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

Xabier M. Aretxabaleta, Marina Gonchenko, Nathan L. Harshman, Steven Glenn Jackson, Maxim Olshanii and Grigory E. Astrakharchik
Additional contact information
Xabier M. Aretxabaleta: Departament de Física, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain
Marina Gonchenko: Departament de Matemàtiques, Universitat Politècnica de Catalunya, 08028 Barcelona, Spain
Nathan L. Harshman: Department of Physics, American University, 4400 Massachusetts Ave. NW, Washington, DC 20016, USA
Steven Glenn Jackson: Department of Mathematics, University of Massachusetts Boston, Boston, MA 02125, USA
Maxim Olshanii: Department of Physics, University of Massachusetts Boston, Boston, MA 02125, USA
Grigory E. Astrakharchik: Departament de Física, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain

Mathematics, 2020, vol. 8, issue 4, 1-31

Abstract: 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.

Keywords: Galperin billiards; calculating pi; three-body problem; solvable model; integrability; superintegrability; irrational bases (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/4/509/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/4/509/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:4:p:509-:d:340670

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().