A Coriolis Tutorial
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Cen, Julia.Pdf
City Research Online City, University of London Institutional Repository Citation: Cen, J. (2019). Nonlinear classical and quantum integrable systems with PT -symmetries. (Unpublished Doctoral thesis, City, University of London) This is the accepted version of the paper. This version of the publication may differ from the final published version. Permanent repository link: https://openaccess.city.ac.uk/id/eprint/23891/ Link to published version: Copyright: City Research Online aims to make research outputs of City, University of London available to a wider audience. Copyright and Moral Rights remain with the author(s) and/or copyright holders. URLs from City Research Online may be freely distributed and linked to. Reuse: Copies of full items can be used for personal research or study, educational, or not-for-profit purposes without prior permission or charge. Provided that the authors, title and full bibliographic details are credited, a hyperlink and/or URL is given for the original metadata page and the content is not changed in any way. City Research Online: http://openaccess.city.ac.uk/ [email protected] Nonlinear Classical and Quantum Integrable Systems with PT -symmetries Julia Cen A Thesis Submitted for the Degree of Doctor of Philosophy School of Mathematics, Computer Science and Engineering Department of Mathematics Supervisor : Professor Andreas Fring September 2019 Thesis Commission Internal examiner Dr Olalla Castro-Alvaredo Department of Mathematics, City, University of London, UK External examiner Professor Andrew Hone School of Mathematics, Statistics and Actuarial Science University of Kent, UK Chair Professor Joseph Chuang Department of Mathematics, City, University of London, UK i To my parents. -
WUN2K for LECTURE 19 These Are Notes Summarizing the Main Concepts You Need to Understand and Be Able to Apply
Duke University Department of Physics Physics 361 Spring Term 2020 WUN2K FOR LECTURE 19 These are notes summarizing the main concepts you need to understand and be able to apply. • Newton's Laws are valid only in inertial reference frames. However, it is often possible to treat motion in noninertial reference frames, i.e., accelerating ones. • For a frame with rectilinear acceleration A~ with respect to an inertial one, we can write the equation of motion as m~r¨ = F~ − mA~, where F~ is the force in the inertial frame, and −mA~ is the inertial force. The inertial force is a “fictitious” force: it's a quantity that one adds to the equation of motion to make it look as if Newton's second law is valid in a noninertial reference frame. • For a rotating frame: Euler's theorem says that the most general mo- tion of any body with respect to an origin is rotation about some axis through the origin. We define an axis directionu ^, a rate of rotation dθ about that axis ! = dt , and an instantaneous angular velocity ~! = !u^, with direction given by the right hand rule (fingers curl in the direction of rotation, thumb in the direction of ~!.) We'll generally use ~! to refer to the angular velocity of a body, and Ω~ to refer to the angular velocity of a noninertial, rotating reference frame. • We have for the velocity of a point at ~r on the rotating body, ~v = ~! ×~r. ~ dQ~ dQ~ ~ • Generally, for vectors Q, one can write dt = dt + ~! × Q, for S0 S0 S an inertial reference frame and S a rotating reference frame. -
Weak Galilean Invariance As a Selection Principle for Coarse
Weak Galilean invariance as a selection principle for coarse-grained diffusive models Andrea Cairolia,b, Rainer Klagesb, and Adrian Bauleb,1 aDepartment of Bioengineering, Imperial College London, London SW7 2AZ, United Kingdom; and bSchool of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved April 16, 2018 (received for review October 2, 2017) How does the mathematical description of a system change in dif- in the long-time limit, hX 2i ∼ t β with β = 1, for anomalous dif- ferent reference frames? Galilei first addressed this fundamental fusion it scales nonlinearly with β 6=1. Anomalous dynamics has question by formulating the famous principle of Galilean invari- been observed experimentally for a wide range of physical pro- ance. It prescribes that the equations of motion of closed systems cesses like particle transport in plasmas, molecular diffusion in remain the same in different inertial frames related by Galilean nanopores, and charge transport in amorphous semiconductors transformations, thus imposing strong constraints on the dynam- (5–7) that was first theoretically described in refs. 8 and 9 based ical rules. However, real world systems are often described by on the continuous-time random walk (CTRW) (10). Likewise, coarse-grained models integrating complex internal and exter- anomalous diffusion was later found for biological motion (11– nal interactions indistinguishably as friction and stochastic forces. 13) and even human movement (14). Recently, it was established Since Galilean invariance is then violated, there is seemingly no as a ubiquitous characteristic of cellular processes on a molec- alternative principle to assess a priori the physical consistency of a ular level (15). -
Navier-Stokes Equation
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PHYSICS of ARTIFICIAL GRAVITY Angie Bukley1, William Paloski,2 and Gilles Clément1,3
