Possible conserved quantity for the Hénon-Heiles problem

Finkler, Paul; Jones, C. E.; Sowell, Glenn A.

doi:10.1103/physreva.42.1931

Cited by 7 publications

(5 citation statements)

References 7 publications

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“…However, some solutions of are subject to a third integral (Hénon & Heiles 1964), as has been demonstrated numerically by Fukushige & Heggie (2000), who calculated a Poincaré surface of section. In principle, one may obtain power‐series expansions of such third integrals, see, for example, the original works by Gustavson (1966) and Finkler, Jones & Sowell (1990) and the review in Moser (1968). Also, third integrals can be related to the existence of Killing tensor fields which are well known in General Relativity (Clementi & Pettini 2002).…”

Section: The Tidal Approximationmentioning

confidence: 99%

“…In principle, one may obtain power series expansions of such third integrals, see, e.g. the original works by Gustavson (1966) and Finkler, Jones & Sowell (1990) and the review in Moser (1968). Also, third integrals can be related to the existence of Killing tensor fields which are well-known in General Relativity (Clementi & Pettini 2002).…”

Section: The Tidal Approximationmentioning

confidence: 99%

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Escape from the vicinity of fractal basin boundaries of a star cluster

Ernst

Just

Spurzem

et al. 2007

Monthly Notices of the Royal Astronomical Society

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The dissolution process of star clusters is rather intricate for theory. We investigate it in the context of chaotic dynamics. We use the simple Plummer model for the gravitational field of a star cluster and treat the tidal field of the Galaxy within the tidal approximation. That is, a linear approximation of tidal forces from the Galaxy based on epicyclic theory in a rotating reference frame. The Poincar\'e surfaces of section reveal the effect of a Coriolis asymmetry. The system is non-hyperbolic which has important consequences for the dynamics. We calculated the basins of escape with respect to the Lagrangian points $L_1$ and $L_2$. The longest escape times have been measured for initial conditions in the vicinity of the fractal basin boundaries. Furthermore, we computed the chaotic saddle for the system and its stable and unstable manifolds. The chaotic saddle is a fractal structure in phase space which has the form of a Cantor set and introduces chaos into the system.Comment: Accepted by MNRAS, Figures have lower qualit

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Section: The Tidal Approximationmentioning

confidence: 99%

Section: The Tidal Approximationmentioning

confidence: 99%

Escape from the vicinity of fractal basin boundaries of a star cluster

Ernst

Just

Spurzem

et al. 2007

Monthly Notices of the Royal Astronomical Society

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“…A knowledge of these will be extremely important in developing our method for determining a conserved quantity for this potential. The symmetries are these: (1) time reversal; (2) reflection through the origin in the x variable (i.e. , x goes tox); and (3) invariance under rotations in the x-y plane by multiples of 120'.…”

Section: Definition Of the Variables And The Equations Of Motionmentioning

confidence: 99%

“…, Ref. [2], which will hereafter be referred to as FJSI). Chaotic trajectories produce Poincare sections which are area filling, whereas nonchaotic trajectories produce one-dimensional closed-curve Poincare sections [1,2].…”

Section: Introductionmentioning

confidence: 99%

“…Area-filling Poincare sections suggest the absence of such a quantity. Approximately conserved quantities have been found and are expressed in power-series expansions in terms of the coordinates and velocities [2,3]. Such quantities were originally studied by Gustavson [3], whose method brought the Hamiltonian into normal form by a succession of canonical transformations.…”

Section: Introductionmentioning

confidence: 99%

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Construction of a quasiconserved quantity in the Hénon-Heiles problem using a single set of variables

1991

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The problem of finding the coefficients of a simple series expansion for a quasiconserved quantity K for the Henon-Heiles Hamiltonian H using a single set of variables is solved. In the past, this type of approach has been problematic because the solution to the equations determining the coefficients in the expansion is not unique. As a result, the existence of a consistent expression for K to all orders had not previously been established. We show how to deal with this arbitrariness in the expansion coefficients for K in a consistent way. Due to this arbitrariness, we find a class of expansions for K, in contrast to the single unique expansion for K generated by the normal-form approach of Gustavson [Astron. J. 71, 670 (1966)]. It may be possible to devise a criterion for deciding which one of our expansions is "optimally convergent, " although we do not deal with this question here. We proceed by introducing a single set of dynamic variables that have simple symmetry properties and that also "diagonalize" the problem of finding the coefficients of K. No canonical transformations are required. A straightforward constructive procedure is given for generating the power series to any order for quantities having the symmetry of the Hamiltonian that -are formally conserved. This leads to a very practical method for calculating a quasiconserved quantity in the Henon-Heiles problem. A comparison is made through several orders of the terms generated by this approach and those generated in the original Gustavson expansion in normal form.

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Semiclassical quantization of a nonintegrable system: Pushing the Fourier method into the chaotic regime

Sohlberg

Shirts

1994

The Journal of Chemical Physics

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Semiclassical quantization of vibrational systems using fastFourier transform methods: Application to HDO stretches J. Chem. Phys. 94, 6036 (1991); 10.1063/1.460441 EBK quantization of nonseparable systems: A Fourier transform methodSemiclassical Einstein-Brillouin-Keller (EBK) quantization of the nonintegrable Henon-Heiles Hamiltonian succeeds using the Fourier transform method of Martens and Ezra. Two innovations are required for this success: (1) the use of tunneling corrected quantizing actions obtained from an approximate, one-dimensional Hamiltonian and (2) exploitation of intermediate-time approximate quasiperiodicity or "vague tori" wherein the Fourier transform of chaotic motion over 10-100 vibrational periods allows the determination of frequencies and amplitudes which approximate motion during the time interval. Approximate tori, actions, and EBK energy levels are then straightforward. We use an interpolation method to smooth over small resonance zones that are not expected to be important quantum mechanically.

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Possible conserved quantity for the Hénon-Heiles problem

Cited by 7 publications

References 7 publications

Escape from the vicinity of fractal basin boundaries of a star cluster

Escape from the vicinity of fractal basin boundaries of a star cluster

Construction of a quasiconserved quantity in the Hénon-Heiles problem using a single set of variables

Semiclassical quantization of a nonintegrable system: Pushing the Fourier method into the chaotic regime

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