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.
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In this paper, by “symmetry” we will always mean “Lie point symmetry,” even when not explicitly stated.
In fact, instead of V, it suffices to take the projection of V to the (x, A, Φ)-space. Since, however, this would slightly simplify just V8 by leaving out tt, in order to spare notation, we continue to consider the whole of V.
