now, so that it is worthwhile exploring theoretical alternatives. We shall prove that an idea from Cox and Hill's paper, when applied not in the context of extended relativity but in the framework of F. R. Tangherlini's absolute synchronization, allows the problem of tachyon localization for spinless particles to be solved.
The paper gives the solution and analysis of projectile motion in a vacuum if the launch and
impact heights are not equal. Formulas for maximum horizontal range and the corresponding
angle are derived. An Excel application that simulates the motion is presented. The result of
an experiment is given in which 38 secondary school students developed the application and
made investigation into the system. A questionnaire survey was carried out to find out
whether the students found the lessons interesting, learned new skills and wanted to model
projectile motion in the air as an example of more realistic motion. The results are discussed.
The article presents a way of solving the problem of planetary motion, or, the Kepler problem, without using the transition. The governing equation is solved for the components of the velocity vector in Cartesian coordinates. Substitution for speed in the law of energy conservation yields the equation of the trajectory. A time implicit closed formula for the azimuth is derived. An Excel application is presented that simulates the motion by solving the azimuth equation numerically without using programming.
Many textbooks formulate the Kepler's third law in terms of a mean distance between the Sun and each of the planets, but nothing is said about what kind of mean is. In this work we calculate three diferent means for this distance, and verify that only one is compatible with the referred law. We also calculate and compare the three means for all relevant physical quantities.
We consider a relativistic particle under the action of a time-periodic central force field in the plane. When it is attractive
at a given level there are many subharmonic and quasi-periodic motions.
In this paper, using Leray–Schauder degree arguments, critical point theory for lower semicontinuous functionals and the method of lower and upper solutions, we give existence results for periodic problems involving the relativistic operator
with
. In particular we show that in this case we have non-resonance, that is periodic problem
has at least one solution for any continuous function
. Then, we consider Brillouin and Mathieu-Duffing type equations for which
.
The orbital precession of a charged test body interacting with a charged and massive body, fixed
in the origin, is obtained by means of an approximate solution to the motion equations for such a system.
Once the precession is known, in order to fit the Mercury perihelion precession uncertainty, some analysis
regarding the possible charges for the Sun and Mercury is made.
The analytic solution of a body falling an arbitrary distance toward a gravitational source is presented. This problem has apparently received little attention in textbooks. The solution can be extended smoothly through the singularity at the origin to form a class of trajectories that we call singular orbits. Trajectories in phase space have singular branches, yet cycle in a finite time. We calculate the period of oscillation about the singularity confirming that the singular orbits obey Kepler's third law. Viewed as geodesics in a Newtonian 1+1 spacetime, the geodesic deviation equation is derived and solved analytically, providing an excellent example of curvature physics in Newtonian spacetime. The results are used to estimate the duration during which a freely falling local frame can be considered inertial. A numerical investigation with damping included shows that a final state is reached in which the particle is confined to the origin, but acquires infinite speed during each pass. We show via some examples, the pedagogical applications of the solution.
We study the gravity assist in the general case, i.e. when the spacecraft is not in a coplanar motion with respect to the planet's orbit. Our derivation is based on Kepler's planetary motion and Galilean addition of velocities, subjects covered in introductory physics courses. The main purpose of this paper is to illustrate how the gravity assist can be used to deviate a spacecraft outside its original plane of motion. As an example, we use the NASA-ESA's Ulysses mission to "test" our simple model. Comment: 12 pages, 7 figures
ResearchGate has not been able to resolve any references for this publication.