Algorithmic regularization of the few-body problem

Mikkola, Seppo; Tanikawa, Kiyotaka

doi:10.1046/j.1365-8711.1999.02982.x

Cited by 158 publications

(117 citation statements)

References 4 publications

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“…Leapfrog for the chain system: The leapfrog method, which is a second-order symplectic integration method and is applied in the relative frame after the masses are relabeled N 1, 2, ,  along the chain as described in Section 2.1 (Mikkola & Tanikawa 1999b). Although it is not based on KS regularization, it can be used successfully for collision orbits.…”

Section: Regularized Integration Methodsmentioning

confidence: 99%

An Efficient Conservative Integrator With a Chain Regularization for the Few-Body Problem

Minesaki

2015

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We design an efficient orbital integration scheme for the general N-body problem that preserves all the conserved quantities except the angular momentum. This scheme is based on the chain concept and is regarded as an extension of a d'Alembert-type scheme for constrained Hamiltonian systems. It also coincides with the discretetime general three-body problem for particle number N = 3. Although the proposed scheme is only second-order accurate, it can accurately reproduce some periodic orbits, which cannot be done with generic geometric numerical integrators.

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Section: Regularized Integration Methodsmentioning

confidence: 99%

An Efficient Conservative Integrator With a Chain Regularization for the Few-Body Problem

Minesaki

2015

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“…Equations need not to be regularized. [Logarithmic Hamiltonian (logH) or Time Transformed Leapfrog (TTL), Mikkola & Tanikawa (1999), Preto and Tremaine (1999), Mikkola and Aarseth(2002)] -In all the alternatives high precision can be obtained with the help of an extrapolation method [Gragg /Bulirsh-Stoer] -Very old starting points: Sundman's time transformation t = r (for the two-body problem), Levi-Civita's two-dimensional coordinate transformation x + iy = (Q 1 + iq 2 ) 2 . Kustaanheimo-Stiefel (1965) transformation from four dimensional space to three dimensions made finally regularization possible for stellar dynamics simulations.…”

Section: Seppo Mikkola Tuorla Observatory Turku Finlandmentioning

confidence: 99%

“…The most recent advances in this field started in 1999 when Mikkola and Tanikawa (1999) as well as Preto and Tremaine (1999) invented the logarithmic Hamiltonian, which together with the leapfrog algorithm, produces regular results, in fact correct trajectory for the two-body problem. This method is useful also for the N-Body problem since during close approaches the problem reduces essentially to a two-body problem.…”

Section: Seppo Mikkola Tuorla Observatory Turku Finlandmentioning

confidence: 99%

Division a Commission 7: Celestial Mechanics and Dynamical Astronomy

Morbidelli¹,

Beaugé²,

Knežević³

et al. 2015

Proc. IAU

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In order to mark a distinction with the traditional triennial reports, for this legacy issue we have asked our present and past OC members, as well as a few other outstanding members of the Celestial Mechanics community, to write a short essay on “recent highlights and the future of Celestial Mechanics”. Below we collect the contributions of the people who responded to our invitation. As it is natural, each of them interpreted their task differently. Some produced a dissertation on broad and general aspects, others focused on a specific topic of their interest. Some considered that their role was to provide a detailed review, with a list of key references, others preferred to mention the topics for which progress has been significant but without quoting any references, implicitly considering that this progress was possible thanks to the collective efforts of many scientists, and not just a few. This is great, as we appreciate the diversity of attitudes and opinions.

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“…This 3-d method involves the global regularization scheme described in [19] as well as a timetransformed leapfrog propagation technique [20] in conjunction with the Bulirsch-Stoer method [21,22]. This technique has been developed in the context of gravitational few-body systems [20,23,24] and we currently adopt it to treat strongly-driven molecules. The advantage of this latter propagation technique over the one we previously used in [15,18] is that it is numerically more robust with a smaller propagation error.…”

Section: Introductionmentioning

confidence: 99%

Toolkit for semiclassical computations for strongly driven molecules: Frustrated ionization ofH2driven by elliptical laser fields

2014

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We study the formation of highly excited neutral atoms during the break-up of strongly-driven molecules. Past work on this significant phenomenon has shown that during the formation of highly excited neutral atoms (H * ) during the break-up of H2 in a linear laser field the electron that escapes does so either very quickly or after remaining bound for a few periods of the laser field. Here, we address the electron-nuclear dynamics in H * formation in elliptical laser fields, through Coulomb explosion. We show that with increasing ellipticity two-electron effects are effectively "switchedoff". We perform these studies using a toolkit we have developed for semiclassical computations for strongly-driven multi-center molecules. This toolkit includes the formulation of the probabilities of strong-field phenomena in a transparent way. This allows us to identify the shortcomings of currently used initial phase space distributions for the electronic degrees of freedom. In addition, it includes a 3-dimensional method for time-propagation that fully accounts for the Coulomb singularity. This technique has been previously developed in the context of celestial mechanics and we currently adopt it to strongly-driven systems. Moreover, we allow for tunneling during the time-propagation. We find that this is necessary in order to accurately describe the fragmentation of strongly-driven molecules.

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Algorithmic regularization of the few-body problem

Cited by 158 publications

References 4 publications

An Efficient Conservative Integrator With a Chain Regularization for the Few-Body Problem

An Efficient Conservative Integrator With a Chain Regularization for the Few-Body Problem

Division a Commission 7: Celestial Mechanics and Dynamical Astronomy

Toolkit for semiclassical computations for strongly driven molecules: Frustrated ionization ofH2driven by elliptical laser fields

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