Some possible Lagrangians for the quadratically damped system

In this paper, some possible Lagrangians for the quadratically damped systems are investigated. The corresponding classical Hamiltonians are investigated with Hamilton equations. The quantum Hamiltonians are also constructed so that they may be Hermitian. As an example, the quantum mechanics for the harmonic oscillator with a small quadratic damping is discussed.

Mei Symmetry and Conservation Laws for Time-Scale Nonshifted Hamilton Equations

The Mei symmetry and conservation laws for time-scale nonshifted Hamilton equations are explored, and the Mei symmetry theorem is presented and proved. Firstly, the time-scale Hamilton principle is established and extended to the nonconservative case. Based on the Hamilton principles, the dynamic equations of time-scale nonshifted constrained mechanical systems are derived. Secondly, for the time-scale nonshifted Hamilton equations, the definitions of Mei symmetry and their criterion equations are given. Thirdly, Mei symmetry theorems are proved, and the Mei-type conservation laws in time-scale phase space are driven. Two examples show the validity of the results.

Noether Symmetry Method for Hamiltonian Mechanics Involving Generalized Operators

Based on the generalized operators, Hamilton equation, Noether symmetry, and perturbation to Noether symmetry are studied. The main contents are divided into four parts, and every part includes two generalized operators. Firstly, Hamilton equations within generalized operators are established. Secondly, the Noether symmetry method and conserved quantity are studied. Thirdly, perturbation to the Noether symmetry and adiabatic invariant are presented. And finally, two applications are presented to illustrate the methods and results.

Tunneling Quantum Dynamics in Ammonia

Ammonia is a well-known example of a two-state system and must be described in quantum-mechanical terms. In this article, we will explain the tunneling phenomenon that occurs in ammonia molecules from the perspective of trajectory-based quantum dynamics, rather than the usual quantum probability perspective. The tunneling of the nitrogen atom through the potential barrier in ammonia is not merely a probability problem; there are underlying reasons and mechanisms explaining why and how the tunneling in ammonia can happen. Under the framework of quantum Hamilton mechanics, the tunneling motion of the nitrogen atom in ammonia can be described deterministically in terms of the quantum trajectories of the nitrogen atom and the quantum forces applied. The vibrations of the nitrogen atom about its two equilibrium positions are analyzed in terms of its quantum trajectories, which are solved from the Hamilton equations of motion. The vibration periods are then computed by the quantum trajectories and compared with the experimental measurements.

The Quantization of Gravity: Quantization of the Hamilton Equations

We quantize the Hamilton equations instead of the Hamilton condition. The resulting equation has the simple form −Δu=0 in a fiber bundle, where the Laplacian is the Laplacian of the Wheeler–DeWitt metric provided n≠4. Using then separation of variables, the solutions u can be expressed as products of temporal and spatial eigenfunctions, where the spatial eigenfunctions are eigenfunctions of the Laplacian in the symmetric space SL(n,R)/SO(n). Since one can define a Schwartz space and tempered distributions in SL(n,R)/SO(n) as well as a Fourier transform, Fourier quantization can be applied such that the spatial eigenfunctions are transformed to Dirac measures and the spatial Laplacian to a multiplication operator.

Hamiltonian Formulation for Continuous Third-order Systems Using Fractional Derivatives

We constructed the Hamiltonian formulation of continuous field systems with third order. A combined Riemann–Liouville fractional derivative operator is defined and a fractional variational principle under this definition is established. The fractional Euler equations and the fractional Hamilton equations are obtained from the fractional variational principle. Besides, it is shown that the Hamilton equations of motion are in agreement with the Euler-Lagrange equations for these systems. We have examined one example to illustrate the formalism. Keywords: Fractional derivatives, Lagrangian formulation, Hamiltonian formulation, Euler-lagrange equations, Third-order lagrangian.