Chapter 2 PHYSICS OF ARTIFICIAL GRAVITY Angie Bukley1, William Paloski,2 and Gilles Clément1,3 1 Ohio University, Athens, Ohio, USA 2 NASA Johnson Space Center, Houston, Texas, USA 3 Centre National de la Recherche Scientifique, Toulouse, France This chapter discusses potential technologies for achieving artificial gravity in a space vehicle. We begin with a series of definitions and a general description of the rotational dynamics behind the forces ultimately exerted on the human body during centrifugation, such as gravity level, gravity gradient, and Coriolis force. Human factors considerations and comfort limits associated with a rotating environment are then discussed. Finally, engineering options for designing space vehicles with artificial gravity are presented. Figure 2-01. Artist's concept of one of NASA early (1962) concepts for a manned space station with artificial gravity: a self- inflating 22-m-diameter rotating hexagon. Photo courtesy of NASA. 1 ARTIFICIAL GRAVITY: WHAT IS IT? 1.1 Definition Artificial gravity is defined in this book as the simulation of gravitational forces aboard a space vehicle in free fall (in orbit) or in transit to another planet. Throughout this book, the term artificial gravity is reserved for a spinning spacecraft or a centrifuge within the spacecraft such that a gravity-like force results. One should understand that artificial gravity is not gravity at all. Rather, it is an inertial force that is indistinguishable from normal gravity experience on Earth in terms of its action on any mass. A centrifugal force proportional to the mass that is being accelerated centripetally in a rotating device is experienced rather than a gravitational pull. -
SMU PHYSICS 1303: Introduction to Mechanics
SMU PHYSICS 1303: Introduction to Mechanics Stephen Sekula1 1Southern Methodist University Dallas, TX, USA SPRING, 2019 S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 1 Outline Conservation of Energy S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 2 Conservation of Energy Conservation of Energy NASA, “Hipnos” by Molinos de Viento and available under Creative Commons from Flickr S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 3 Conservation of Energy Key Ideas The key ideas that we will explore in this section of the course are as follows: I We will come to understand that energy can change forms, but is neither created from nothing nor entirely destroyed. I We will understand the mathematical description of energy conservation. I We will explore the implications of the conservation of energy. Jacques-Louis David. “Portrait of Monsieur de Lavoisier and his Wife, chemist Marie-Anne Pierrette Paulze”. Available under Creative Commons from Flickr. S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 4 Conservation of Energy Key Ideas The key ideas that we will explore in this section of the course are as follows: I We will come to understand that energy can change forms, but is neither created from nothing nor entirely destroyed. I We will understand the mathematical description of energy conservation. I We will explore the implications of the conservation of energy. Jacques-Louis David. “Portrait of Monsieur de Lavoisier and his Wife, chemist Marie-Anne Pierrette Paulze”. Available under Creative Commons from Flickr. S. Sekula (SMU) SMU — PHYS 1303 SPRING, 2019 4 Conservation of Energy Key Ideas The key ideas that we will explore in this section of the course are as follows: I We will come to understand that energy can change forms, but is neither created from nothing nor entirely destroyed. -
Vorticity Production Through Rotation, Shear, and Baroclinicity
A&A 528, A145 (2011) Astronomy DOI: 10.1051/0004-6361/201015661 & c ESO 2011 Astrophysics Vorticity production through rotation, shear, and baroclinicity F. Del Sordo1,2 and A. Brandenburg1,2 1 Nordita, AlbaNova University Center, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden e-mail: [email protected] 2 Department of Astronomy, AlbaNova University Center, Stockholm University, 10691 Stockholm, Sweden Received 31 August 2010 / Accepted 14 February 2011 ABSTRACT Context. In the absence of rotation and shear, and under the assumption of constant temperature or specific entropy, purely potential forcing by localized expansion waves is known to produce irrotational flows that have no vorticity. Aims. Here we study the production of vorticity under idealized conditions when there is rotation, shear, or baroclinicity, to address the problem of vorticity generation in the interstellar medium in a systematic fashion. Methods. We use three-dimensional periodic box numerical simulations to investigate the various effects in isolation. Results. We find that for slow rotation, vorticity production in an isothermal gas is small in the sense that the ratio of the root-mean- square values of vorticity and velocity is small compared with the wavenumber of the energy-carrying motions. For Coriolis numbers above a certain level, vorticity production saturates at a value where the aforementioned ratio becomes comparable with the wavenum- ber of the energy-carrying motions. Shear also raises the vorticity production, but no saturation is found. When the assumption of isothermality is dropped, there is significant vorticity production by the baroclinic term once the turbulence becomes supersonic. In galaxies, shear and rotation are estimated to be insufficient to produce significant amounts of vorticity, leaving therefore only the baroclinic term as the most favorable candidate. -
Assessment of DUACS Sentinel-3A Altimetry Data in the Coastal Band of the European Seas: Comparison with Tide Gauge Measurements
