Group classification of charged particle motion in stationary electromagnetic fields

Abstract

In this paper we classify in terms of Lie point symmetries the three-dimensional nonrelativistic motion of charged particles in arbitrary time-independent electromagnetic fields. The classification is made on the ground of equivalence transformations, and, when the system is nonlinear and particularly for inhomogeneous and curved magnetic fields, it is also complete. Using the homogeneous Maxwell's equations as auxiliary conditions for consistency, in which case the system amounts to a Lagrangian of three degrees of freedom with velocity-dependent potentials, the equivalence group stays the same. Therefore, instead of the actual fields, the potentials are equally employed and their gauge invariance results in an infinite-dimensional equivalence algebra, which nevertheless projects to finite-dimensional symmetry algebras. Subsequently, optimal systems of equivalence subalgebras are obtained that lead to one-, two-and three-parameter extended symmetry groups, besides the obvious time translations. Finally, based on symmetries of Noether type, aspects of complete integrability are discussed, as well.1. In this paper, by "symmetry" we will always mean "Lie point symmetry", even when not explicitly stated.

Lie symmetries for systems of evolution equations

“…The reduction process is the main application of Lie symmetries, however, it is not a univocal approach. Symmetries can be used for the determination of conservation currents [3], for the classification of differential equations [4][5][6][7][8][9][10] and for the reconnaissance of some well-known systems [11][12][13][14][15].In the recent literature, it has been shown that there is a close relation between the Lie symmetries of a second order differential equation and the geometry of the space where motion occurs. For example, the conservation of energy and angular momentum in Newtonian Physics is a result of the Lie point symmetries, generated by the Killing vectors of translations and rotations respectively.…”

“…The reduction process is the main application of Lie symmetries, however, it is not a univocal approach. Symmetries can be used for the determination of conservation currents [3], for the classification of differential equations [4][5][6][7][8][9][10] and for the reconnaissance of some well-known systems [11][12][13][14][15].…”

Dynamics of dipole in a stationary non-homogeneous electromagnetic field

“…Fact that the existence of a first integral linear in momenta is connected with certain symmetry of the system is well known. For a single charged particle motion in stationary electromagnetic fields the results of group symmetry analysis are known, see, for example 12 , 13 . However system considered in this paper is much more complicated – it has five degrees of freedom and the form of electromagnetic field is not specified.…”

Dynamics of charged particles in electromagnetic fields with applications in fusion devices

“…Parts of this thesis, chapters 2 and 5, have been published in international scientific peerreviewed journals, namely [62] and [63], respectively. Others, chapters 3 and 6, which correspond to [61] and [60] accordingly, will soon be submitted.…”

“…It gives me great pleasure to acknowledge the significance of my scientific interaction with Dr Stelios Dimas, who introduced me to the theory of equivalence transformations. Without his contribution, chapter 6 and [60] accordingly would not even begun to exist. Stelio, thank you for all the time and effort you put through from the other side of the world to get into my head over skype how classifying equations should be treated.…”