Canonical transformations

2020 ◽  
pp. 59-66
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Harmonic Oscillator ◽  
Canonical Transformation ◽  
Hamiltonian Function ◽  
Harmonic Oscillators ◽  
Nonlinear Coupling ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Anharmonic Oscillations ◽  
The Given

This chapter addresses the canonical transformation defined by the given generating function, the rotation in the phase space as a canonical transformation, and themovement of the system as a canonical transformation. The chapter also discusses using the canonical transformations for solving the problems of the anharmonic oscillations and using the canonical transformation to diagonalize the Hamiltonian function of an anisotropic charged harmonic oscillator in a magnetic field. Finally, the chapter addresses the canonical variables which reduce the Hamiltonian function of the harmonic oscillator to zero and using them for consideration of the system of the harmonic oscillators with the weak nonlinear coupling.

Canonical transformations

2020 ◽  
pp. 317-337
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Harmonic Oscillator ◽  
Canonical Transformation ◽  
Hamiltonian Function ◽  
Harmonic Oscillators ◽  
Nonlinear Coupling ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Anharmonic Oscillations ◽  
The Given

This chapter addresses the canonical transformation defined by the given generating function, the rotation in the phase space as a canonical transformation, and themovement of the system as a canonical transformation. The chapter also discusses using the canonical transformations for solving the problems of the anharmonic oscillations and using the canonical transformation to diagonalize the Hamiltonian function of an anisotropic charged harmonic oscillator in a magnetic field. Finally, the chapter addresses the canonical variables which reduce the Hamiltonian function of the harmonic oscillator to zero and using them for consideration of the system of the harmonic oscillators with the weak nonlinear coupling.

scholarly journals Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

2019 ◽  
Vol 23 (01) ◽  
pp. 1950084
Author(s):  
Wenmin Gong
Keyword(s):  
Magnetic Field ◽  
Periodic Orbits ◽  
Homotopy Class ◽  
Linear Growth ◽  
Floer Homology ◽  
Hamiltonian Function ◽  
Universal Cover ◽  
Zero Section ◽  
Free Homotopy Class ◽  
The Given

In this paper, we establish the existence of periodic orbits belonging to any [Formula: see text]-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact [Formula: see text]-form [Formula: see text] admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any non-trivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran–Polterovich–Salamon capacity.

QUANTUM MECHANICS WITH p-ADIC NUMBERS

1989 ◽  
Vol 04 (19) ◽  
pp. 5133-5147 ◽  
Author(s):  
YANNICK MEURICE
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Heisenberg Group ◽  
Canonical Transformation ◽  
Free Particle ◽  
Complex Hilbert Space ◽  
Canonical Transformations ◽  
Linear Canonical Transformation

We discuss unitary realizations of the Heisenberg group and the linear canonical transformations over a complex Hilbert space but with dynamical variables on a p-adic field Qp. For all p, an appropriate choice of phase turns the realization of the linear canonical transformation into a representation up to a sign of SL (2, Qp). We give the spectra of the subgroups corresponding to the free particle and the harmonic oscillator. We discuss briefly the possibility of an adelic interpretation.

Формирование изображений электрических сигналов преобразователя магнитного поля при гистерезисной интерференции для контроля металлов в импульсных магнитных полях

2020 ◽  
pp. 38-45
Author(s):  
В.В. Павлюченко ◽  
Е.С. Дорошевич
Keyword(s):  
Magnetic Field ◽  
Aluminum Plate ◽  
Thickness Variation ◽  
Magnetic Carrier ◽  
Electric Voltage ◽  
Total Strength ◽  
Magnetic Field Transducer ◽  
The Given ◽  
The Magnetic Field

Based on the developed methods of hysteresis interference, the calculated dependences U(x) of the electric voltage taken from the magnetic field transducer on the x coordinate were obtained. A magnetic carrier with an arctangent characteristic was exposed to a series of bipolar pulses of the magnetic field of a linear inductor of one, two, three, four, five and fifteen pulses. An algorithm is presented for the sequence of changes in the magnitude of the total strength of the magnetic field pulses on the surface of an aluminum plate, which provides the same amplitude of hysteresis oscillations of the electric voltage and makes it possible to obtain a linear difference dependence U(x) for wedge-shaped and flat aluminum samples. The results obtained make it possible to increase the accuracy and efficiency of control of the thickness of the object and its thickness variation in the given directions, as well as the defects of the object.

