Construction of Invariant Tori around Closed Orbits
نویسنده
چکیده
The approach and methods introduced by McGill & Binney and Kaasalainen & Binney (Papers I and III), for the construction of phase-space tori that are approximate invariant tori of a given Hamiltonian, are generalized to include motion`trapped' around general closed orbits. This is accomplished by introducing point transformations that map the connguration space around a closed orbit in the target potential to one in a toy potential for which action-angle coordinates are known. This approach opens up the possibility of constructing tori for an arbitrary orbit family. The method is illustrated by applying it to thèbanana' and``sh' minor-orbit families in the planar logarithmic potential.
منابع مشابه
Multiple Poincaré sections method for finding the quasiperiodic orbits of the restricted three body problem
A new fully numerical method is presented which employs multiple Poincaré sections to find quasiperiodic orbits of the Restricted Three-Body Problem (RTBP). The main advantages of this method are the small overhead cost of programming and very fast execution times, robust behavior near chaotic regions that leads to full convergence for given family of quasiperiodic orbits and the minimal memory...
متن کاملAre ghost surfaces quadratic-flux-minimizing?
Two candidates for “almost-invariant” toroidal surfaces passing through magnetic islands, namely quadratic-flux-minimizing (QFMin) surfaces and ghost surfaces, use families of periodic pseudo-orbits (i.e. paths for which the action is not exactly extremal). QFMin pseudo-orbits, which are coordinate-dependent, are field lines obtained from a modified magnetic field, and ghost-surface pseudo-orbi...
متن کاملShadowing Orbits for Transition Chains of Invariant Tori Alternating with Birkhoff Zones of Instability
We consider a dynamical system that exhibits transition chains of invariant tori alternating with Birkhoff zones of instability in a 2-dimensional center manifold. It is known that there exist orbits that shadow the transition chains. It is also known that there exist orbits that cross the Birkhoff zones of instability. We describe a topological mechanism that allows one to join together the tw...
متن کاملStructure and Breakdown of Invariant Tori in a 4-d Mapping Model of Accelerator Dynamics
We study sequences of periodic orbits and the associated phase space dynamics in a 4-D symplectic map of interest to the problem of beam stability in circular particle accelerators. The increasing period of these orbits is taken from a sequence of rational approximants to an incommensurate pair of irrational rotation numbers of an invariant torus. We find stable (elliptic– elliptic) periodic or...
متن کاملRenormalization of Vector Fields
These notes cover some of the recent developments in the renormalization of quasiperiodic flows. This includes skew flows over tori, Hamiltonian flows, and other flows on T×R. After stating some of the problems and describing alternative approaches, we focus on the definition and basic properties of a single renormalization step. A second part deals with the construction of conjugacies and inva...
متن کامل