Symmetries and invariants for some cases involving charged particles and general electromagnetic fields: a brief review

Ritter, O.

doi:10.1590/s0103-97332000000200029

Cited by 8 publications

(8 citation statements)

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“…The first system, (1.1), has been thoroughly studied in the literature [5,6,7,8,10,12,14,23,25,26,28]. The main result -as shown first by Poincaré -is that for every solution r, there is a cone with vertex at the origin on which r is a geodesic (it follows from (1.1) that |9 r| is constant).…”

Section: Introductionmentioning

confidence: 99%

Particle Motion in Monopoles and Geodesics on Cones

Mayrand¹

2014

SIGMA

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Abstract. The equations of motion of a charged particle in the field of Yang's SU(2) monopole in 5-dimensional Euclidean space are derived by applying the Kaluza-Klein formalism to the principal bundle R 8 \ {0} → R 5 \ {0} obtained by radially extending the Hopf fibration S 7 → S 4 , and solved by elementary methods. The main result is that for every particle trajectory r : I → R 5 \ {0}, there is a 4-dimensional cone with vertex at the origin on which r is a geodesic. We give an explicit expression of the cone for any initial conditions.

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Section: Introductionmentioning

confidence: 99%

Particle Motion in Monopoles and Geodesics on Cones

Mayrand¹

2014

SIGMA

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“…This case has already been investigated in [8]. So, in light of ( 16), equations (15) trivially result in…”

Section: Lie Point Symmetriesmentioning

confidence: 85%

“…Our intention is to cover a variety of applications, in which the electromagnetic field may be quite complicated, such as those occurring in plasma physics and fusion devices. A series of simpler problems for particular choices of the functions B and E is presented in [8].…”

Section: Introductionmentioning

confidence: 99%

Symmetries of charged particle motion under time-independent electromagnetic fields

Kallinikos¹,

Meletlidou²

2013

J. Phys. A: Math. Theor.

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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.

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“…Since then, this system has been thoroughly studied by many people, see Ref. [3,4,5,6,7,8,9,19,21,22]. Poincaré firstly showed that for every solution r, there is a cone with vertex at the origin on which r is a geodesic.…”

Section: Introductionmentioning

confidence: 99%