Shape-invariant orbits and their Laplace-Runge-Lenz vectors for a class of “Double Potentials”

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Abstract

We derive exact E = 0 classical solutions for the following class of Hamiltonians with “double potentials” (Formula presented), where (Formula presented). For μ = –1/2 and μ = –1 the HD yields the Kepler and oscillator systems for E ≠ 0, respectively. The classical orbits of HD are shape invariant for a wide range of γ and λ, in the sense that each maximum of their orbits r(φ)is followed by a minimum after an angular shift of Δφ = π/2μ. We map the LRL vector M:= (M1,M2) of the Kepler problem to a complex expression Mμ ∈ ℂ, which is conserved for every μ. We use Mμ to derive a general expression for the orbit r(φ,μ;γ,λ) for all μ ≠ 0. We also contrast the limit of the above orbits as λ → 0 with those considered by Daboul and Nieto for the power-law potentials VP:= –γ/r2+2μ.

Original languageEnglish
Title of host publicationLie Theory and Its Applications in Physics, 2013
EditorsVladimir Dobrev
PublisherSpringer New York LLC
Pages551-559
Number of pages9
ISBN (Electronic)9784431552840
DOIs
StatePublished - 1 Jan 2014
Event10th Workshop on Lie Theory and Its Applications in Physics, LT 2013 - , Bulgaria
Duration: 17 Jun 201323 Jun 2013

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume111
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

Conference10th Workshop on Lie Theory and Its Applications in Physics, LT 2013
Country/TerritoryBulgaria
Period17/06/1323/06/13

ASJC Scopus subject areas

  • General Mathematics

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