Symmetries of charged particle motion under time-independent electromagnetic fields

Kallinikos, Nikos; Meletlidou, E.

doi:10.1088/1751-8113/46/30/305202

Cited by 17 publications

(24 citation statements)

References 20 publications

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“…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.…”

mentioning

confidence: 99%

Lie symmetries for systems of evolution equations

Paliathanasis

Tsamparlis

2018

Journal of Geometry and Physics

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The Lie symmetries for a class of systems of evolution equations are studied. The evolution equations are defined in a bimetric space with two Riemannian metrics corresponding to the space of the independent and dependent variables of the differential equations. The exact relation of the Lie symmetries with the collineations of the bimetric space is determined.Lie symmetries is a powerful method for the determination of solutions in the theory of differential equations. A Lie symmetry is important as it provides invariants which can be used to write a new differential equation with less degree of freedom. Furthermore, a solution of such an equation is also a solution of the original differential equation [1,2]. 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 general result for a holonomic autonomous dynamical system moving in a Riemannian space is that the Lie point symmetries of the equations of motion *

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mentioning

confidence: 99%

Lie symmetries for systems of evolution equations

Paliathanasis

Tsamparlis

2018

Journal of Geometry and Physics

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“…. , 8 on the (t, x)-space are none other than the symmetries found in [13] for inhomogeneous and curved magnetic fields. Apart from v 0 , they constitute the symmetry generator (see equation (21)…”

Section: Equivalence Transformations In Terms Of Fieldsmentioning

confidence: 86%

“…and since the symmetries of the equivalence group are the only symmetries for inhomogeneous and curved magnetic fields it is of no surprise that they are none other than the ones found in [13] (see equations (26)-(27) therein) from the symmetry condition. Of course, the general solution to the above system for the electromagnetic potential was given there in terms of the symmetry generator.…”

Section: Classifying Equationsmentioning

confidence: 99%

Group classification of charged particle motion in stationary electromagnetic fields

Kallinikos

2017

Journal of Mathematical Physics

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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.

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“…where by substitute in (21) it follows Application of the differential invariants of the autonomous symmetry vector ∂ ξ in (23) lead to the nonlinear…”

Section: Travel-wave Solutionmentioning

confidence: 99%

Lie Symmetries and Similarity Solutions for Rotating Shallow Water

Paliathanasis

2019

Zeitschrift Für Naturforschung A

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We study a nonlinear system of partial differential equations which describe rotating shallow water with an arbitrary constant polytropic index γ for the fluid. In our analysis we apply the theory of symmetries for differential equations and we determine that the system of our study is invariant under a five dimensional Lie algebra. The admitted Lie symmetries form the {2A1 ⊕s 2A1}⊕s A1 Lie algebra for γ = 1 and 2A1 ⊕s 3A1 for γ = 1. The application of the Lie symmetries is performed with the derivation of the corresponding zero-order Lie invariants which applied to reduce the system of partial differential equations into integrable systems of ordinary differential equations. For all the possible reductions the algebraic or closed-form solutions are presented. Travel-wave and scaling solutions are also determined.where h = h (t, x) denotes the height of the fluid surface, u = u (t, x) denotes the velocity component in the x-direction and v = v (t, x) is the other horizontal velocity component that is in the direction orhtogonal to the x−direction [29]. Parameter γ is the polytropic parameter of the fluid, where in this work is assumed to be 0 ≤ γ. The system (1)-(3) is important for the study of atmospheric phenomena like geostrophic adjustment

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Symmetries of charged particle motion under time-independent electromagnetic fields

Cited by 17 publications

References 20 publications

Lie symmetries for systems of evolution equations

Lie symmetries for systems of evolution equations

Group classification of charged particle motion in stationary electromagnetic fields

Lie Symmetries and Similarity Solutions for Rotating Shallow Water

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