Nonlinear realizations, the orbit method and Kohn's theorem (original) (raw)

Nonlinear realizations, the orbit method and

2016

The orbit method is used to describe the centre of mass motion of the system of particles with fixed charge to mass ratio moving in homogeneous magnetic field and confined by harmonic potential. The nonlinear action of symmetry group on phase space is identified and compared with the one obtained with the help of Eisenhart-Duval lift.

Nonlinear realizations, the orbit method and Kohnʼs theorem

Physics Letters B, 2012

Symmetry analysis for a charged particle in a certain varying magnetic field

2003

We analyze the classical equations of motion for a particle moving in the presence of a static magnetic field applied in the z direction, which varies as 1 x 2. We find the symmetries through Lie's method of group analysis. In the corresponding quantum mechanical case, the method of spectrum generating su(1, 1) algebra is used to find the energy levels for the Schroedinger equation without explicitly solving the equation. The Lie point symmetries are enumerated. We also find that for specific eigenvalues the vector field contains 1 x ∂ ∂x and 1 x 2 ∂ ∂x type of terms and a finite Lie product of the generators do not close.

Center of the charged particle orbit for any linear gauge

The equation of motion for a charged particle moving in the ndimensional constant magnetic filed is obtained for any linear gauge and any metric tensor by generalization of Johnson and Lipmann's approach. It allows to consider the magnetic orbits in the n-dimensional space. It is shown that the movement of a particle can always be decomposed into a number of two-dimensional cyclotronic motions and a free particle part.

Symmetries of charged particle motion under time-independent electromagnetic fields

Journal of Physics A: Mathematical and Theoretical, 2013

A symmetry analysis is presented for the three-dimensional nonrelativistic motion of charged particles in arbitrary stationary electromagnetic fields. The general form of the Lie point symmetries is found along with the fields that respect them, considering non-trivial cases of physical interest. The restrictions placed upon the electromagnetic field yield five classes of solutions, expressed in terms of the vector and scalar potentials. The Noether type symmetries are also investigated and their corresponding invariants are found. A second integral of motion, besides the Hamiltonian, results in three general cases. Finally, a relation between the symmetries of the charged particle motion and the symmetries of the magnetic field lines is established.

Illustrating dynamical symmetries in classical mechanics: The Laplace–Runge–Lenz vector revisited

American Journal of Physics, 2003

The inverse square force law admits a conserved vector that lies in the plane of motion. This vector has been associated with the names of Laplace, Runge, and Lenz, among others. Many workers have explored aspects of the symmetry and degeneracy associated with this vector and with analogous dynamical symmetries. We define a conserved dynamical variable α that characterizes the orientation of the orbit in two-dimensional configuration space for the Kepler problem and an analogous variable β for the isotropic harmonic oscillator. This orbit orientation variable is canonically conjugate to the angular momentum component normal to the plane of motion. We explore the canonical one-parameter group of transformations generated by α (β). Because we have an obvious pair of conserved canonically conjugate variables, it is desirable to use them as a coordinate-mometum pair. In terms of these phase space coordinates, the form of the Hamiltonian is nearly trivial because neither member of the pair can occur explicitly in the Hamiltonian. From these considerations we gain a simple picture of dynamics in phase space. The procedure we use is in the spirit of the Hamilton-Jacobi method.

On orbits for a particular case of axial symmetry

Serbian Astronomical Journal, 2009

A particular case of steady state and axial symmetry -the potential formula proposed by Miyamoto and Nagai - is studied. A number of orbits of a bound test particle is determined numerically, with both, the potential parameters and initial conditions, varied. Unlike special cases, such as nearly circular and nearly planar orbits, in the case of 'truly spatial orbits' the time dependence of the coordinates becomes very complicated and a mathematical treatment including any known periodic functions is hardly possible. Bearing in mind that orbits studied in the present paper are determined by three elements, the authors propose the mean values over time of the squares of velocity components to characterize them.

Quantum Orbit Method in the Presence of Symmetries

Symmetry, 2021

We review some of the main achievements of the orbit method, when applied to Poisson– Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider C∗-algebra quantization obtained through groupoid techniques, and we try to put the results obtained in algebraic or representation theoretical contexts in relation with groupoid quantization.

Revolution of Charged Particles Round a Centre of Force of Attraction

2006

Charged particles of masses nm revolve in orbits through angle ψ at distances nr from a centre of force of attraction, with angular momenta nL perpendicular to the plane of the orbit, where n is an integer greater than 0 and m is the electronic mass. The equation of motion of the nth orbit of revolution is derived revealing that excited particles revolve in unclosed ellipses, with emission of radiation at the frequencies of revolution, before settling down, after many cycles of ψ, into equally spaced stable circular planar orbits. In unipolar revolution, radiating particles each of mass nm, carrying the electronic charge –e, revolve in unclosed elliptic orbits round a positively charged nucleus, before settling into circular stable orbits of radii nr1, where r1 is the radius of the first orbit. In bipolar revolution, two radiating particles of the same mass nm and charges e and –e, revolve in unclosed ellipses round a common centre of mass before reverting into circular stable orbits of radii ns1, where s1 is the radius of the first orbit. Discrete masses nm and angular momenta nL lead to quantization of the orbits outside Bohr’s quantum mechanics. Keywords: Angular momentum, force, frequency, mass, orbit of revolution, radiation.

Nonlinear realizations, the orbit method and Kohn's theorem (original) (raw)

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