The Quantum Regularization of Singular Black-Hole Solutions in Covariant Quantum Gravity

An excruciating issue that arises in mathematical, theoretical and astro-physics concerns the possibility of regularizing classical singular black hole solutions of general relativity by means of quantum theory. The problem is posed here in the context of a manifestly covariant approach to quantum gravity. Provided a non-vanishing quantum cosmological constant is present, here it is proved how a regular background space-time metric tensor can be obtained starting from a singular one. This is obtained by constructing suitable scale-transformed and conformal solutions for the metric tensor in which the conformal scale form factor is determined uniquely by the quantum Hamilton equations underlying the quantum gravitational field dynamics.

Time and classical equations of motion from quantum entanglement via the Page and Wootters mechanism with generalized coherent states

We draw a picture of physical systems that allows us to recognize what “time” is by requiring consistency with the way that time enters the fundamental laws of Physics. Elements of the picture are two non-interacting and yet entangled quantum systems, one of which acting as a clock. The setting is based on the Page and Wootters mechanism, with tools from large-N quantum approaches. Starting from an overall quantum description, we first take the classical limit of the clock only, and then of the clock and the evolving system altogether; we thus derive the Schrödinger equation in the first case, and the Hamilton equations of motion in the second. This work shows that there is not a “quantum time”, possibly opposed to a “classical” one; there is only one time, and it is a manifestation of entanglement.

FORMATION OF UNIVERSAL COMPETENCIES IN SOLVING HAMILTON EQUATIONS IN THE COURSE OF THEORETICAL PHYSICS (MODULE “CLASSICAL MECHANICS”)

Исследуется подход к формированию универсальных компетенций в курсе «Теоретическая физика. Модуль: Классическая механика» для студентов бакалавриата на примере раздела, связанного с нахождением закона движения тела. Акцент ставится на подходе к решению уравнений Гамильтона как системы дифференциальных уравнений первого порядка. Модуль «Классическая механика» является начальным этапом изучения теоретической физики. В этом разделе рассматриваются различные подходы к исследованию динамики механических систем, такие как Ньютоновская механика, Лагранжева механика и канонический формализм Гамильтона. Эти подходы являются эквивалентными, но формализм Гамильтона имеет ряд преимуществ. Тематика работы актуальна для студентов педагогических вузов, чьи профессиональные задачи предполагают умение осуществлять поиск, критический анализ и синтез информации, применять системный подход для решения поставленных задач (УК-1), обобщать теоретический материал и применять его к конкретным задачам с конкретными методическими целями. Цель разработки – помочь студентам увидеть общее и различное в решении задач, рассматривая предложенные задачи с единой позиции. Методическая задача состоит в формировании компетенций группы УК-1 при решении предлагаемых заданий. At present, the competence-based approach is dominant in education, since it presupposes, first of all, not the self-valuable assimilation of knowledge by students, but the opportunity to use this knowledge in the learning process to solve urgent problems. The most important feature of modern education is its universality. There are universal competences in all modules of the educational program and in various activities. This work is devoted to the formation of universal competencies in the course “Theoretical Physics. Module: Classical Mechanics” for undergraduate students on the example of a section related to finding the law of body motion. The emphasis is on the approach to solving Hamilton’s equations as a system of first-order differential equations. The module “Classical Mechanics” is the initial stage of the study of theoretical physics. This section discusses various approaches to the study of the dynamics of mechanical systems, such as Newtonian mechanics, Lagrangian mechanics, and Hamilton’s canonical formalism. These approaches are equivalent, but Hamilton’s formalism has several advantages. The topic of the work is relevant for students of pedagogical universities, whose professional tasks involve the ability to search, critical analysis and synthesis of information, apply a systematic approach to solving the assigned tasks (Universal Competencies-1), generalize theoretical material and apply it to specific tasks with specific methodological goals. The purpose of the development is to help students see the similar and different points in problem solving, considering the proposed problems from a unified position. The methodological task is to form the competencies of the UC-1 group when solving the proposed tasks.