scholarly journals The Dynamics of Digits: Calculating Pi with Galperin’s Billiards

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 509
Author(s):  
Xabier M. Aretxabaleta ◽  
Marina Gonchenko ◽  
Nathan L. Harshman ◽  
Steven Glenn Jackson ◽  
Maxim Olshanii ◽  
...  
Keyword(s):  
Irrational Number ◽  
Type Model ◽  
Hard Wall ◽  
Body System ◽  
Mass Ratio ◽  
Specific Mass ◽  
One Dimensional ◽  
Number Π ◽  
The Masses ◽  
Three Body

In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number π . This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of π in a base determined by the ratio of the masses of the two particles. This base can be any integer, but it can also be an irrational number, or even the base can be π itself. This article reviews previous results for Galperin billiards and then pushes these results farther. We provide a complete explicit solution for the balls’ positions and velocities as a function of the collision number and time. We demonstrate that Galperin billiard can be mapped onto a two-particle Calogero-type model. We identify a second dynamical invariant for any mass ratio that provides integrability for the system, and for a sequence of specific mass ratios we identify a third dynamical invariant that establishes superintegrability. Integrability allows us to derive some new exact results for trajectories, and we apply these solutions to analyze the systematic errors that occur in calculating the digits of π with Galperin billiards, including curious cases with irrational number bases.

Using the Mass Ratio to Induce Band Gaps in a 1D Array of Nonlinearly Coupled Oscillators

2012 ◽  
Author(s):  
Brian P. Bernard ◽  
Jeffrey W. Peyser ◽  
Brian P. Mann ◽  
David P. Arnold
Keyword(s):  
Coupled Oscillators ◽  
Stable Equilibrium ◽  
Equations Of Motion ◽  
Band Gaps ◽  
Dimensional System ◽  
Mass Ratio ◽  
Frequency Range ◽  
One Dimensional ◽  
Propagation Zone ◽  
The Masses

A one dimensional system of nonlinearly coupled magnetic oscillators has been studied. After deriving the equations of motion for each oscillator, the system is linearized about a stable equilibrium and studied using an assumed solution form for a traveling wave. Wave propagation and attenuation regions are predicted by reducing the system of equations to a standard eigenvalue problem. Through evaluating these equations across the entire irreducible Brillouin zone, it is determined that when the masses of each oscillator are identical, the entire frequency range of the system is a propagation zone. By varying the masses comprising a unit cell, band gaps are observed. It is shown that the mass ratio can be used to guide both the size and location of these band gaps. Numerical simulations are performed to support our analytical findings.

scholarly journals Universality in a one-dimensional three-body system

2019 ◽  
Vol 100 (1) ◽  
Author(s):  
Lucas Happ ◽  
Matthias Zimmermann ◽  
Santiago I. Betelu ◽  
Wolfgang P. Schleich ◽  
Maxim A. Efremov
Keyword(s):  
Body System ◽  
One Dimensional ◽  
Three Body

Weighing the antiproton by parts-per-billion-scale laser spectroscopy of antiprotonic helium

2007 ◽  
Vol 85 (5) ◽  
pp. 453-460 ◽  
Author(s):  
R S Hayano
Keyword(s):  
Laser Spectroscopy ◽  
Continuous Wave ◽  
Frequency Comb ◽  
Antiprotonic Helium ◽  
Electron Mass ◽  
Helium Nucleus ◽  
Body System ◽  
Mass Ratio ◽  
14.20 Dh ◽  
Three Body

A femtosecond optical frequency comb and continuous-wave pulse-amplified laser were used to measure 12 transition frequencies of antiprotonic helium (metastable three-body system consisting of an antiproton, an electron, and a helium nucleus) to fractional precisions of (9–16) × 10–9. One of these is between two states having microsecond-scale lifetimes hitherto unaccessible to our precision laser spectroscopy method. Comparisons with three-body QED calculations yielded an antiproton-to-electron mass ratio of [Formula: see text] = 1836.152 674(5).PACS Nos.: 36.10.–k, 14.20.Dh, 32.70.Jz

