An orbit-preserving discretization of the classical Kepler problem

We present a remarkable discretization of the classical Kepler problem which preserves its trajectories and all integrals of motion. The points of any discrete orbit belong to an appropriate continuous trajectory.Comment: 7 page

Additional Constants of Motion for a Discretization of the Calogero--Moser Model

The maximal super-integrability of a discretization of the Calogero--Moser model introduced by Nijhoff and Pang is presented. An explicit formula for the additional constants of motion is given.Comment: 7 pages, no figure

A direct bonded fixed partial dental prosthesis: A clinical report

A direct bonded fixed partial dental prosthesis, with a composite resin denture tooth as a pontic, a tri-n-butylborane initiated adhesive resin, and screw posts for reinforcement, was still functioning after an observation period of 20 years. The prosthesis was found to be reliable for long-term clinical use when chemically and mechanically reinforced

Quasi-conservative Integration Method for Restricted Three-body Problem

The simplest restricted three-body problem, in which two massive points and a massless point particle attract one another according to Newton’s law of inverse squares, has pulsating Hill’s regions where the massless particle moves inside the closed regions surrounding only one of the massive points. Until now, no numerical integrator is known to maintain these regions, making it challenging to reproduce the phenomenon of gravitational capture of massless particles. In this article, we propose a second-order integrator that preserves Hill’s regions to accurately simulate this phenomenon. Our integrator is based on a logarithmic Hamiltonian leapfrog method developed by Mikkola and Tanikawa and features a parameter that is adjusted to preserve a second-order approximation of an invariant integration relation of this restricted three-body problem. We analytically and numerically clarify that this integrator has the following properties: (i) it retains the collinear and triangular Lagrangian solutions regardless of the eccentricity of the relative orbit of the two massive points, (ii) it has the same Hill stability criterion for satellite-type motion of the massless point particle as the original problem, and (iii) it conserves the Jacobi integral for zero eccentricity

特解を保つ重力3 体問題の差分化

九州大学応用力学研究所研究集会報告 No.21ME-S7 「非線形波動研究の現状と将来 : 次の10 年への展望」RIAM Symposium No.21ME-S7 Current and Future Research on Nonlinear Waves : Perspectives for the Next Decade変数変換によって，一般3 体問題が拘束力学系と見なせることを示す。さらに，この拘束系にエネルギー保存型差分法を適用する。得られた差分系は以下の3 つの条件を満足する：(i)Hamiltonian，全線運動量，重心の位置を保つ，(ii) 特異点消去によって，数値計算誤差の増大が少ない，(iii) 高精度に線形安定領域の境界付近での解軌道を再現する

正則化法と全保存型差分法を用いた重力N 体問題のシミュレーション

九州大学応用力学研究所研究集会報告 No.20ME-S7 「非線形波動の数理と物理」RIAM Symposium No.20ME-S7 Mathematics and Physics in Nonlinear waves重力N 体問題はN(N ? 1)/2 個の摂動2 体問題と見なせることが知られている．本講演では，正則化法で摂動Kepler 問題が持つ特異点を消去した後に，エネルギー保存差分法，多様体修正法を適用することで，重力N 体問題が持つ全ての保存量を保つスキームを構成する．このスキームは保存量だけでなく，力学的に安定なLagrange 平衡解を高精度に再現する