Optical mechanical analogy and nonlinear nonholonomic constraints

Bloch, Anthony M.; Rojo, A. G.

doi:10.1103/physreve.93.023005

Cited by 7 publications

(9 citation statements)

References 20 publications

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Section: Literature Review and Problem Statementmentioning

confidence: 99%

“…Optical-mechanical analogy has remained relevant up to now [8][9][10]. Study [8] shows the existence of connection between the trajectories of particles under the action of nonholonomic constraint, and the trajectories of light rays with a variable refractive index. Article [9] provides a proof of the existence of a new optical-mechanical analogy between the equation of rotational motion of the body in mechanics (taking into account the principle of relativity) and the first pair of Maxwell's equations.…”

Section: Literature Review and Problem Statementmentioning

confidence: 99%

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A method of V-function: ultimate solution to the direct and inverse problems of dynamics for a hydrogen-like atom

Valishin

Моисеев

2017

EEJET

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На базі методу V-функції здійснюється продовження оптико-механічної аналогії. Розглядається траєктор-но-хвильовий рух частинки. Вказується на наявність квантування енергії частинки, рішення без частинки в разі прямолінійного рівномірного руху з постійною швид-кістю. Досліджуються властивості хвильової природи руху електрона в водородоподібному атомі як рішення прямої задачі. Показується спосіб знаходження кінцевого рішення стаціонарного хвильового рівняння Ключові слова: варіаційний принцип, пряма задача динаміки, зворотна задача динаміки, оптико-механіч-на аналогія, хвильовий рух, траєкторний рух, хвильова функція, хвильове рівняння На базе метода V-функции осуществляется продол-жение оптико-механической аналогии. Рассматривается траекторно-волновое движение частицы. Указывается на наличие квантования энергии частицы, решения без частицы в случае прямолинейного равномерного движе-ния с постоянной скоростью. Исследуются свойства волновой природы движения электрона в водородоподоб-ном атоме как решение прямой задачи. Показывается способ нахождения конечного решения стационарного волнового уравнения Ключевые слова: вариационный принцип, прямая задача динамики, обратная задача динамики, опти-ко-механическая аналогия, волновое движение, траек-торное движение, волновая функция, волновое уравнение

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Section: Literature Review and Problem Statementmentioning

confidence: 99%

Section: Literature Review and Problem Statementmentioning

confidence: 99%

A method of V-function: ultimate solution to the direct and inverse problems of dynamics for a hydrogen-like atom

Valishin

Моисеев

2017

EEJET

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“…The optical-mechanical analogy, as expressed by Hamilton, refers to the isomorphism between the trajectories of a particle moving in a potential, and that of a light ray propagating through a medium. [26] It stems from analogous conservation laws and can be expressed as an equivalence between the momentum of a particle and the refraction index of a light ray:…”

Section: The Optical-mechanical Analogymentioning

confidence: 99%

The “Chemical Mechanics” of the Periodic Table

Ceulemans¹,

Thyssen²

2018

Mendeleev to Oganesson

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In 1969, the centennial of Mendeleev’s discovery of the periodic table was commemorated by an international conference devoted to the periodicity and symmetry of the elementary structure of matter. The conference was held in the Vatican and brought together a selected audience of first-rate atomic and nuclear scientists. In 1971, the proceedings were published in a joint publication [1] of the Academy of Sciences of Torino and the National Academy in Rome. Among the many interesting contributions, the American cosmologist John Archibald Wheeler described a mind-boggling journey from “Mendeleev’s atom to the collapsing star.” According to Wheeler [2], Mendeleev was convinced that the atom is not “deathlike inactivity” but a dynamic reality and Mendeleev expressed his hope that the discovery of an orderly pattern would “hasten the advent of a true chemical mechanics.” This hope has certainly been met by Schrödinger’s wave mechanics, which provides an accurate tool to simulate the properties of the elements. However, the overall structure and symmetry of the periodic table continues to defy understanding. The quest for an effective universal force law at the basis of the mechanics of multi-electron atoms forms the topic of this contribution. The search for central force laws should start with Bertrand’s theorem in classical mechanics. In 1873, the French mathematician Joseph Louis Bertrand presented to the Paris Academy a short note [3, 4] on central force laws that give rise to stable orbits. For a proper understanding of the research question which Bertrand was addressing, we start from an everyday experiment. A mass attached to a string can easily be swept around in a perfectly circular orbit by simply pulling on the string. The only requirement is that the force should be fixed and directed toward the center of the orbit. If we want the mass to go faster, we simply have to pull harder.

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“…Here, we will demonstrate with a null geodesic in isotropic space how refractive phenomena can arise from a gravitational metric, as shown in [21]. Suppose we have an isotropic space-time metric given by:…”

Section: Preliminaries: Null-geodesicsmentioning

confidence: 99%

Dynamical systems of null geodesics and solutions of Tomimatsu-Sato 2

Chanda,

Guha

2017

Preprint

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We have studied optical metrics via null geodesics and optical-mechanical formulation of classical mechanics, and described the geometry and optics of mechanical systems with drag dependent quadratically on velocity. Then we studied null geodesics as a central force system, deduced the related Binet's equation applied the analysis to other solutions of Einstein's equations in spherically symmetric spaces, paying special attention to the Tomimatsu-Sato metric. Finally, we examined the dualities between different systems arising from conformal transformations that preserve the Jacobi metric.

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Optical mechanical analogy and nonlinear nonholonomic constraints

Cited by 7 publications

References 20 publications

A method of V-function: ultimate solution to the direct and inverse problems of dynamics for a hydrogen-like atom

A method of V-function: ultimate solution to the direct and inverse problems of dynamics for a hydrogen-like atom

The “Chemical Mechanics” of the Periodic Table

Dynamical systems of null geodesics and solutions of Tomimatsu-Sato 2

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