TY - GEN

T1 - Shape-invariant orbits and their Laplace-Runge-Lenz vectors for a class of “Double Potentials”

AU - Daboul, Jamil

N1 - Publisher Copyright:
© Springer Japan 2014.

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84922129542&partnerID=8YFLogxK

U2 - 10.1007/978-4-431-55285-7_42

DO - 10.1007/978-4-431-55285-7_42

M3 - Conference contribution

AN - SCOPUS:84922129542

T3 - Springer Proceedings in Mathematics and Statistics

SP - 551

EP - 559

BT - Lie Theory and Its Applications in Physics, 2013

A2 - Dobrev, Vladimir

PB - Springer New York LLC

T2 - 10th Workshop on Lie Theory and Its Applications in Physics, LT 2013

Y2 - 17 June 2013 through 23 June 2013

ER -