The dynamics of digits: Calculating pi with Galperin's billiards

In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number $\pi$. This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of $\pi$ 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 $\pi$ 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 $\pi$ with Galperin billiards, including curious cases with irrational number bases.Comment: 30 pages, 13 figure

Conformational aspects of polymorphs and phases of 2-propyl-1H-benzimidazole

This paper reports on the polymorphism of 2-propyl-1H-benzimidazole (2PrBzIm) induced by temperature change. Upon heating, an irreversible reconstructive-type phase transition at T = 384 K from the ordered form I (P212121) to a new polymorph, form II HT (Pcam), was observed. The structural transformation between forms I and II involves significant changes in the crystal packing, as well as a key conformational variation around the propyl chain of the molecule. After the first irreversible phase transition, the II HT form undergoes two further (reversible) phase transitions upon cooling at 361 K (II RT) and 181 K (II LT). All three phases (forms II HT, II RT and II LT) have almost identical crystal packing and, given the reversibility of the conversions as a function of temperature, they are referred to as form II temperature phases. They differ, however, with respect to conformational variations around the propyl chain of 2PrBzIm. Energy calculations of the gas-phase conformational energy landscape of this compound about its flexible bonds allowed us to classify the observed conformational variations of all forms into changes and adjustments of conformers. This reveals that forms I and II are related by conformational change, and that two of the form II phases (HT and RT) are related by conformational adjustment, whilst the other two (RT and LT) are related by conformational change. We introduce the term 'conformational phases' for different crystal phases with almost identical packing but showing changes in conformation.This work has been financed by the MINECO (Project Nos. CTQ2014-56833-R, CTQ2015-63997-C2-2-P) and Comunidad Autónoma de Madrid (Project No. S2013/MIT2841, Fotocarbon), the Basque Government (Project No. IT779-13) and the CICE/JA-FEDER-UJA: Plan de Fortalecimiento de las Capacidades de I+D+i/2014–15 and projects UJA2013/08/03, UJA2015/08/07 and UJA2016/08/15. The authors are grateful for the technical and human support provided by SGIker (UPV/EHU), CICT (UJA) and European funding (ERDFand ESF). M. M. Quesada-Moreno thanks the University of Jaén for both pre- and post-doctoral fellowships.Peer Reviewe