QUASI-ONE-DIMENSIONAL SEPARATION TYPE MODEL OF PSEUDO-SHOCK IN A DUCT

2013 ◽  
Vol 44 (5) ◽  
pp. 639-664 ◽  
Author(s):  
Evgeniy Aleksandrovich Meshcheryakov ◽  
Violetta Vasilievna Yashina
Keyword(s):  
Type Model ◽  
One Dimensional

scholarly journals Nekhoroshev Stability in Quasi-Integrable Degenerate Hamiltonian Systems

1999 ◽  
Vol 172 ◽  
pp. 443-444
Author(s):  
Massimiliano Guzzo
Keyword(s):  
Degenerate System ◽  
Escape Time ◽  
Kam Theorem ◽  
Mass Ratio ◽  
Three Body Problem ◽  
Arbitrary Potential ◽  
Hamiltonian Perturbation Theory ◽  
Three Body ◽  
Nekhoroshev Stability ◽  
Degenerate Hamiltonian Systems

Many classical problems of Mechanics can be studied regarding them as perturbations of integrable systems; this is the case of the fast rotations of the rigid body in an arbitrary potential, the restricted three body problem with small values of the mass-ratio, and others. However, the application of the classical results of Hamiltonian Perturbation Theory to these systems encounters difficulties due to the presence of the so-called ‘degeneracy’. More precisely, the Hamiltonian of a quasi-integrable degenerate system looks likewhere (I, φ) є U × Tn, U ⊆ Rn, are action-angle type coordinates, while the degeneracy of the system manifests itself with the presence of the ‘degenerate’ variables (p, q) є B ⊆ R2m. The KAM theorem has been applied under quite general assumptions to degenerate Hamiltonians (Arnold, 1963), while the Nekhoroshev theorem (Nekhoroshev, 1977) provides, if h is convex, the following bounds: there exist positive ε0, a0, t0 such that if ε < ε0 then if where Te is the escape time of the solution from the domain of (1). An escape is possible because the motion of the degenerate variables can be bounded in principle only by , and so over the time they can experience large variations. Therefore, there is the problem of individuating which assumptions on the perturbation and on the initial data allow to control the motion of the degenerate variables over long times.

On the Stability of the Linearly Related Modes of Certain Nonlinear Two-Degree-of-Freedom Systems

1961 ◽  
Vol 28 (1) ◽  
pp. 71-77 ◽  
Author(s):  
C. P. Atkinson
Keyword(s):  
Nonlinear Differential Equations ◽  
Mass Ratio ◽  
Positive Mass ◽  
Fourth Degree ◽  
Real Roots ◽  
General Stability ◽  
The Stability ◽  
The Masses ◽  
Spring Constants ◽  
Two Degree Of Freedom

This paper presents a method for analyzing a pair of coupled nonlinear differential equations of the Duffing type in order to determine whether linearly related modal oscillations of the system are possible. The system has two masses, a coupling spring and two anchor springs. For the systems studied, the anchor springs are symmetric but the masses are not. The method requires the solution of a polynomial of fourth degree which reduces to a quadratic because of the symmetric springs. The roots are a function of the spring constants. When a particular set of spring constants is chosen, roots can be found which are then used to set the necessary mass ratio for linear modal oscillations. Limits on the ranges of spring-constant ratios for real roots and positive-mass ratios are given. A general stability analysis is presented with expressions for the stability in terms of the spring constants and masses. Two specific examples are given.

Multiple impacts and energy transfer in a three-body system for noncollinear collisions

1993 ◽  
Vol 87 (3) ◽  
pp. 195-213 ◽  
Author(s):  
Vladimir M. Azriel ◽  
Lev Yu. Rusin ◽  
Mikhail B. Sevryuk
Keyword(s):  
Energy Transfer ◽  
Body System ◽  
Multiple Impacts ◽  
Three Body
Sign in / Sign up

Export Citation Format

Share Document