The Hamiltonian equations of motion. Poisson brackets

2020 ◽  
pp. 305-316
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Hydrogen Atom ◽  
Particle Velocity ◽  
Conservation Law ◽  
Equations Of Motion ◽  
Hamiltonian Function ◽  
Poisson Brackets ◽  
Hamiltonian Equations ◽  
Hidden Symmetry

This chapter addresses invariance of the Hamiltonian function under a given transformation and the conservation law, the Hamiltonian function for the beam of light, the motion of a charged particle in a nonuniform magnetic field, and the motion of electrons in a metal or semiconductor. The chapter also discusses the Poisson brackets and the model of the electron and nuclear paramagnetic resonances, the Poisson brackets for the components of the particle velocity, and the “hidden symmetry” of the hydrogen atom.

The Hamiltonian equations of motion. Poisson brackets

2020 ◽  
pp. 54-58
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Hydrogen Atom ◽  
Particle Velocity ◽  
Conservation Law ◽  
Equations Of Motion ◽  
Hamiltonian Function ◽  
Poisson Brackets ◽  
Hamiltonian Equations ◽  
Hidden Symmetry

This chapter addresses invariance of the Hamiltonian function under a given transformation and the conservation law, the Hamiltonian function for the beam of light, the motion of a charged particle in a nonuniform magnetic field, and the motion of electrons in a metal or semiconductor. The chapter also discusses the Poisson brackets and the model of the electron and nuclear paramagnetic resonances, the Poisson brackets for the components of the particle velocity, and the “hidden symmetry” of the hydrogen atom.

Two-dimensional motion of a parabolically confined charged particle in a perpendicular magnetic field

Open Physics ◽  
2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Orion Ciftja
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Equations Of Motion ◽  
Complex Variables ◽  
Perpendicular Magnetic Field ◽  
Two Dimensional ◽  
Cyclotron Motion ◽  
Lissajous Figures ◽  
Long Time ◽  
Concentric Circles

AbstractThe classical two-dimensional motion of a parabolically confined charged particle in presence of a perpendicular magnetic is studied. The resulting equations of motion are solved exactly by using a mathematical method which is based on the introduction of complex variables. The two-dimensional motion of a parabolically charged particle in a perpendicular magnetic field is strikingly different from either the two-dimensional cyclotron motion, or the oscillator motion. It is found that the trajectory of a parabolically confined charged particle in a perpendicular magnetic field is closed only for particular values of cyclotron and parabolic confining frequencies that satisfy a given commensurability condition. In these cases, the closed paths of the particle resemble Lissajous figures, though significant differences with them do exist. When such commensurability condition is not satisfied, path of particle is open and motion is no longer periodic. In this case, after a sufficiently long time has elapsed, the open paths of the particle fill a whole annulus, a region lying between two concentric circles of different radii.

On gyroresonance

1968 ◽  
Vol 2 (1) ◽  
pp. 59-64 ◽  
Author(s):  
M. J. Laird
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Pitch Angle ◽  
Uniform Magnetic Field ◽  
Equations Of Motion ◽  
Circularly Polarized ◽  
Simple Picture

The motion of a charged particle in the field of a plane circularly polarized wave propagating along a uniform magnetic field B0 is investigated. For the wave magnetic field small compared with B0, the equations of motion simplify to those of the pendulum, and a simple picture of what happens for particles near gyro- resonance results. Expressions are found for the amplitude and period of the pitch angle oscillations. Departures from uniformity and possible applications to the magnetosphere are briefly discussed.

scholarly journals DIRAC BRACKETS FROM MAGNETIC BACKGROUNDS

2007 ◽  
Vol 04 (04) ◽  
pp. 523-532 ◽  
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Phase Space ◽  
Symplectic Form ◽  
Weak Magnetic Field ◽  
Complex Geometry ◽  
Poisson Brackets ◽  
Generalized Complex ◽  
Dirac Brackets ◽  
Classical Phase Space

In symplectic mechanics, the magnetic term describing the interaction between a charged particle and an external magnetic field has to be introduced by hand. On the contrary, in generalized complex geometry, such magnetic terms in the symplectic form arise naturally by means of B-transformations. Here we prove that, regarding classical phase space as a generalized complex manifold, the transformation law for the symplectic form under the action of a weak magnetic field gives rise to Dirac's prescription for Poisson brackets in the presence of constraints.

scholarly journals Die Bewegung geladener Teilchen im Magnetfeld eines geraden stromdurchflossenen Drahtes

1959 ◽  
Vol 14 (1) ◽  
pp. 47-54 ◽  
Author(s):  
F. Hebtweck
Keyword(s):  
Magnetic Field ◽  
Angular Momentum ◽  
Charged Particle ◽  
Small Particle ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Particle Energy ◽  
Infinite Length ◽  
Electric Line ◽  
The Magnetic Field

The motion of a charged particle in the magnetic field of a straight electric line current of infinite length is investigated. Using the numerical solutions of the equations of motion the drift velocity of the particle along the wire is calculated. For small particle energies it turns out to be in agreement with ALFVÉN'S approximation. Also the limits of the region to which the particle is confined are calculated for different values of the particle energy and its angular momentum.

Cyclotron resonance in an inhomogeneous plasma

1972 ◽  
Vol 8 (2) ◽  
pp. 255-260 ◽  
Author(s):  
M. J. Laird
Keyword(s):  
Magnetic Field ◽  
Charged Particle ◽  
Cyclotron Resonance ◽  
Uniform Magnetic Field ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Inhomogeneous Plasma ◽  
Phase Speed ◽  
Hamiltonian Form ◽  
Energy And Momentum

The motion of a charged particle in a transverse wave of varying amplitude, wavelength and phase speed βp, propagating along a uniform magnetic field, together with a longitudinal electric field, is investigated. The equations of motion, in Hamiltonian form, are reduced to a system with two degrees of freedom in which integrable cases appear naturally. It is shown that particles may be locked in resonance with the wave, and expressions are found for the energy and momentum of such particles in terms of βp.

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