This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Kicked rotator" – news · newspapers · books · scholar · JSTOR (May 2010) (Learn how and when to remove this message)

The kicked rotator, also spelled as kicked rotor, is a paradigmatic model for both Hamiltonian chaos (the study of chaos in Hamiltonian systems) and quantum chaos. It describes a free rotating stick (with moment of inertia ${\displaystyle I}$) in an inhomogeneous "gravitation like" field that is periodically switched on in short pulses. The model is described by the Hamiltonian

${\displaystyle {\mathcal {H}}(\theta ,p_{\theta },t)={\frac {p_{\theta }^{2}}{2I}}+K\cos \theta \sum _{n=-\infty }^{\infty }\delta \left({\frac {t}{T}}-n\right)}$,

where ${\displaystyle \theta \in }$ is the angular position of the stick (${\displaystyle \theta =\pi }$ corresponds to the position of the rotator at rest), ${\displaystyle p_{\theta }}$ is the conjugated momentum of ${\displaystyle \theta }$, ${\displaystyle \textstyle K}$ is the kicking strength, ${\displaystyle T}$ is the kicking period and ${\displaystyle \textstyle \delta }$ is the Dirac delta function.

Classical properties

Stroboscopic dynamics

The equations of motion of the kicked rotator write

Theses equations show that between two consecutive kicks, the rotator simply moves freely: the momentum ${\displaystyle p}$ is conserved and the angular position growths linearly in time. On the other hand, during each kick the momentum abruptly jumps by a quantity ${\displaystyle KT\sin \theta }$, where ${\displaystyle \theta }$ is the angular position near the kick. The kicked rotator dynamics can thus be described by the discrete map^[1]where ${\displaystyle \theta _{n}}$ and ${\displaystyle p_{n}}$ are the canonical coordinates at time ${\displaystyle t=nT^{-}}$, just before the ${\displaystyle n}$-th kick. It is usually more convenient to introduce dimensionless momentum ${\textstyle p\rightarrow p/{\frac {I}{T}}}$, time ${\textstyle t\rightarrow t/T}$ and kicking strength ${\textstyle K\rightarrow K/{\frac {I}{T^{2}}}}$ to reduce the dynamics to the single parameter mapknown as Chirikov standard map, with the caveat that ${\displaystyle p_{n}}$is not periodic as in the standard map. However, one can directly see that two rotators with same initial angular position ${\displaystyle \theta _{0}}$ but shifted dimensionless momentum ${\displaystyle p_{0}}$ and ${\textstyle p_{0}+2\pi l}$ (with ${\displaystyle l}$ an arbitrary integer) will have the same exact stroboscopic dynamics, but with dimensionless momentum shifted at any time by ${\textstyle 2\pi l}$ (this is why stroboscopic phase portraits of the kicked rotator are usually displayed in a single momentum cell ${\textstyle p\in }$).

Transition from integrability to chaos

The kicked rotator is a prototype model to illustrate the transition from integrability to chaos in Hamiltonian systems and in particular the Kolmogorov–Arnold–Moser theorem. In the limit ${\displaystyle K=0}$, the system describes the free motion of the rotator, the momentum is conserved (the system is integrable) and the corresponding trajectories are straight lines in the ${\displaystyle (\theta ,p)}$ plane (phase space), that is tori. For small, but non-vanishing perturbation ${\displaystyle K}$Zdroj:https://en.wikipedia.org?pojem=Kicked_rotator
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---