Particle Motion in Monopoles and Geodesics on Cones

Abstract

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.

Particle motion in generalized Dirac’s monopoles of dimension 2k + 1

“…Let C be the cone of aperture ψ directed along L ∈Ṙ n+1 and let V be a m-dimensional subspace containing L. Then, C ∩ V is a (m − 1)-dimensional cone of aperture ψ.The following is our main theorem which extends Theorem 7.1 of Ref [10]…”

“…So r lies on the cone C of aperture ψ = arccos( |λ| √ k| L| ) directed along ⋆ L. From proposition 4.2, we know r lies on the 2-dimensional cone C ∩ [V ].Also we have(r ′′ , r) = ((iξ, r ′ F ), r) = ((iξ, F ), r ′ ∧ r) = −((iξ, r F ), r ′ ) = 0, and (r ′′ , r ′ ) = ((iξ, r ′ F ), r ′ ) = ((iξ, F ), r ′ ∧ r ′ ) = 0.Thus r ′′ is orthogonal to r and r ′ . Then form corollary 6.1 of Ref [10],. we know r is a geodesic on the cone of aperture ψ = arccos( |λ| √ k| L| ) directed along ⋆ L.…”

“…Then form corollary 6.1 of Ref. [10], we know r is a geodesic on the cone of aperture ψ = arccos( |λ| √ k|L| ) directed along ⋆L.…”

“…First of all, we review some definitions and propositions from Mayrand [10]. We write an "n-dimensional cone in R n+1 " for any such cone.…”

“…Recently, Mayrand [10] showed that particles in Yang's monopole (which is a natural generalization of Dirac's monople in R 5 ) follow geodesics on 2-dimensional cones, and hence all solutions can be obtained explicitly.…”