Integrable magnetic geodesic flows on lie groups

Abstract

On Lie group manifolds, we consider right-invariant magnetic geodesic flows associated with 2-cocycles of the corresponding Lie algebras. We investigate the algebra of the integrals of motion of magnetic geodesic flows and also formulate a necessary and sufficient condition for their integrability in quadratures, giving the canonical forms of 2-cocycles for all four-dimensional Lie algebras and selecting integrable cases.

Integrating Klein-Gordon-Fock equations in an external electromagnetic field on Lie groups

“…A proof of the statement is given in [14]. As a note, we emphasize that the proposed particular solution is local.…”

“…where ind [F] g is our notation for the cohomological index of the Lie algebra g of a class [F] ∈ H 2 (g), introduced in [14] ind…”

“…on the bilinear form F. As noted in the preceding section, condition (15) means that the form F is a 2-cocycle of g with values in R. Consequently, the set of invariant closed 2-forms on G is in one-to-one correspondence with the set Z 2 (g). We note that formulas (12) and (14) are tetradic decompositions of the tensors g ij and F ij over the right-invariant tetrad of 1-forms {σ a }.…”

“…A construction of a vector potential corresponding to the strength tensor of a right-invariant electromagnetic field on a Lie group was given in [14]. For the completeness of our exposition, we here present the basic results concerning this question.…”

“…In the KGF equation, we switch on a right-invariant electromagnetic field defined using 2form (14). We assume that the metric is right-invariant, i.e., the corresponding squared length element has form (12).…”