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Erschienen in:
2014 | OriginalPaper | Buchkapitel
Shape-Invariant Orbits and Their Laplace-Runge-Lenz Vectors for a Class of “Double Potentials”
verfasst von : Jamil Daboul
Erschienen in: Lie Theory and Its Applications in Physics
Verlag: Springer Japan
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Abstract
We derive exact E = 0 classical solutions for the following class of Hamiltonians with “double potentials” where For \(\mu = -1/2\) and \(\mu = -1\) the H
D
yields the Kepler and oscillator systems for E ≠ 0, respectively. The classical orbits of H
D
are shape invariant for a wide range of γ and λ, in the sense that each maximum of their orbits \(r(\varphi )\) is followed by a minimum after an angular shift of \(\varDelta \varphi =\pi /2\mu\). We map the LRL vector \(\mathbf{M}:= (M_{1},M_{2})\) of the Kepler problem to a complex expression \(M_{\mu } \in \mathbb{C}\), which is conserved for every μ. We use M
μ
to derive a general expression for the orbit \(r(\varphi,\mu;\gamma,\lambda )\) for all μ ≠ 0. We also contrast the limit of the above orbits as \(\lambda \rightarrow 0\) with those considered by Daboul and Nieto for the power-law potentials \(V _{P}:= -\gamma /r^{2+2\mu }\).
$$\displaystyle{H_{D}:= \frac{\mathbf{p}^{2}} {2m} + V _{D}(\,r),}$$
$$\displaystyle{V _{D}:= - \frac{\gamma } {r^{2+2\mu }} + \frac{\lambda } {r^{2+4\mu }}\;,\ \ \ \forall \ \ 0\neq \mu \in \mathbb{R}.}$$