Hamiltonian Mechanics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Angular Momentum ◽  
Jacobi Equation ◽  
Modern Theory ◽  
Hamiltonian Mechanics ◽  
Lagrangian Mechanics ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Light Waves ◽  
Hamilton Jacobi Equation ◽  
Common Action

Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

Harmonic Oscillators with Nonlinear Damping

2017 ◽  
Vol 27 (11) ◽  
pp. 1730037 ◽  
Author(s):  
J. C. Sprott ◽  
W. G. Hoover
Keyword(s):  
Dynamical Systems ◽  
Harmonic Oscillator ◽  
Strange Attractor ◽  
Nonlinear Damping ◽  
Harmonic Oscillators ◽  
Coexisting Attractors ◽  
Simple Harmonic Oscillator ◽  
Interesting Behavior

Dynamical systems with special properties are continually being proposed and studied. Many of these systems are variants of the simple harmonic oscillator with nonlinear damping. This paper characterizes these systems as a hierarchy of increasingly complicated equations with correspondingly interesting behavior, including coexisting attractors, chaos in the absence of equilibria, and strange attractor/repellor pairs.

Brownian motion of a classical harmonic oscillator in a magnetic field

2008 ◽  
Vol 77 (5) ◽  
Author(s):  
J. I. Jiménez-Aquino ◽  
R. M. Velasco ◽  
F. J. Uribe
Keyword(s):  
Magnetic Field ◽  
Brownian Motion ◽  
Harmonic Oscillator

scholarly journals Integrals of the motion and green functions for time-dependent mass harmonic oscillators

2018 ◽  
Vol 64 (1) ◽  
pp. 30
Author(s):  
Surarit Pepore
Keyword(s):  
Harmonic Oscillator ◽  
Harmonic Oscillators ◽  
Time Dependent ◽  
Green Functions ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Damped Harmonic Oscillator ◽  
System Trajectory ◽  
Dependent Mass ◽  
Motion Method

The application of the integrals of the motion of a quantum system in deriving Green function or propagator is established. The Greenfunction is shown to be the eigenfunction of the integrals of the motion which described initial points of the system trajectory in the phasespace. The explicit expressions for the Green functions of the damped harmonic oscillator, the harmonic oscillator with strongly pulsatingmass, and the harmonic oscillator with mass growing with time are obtained in co-ordinate representations. The connection between theintegrals of the motion method and other method such as Feynman path integral and Schwinger method are also discussed.

scholarly journals Theory of Generalized Canonical Transformations for Birkhoff Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Yi Zhang
Keyword(s):  
Canonical Transformation ◽  
Natural Generalization ◽  
Analytical Mechanics ◽  
Hamiltonian Mechanics ◽  
Canonical Transformations ◽  
Transformation Theory ◽  
Special Cases ◽  
Criterion Equation ◽  
Important Means ◽  
Transformation Formulas

Transformation is an important means to study problems in analytical mechanics. It is often difficult to solve dynamic equations, and the use of variable transformation can make the equations easier to solve. The theory of canonical transformations plays an important role in solving Hamilton’s canonical equations. Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics. This paper attempts to extend the canonical transformation theory of Hamilton systems to Birkhoff systems and establish the generalized canonical transformation of Birkhoff systems. First, the definition and criterion of the generalized canonical transformation for the Birkhoff system are established. Secondly, based on the criterion equation and considering the generating functions of different forms, six generalized canonical transformation formulas are derived. As special cases, the canonical transformation formulas of classical Hamilton’s equations are given. At the end of the paper, two examples are given to illustrate the application of the results.

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