Search results for "Kepler problem"
showing 6 items of 6 documents
Action-Angle Variables
2001
In the following we will assume that the Hamiltonian does not depend explicitly on time; ∂H/∂t = 0. Then we know that the characteristic function W(q i , P i ) is the generator of a canonical transformation to new constant momenta P i , (all Q i , are ignorable), and the new Hamiltonian depends only on the P i ,: H = K = K(P i ). Besides, the following canonical equations are valid: $$ \dot Q_i = \frac{{\partial K}} {{\partial P_i }} = v_i = const. $$ (1) $$ \dot P_i = \frac{{\partial K}} {{\partial Q_i }} = 0. $$ (2)
Chaotization of internal motion of excitons in ultrathin layers by spin–orbit coupling
2018
We show that Rashba spin-orbit coupling (SOC) can generate chaotic behavior of excitons in two-dimensional semiconductor structures. To model this chaos, we study a Kepler system with spin-orbit coupling and numerically obtain a transition to chaos at a sufficiently strong coupling. The chaos emerges since the SOC reduces the number of integrals of motion as compared to the number of degrees of freedom. Dynamically, the dependence of the exciton energy on the spin orientation in the presence of SOC produces an anomalous spin-dependent velocity resulting in chaotic motion. We observe numerically the critical dependence of the dynamics on the initial conditions, where the system can return to…
(2+1)-dimensional Einstein-Kepler problem in the centre-of-mass frame
1999
We formulate and analyze the Hamiltonian dynamics of a pair of massive spinless point particles in (2+1)-dimensional Einstein gravity by anchoring the system to a conical infinity, isometric to the infinity generated by a single massive but possibly spinning particle. The reduced phase space \Gamma_{red} has dimension four and topology R^3 x S^1. \Gamma_{red} is analogous to the phase space of a Newtonian two-body system in the centre-of-mass frame, and we find on \Gamma_{red} a canonical chart that makes this analogue explicit and reduces to the Newtonian chart in the appropriate limit. Prospects for quantization are commented on.
The 2 + 1 Kepler problem and its quantization
2001
We study a system of two pointlike particles coupled to three dimensional Einstein gravity. The reduced phase space can be considered as a deformed version of the phase space of two special-relativistic point particles in the centre of mass frame. When the system is quantized, we find some possibly general effects of quantum gravity, such as a minimal distances and a foaminess of the spacetime at the order of the Planck length. We also obtain a quantization of geometry, which restricts the possible asymptotic geometries of the universe.
On some Riemannian aspects of two and three-body controlled problems
2009
The flow of the Kepler problem (motion of two mutually attracting bodies) is known to be geodesic after the work of Moser [20], extended by Belbruno and Osipov [2, 21]: Trajectories are reparameterizations of minimum length curves for some Riemannian metric. This is not true anymore in the case of the three-body problem, and there are topological obstructions as observed by McCord et al. [19]. The controlled formulations of these two problems are considered so as to model the motion of a spacecraft within the influence of one or two planets. The averaged flow of the (energy minimum) controlled Kepler problem with two controls is shown to remain geodesic. The same holds true in the case of o…
On the discreet spectrum of fractional quantum hydrogen atom in two dimensions
2019
We consider a fractional generalization of two-dimensional (2D) quantum-mechanical Kepler problem corresponding to 2D hydrogen atom. Our main finding is that the solution for discreet spectrum exists only for $\mu>1$ (more specifically $1 < \mu \leq 2$, where $\mu=2$ corresponds to "ordinary" 2D hydrogenic problem), where $\mu$ is the L\'evy index. We show also that in fractional 2D hydrogen atom, the orbital momentum degeneracy is lifted so that its energy starts to depend not only on principal quantum number $n$ but also on orbital $m$. To solve the spectral problem, we pass to the momentum representation, where we apply the variational method. This permits to obtain approximate analytica…