remote sensing Article Assessment of DUACS Sentinel-3A Altimetry Data in the Coastal Band of the European Seas: Comparison with Tide Gauge Measurements Antonio Sánchez-Román 1,* , Ananda Pascual 1, Marie-Isabelle Pujol 2, Guillaume Taburet 2, Marta Marcos 1,3 and Yannice Faugère 2 1 Instituto Mediterráneo de Estudios Avanzados, C/Miquel Marquès, 21, 07190 Esporles, Spain; [email protected] (A.P.); [email protected] (M.M.) 2 Collecte Localisation Satellites, Parc Technologique du Canal, 8-10 rue Hermès, 31520 Ramonville-Saint-Agne, France; [email protected] (M.-I.P.); [email protected] (G.T.); [email protected] (Y.F.) 3 Departament de Física, Universitat de les Illes Balears, Cra. de Valldemossa, km 7.5, 07122 Palma, Spain * Correspondence: [email protected]; Tel.: +34-971-61-0906 Received: 26 October 2020; Accepted: 1 December 2020; Published: 4 December 2020 Abstract: The quality of the Data Unification and Altimeter Combination System (DUACS) Sentinel-3A altimeter data in the coastal area of the European seas is investigated through a comparison with in situ tide gauge measurements. The comparison was also conducted using altimetry data from Jason-3 for inter-comparison purposes. We found that Sentinel-3A improved the root mean square differences (RMSD) by 13% with respect to the Jason-3 mission. In addition, the variance in the differences between the two datasets was reduced by 25%. To explain the improved capture of Sea Level Anomaly by Sentinel-3A in the coastal band, the impact of the measurement noise on the synthetic aperture radar altimeter, the distance to the coast, and Long Wave Error correction applied on altimetry data were checked. -
Role of Regional Ocean Dynamics in Dynamic Sea Level Projections by the End of the 21St Century Over Southeast Asia
EGU21-8618, updated on 25 Sep 2021 https://doi.org/10.5194/egusphere-egu21-8618 EGU General Assembly 2021 © Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License. Role of Regional Ocean Dynamics in Dynamic Sea Level Projections by the end of the 21st Century over Southeast Asia Dhrubajyoti Samanta1, Svetlana Jevrejeva2, Hindumathi K. Palanisamy2, Kristopher B. Karnauskas3, Nathalie F. Goodkin1,4, and Benjamin P. Horton1 1Nanyang Technological University, Singapore ([email protected]) 2Centre for Climate Research Singapore, Singapore 3University of Colorado Boulder, USA 4American Museum of Natural History, USA Southeast Asia is especially vulnerable to the impacts of sea-level rise due to the presence of many low-lying small islands and highly populated coastal cities. However, our current understanding of sea-level projections and changes in upper-ocean dynamics over this region currently rely on relatively coarse resolution (~100 km) global climate model (GCM) simulations and is therefore limited over the coastal regions. Here using GCM simulations from the High-Resolution Model Intercomparison Project (HighResMIP) of the Coupled Model Intercomparison Project Phase 6 (CMIP6) to (1) examine the improvement of mean-state biases in the tropical Pacific and dynamic sea-level (DSL) over Southeast Asia; (2) generate projection on DSL over Southeast Asia under shared socioeconomic pathways phase-5 (SSP5-585); and (3) diagnose the role of changes in regional ocean dynamics under SSP5-585. We select HighResMIP models that included a historical period and shared socioeconomic pathways (SSP) 5-8.5 future scenario for the same ensemble and estimate the projected changes relative to the 1993-2014 period. -
HW 6, Question 1: Forces in a Rotating Frame
HW 6, Question 1: Forces in a Rotating Frame Consider a solid platform, like a \merry-go round", that rotates with respect to an inertial frame of reference. It is oriented in the xy plane such that the axis of rotation is in the z-direction, i.e. ! = !z^, which we can take to point out of the page. Now, let us also consider a person walking on the rotating platform, such that this person is walking radially outward on the platform with respect to their (non-inertial) reference frame with velocity v = vr^. In what follows for Question 1, directions in the non-inertial reference frame (i.e. the frame of the walking person) are defined in the following manner: the x0-direction points directly outward from the center of the merry-go round; the y0 points in the tangential direction of rotation in the plane of the platform; and the z0-direction points out of the page, and in the same direction as the z-direction for the inertial reference frame. (a) Identify all forces. Since the person's reference frame is non-inertial, we need to account for the various fictitious forces given the description above, as well as any other external/reaction forces. The angular rate of rotation is constant, and so the Euler force Feul = m(!_ × r) = ~0. The significant forces acting on the person are: 0 • the force due to gravity, Fg = mg = mg(−z^ ), 2 0 • the (fictitious) centrifugal force, Fcen = m! × (! × r) = m! rx^ , 0 • the (fictitious) Coriolis force, Fcor = 2mv × ! = 2m!v(−y^ ), • the normal force the person experiences from the rotating platform N = Nz^0, and • a friction force acting on the person, f, in the x0y0 plane. -
LECTURES 20 - 29 INTRODUCTION to CLASSICAL MECHANICS Prof
LECTURES 20 - 29 INTRODUCTION TO CLASSICAL MECHANICS Prof. N. Harnew University of Oxford HT 2017 1 OUTLINE : INTRODUCTION TO MECHANICS LECTURES 20-29 20.1 The hyperbolic orbit 20.2 Hyperbolic orbit : the distance of closest approach 20.3 Hyperbolic orbit: the angle of deflection, φ 20.3.1 Method 1 : using impulse 20.3.2 Method 2 : using hyperbola orbit parameters 20.4 Hyperbolic orbit : Rutherford scattering 21.1 NII for system of particles - translation motion 21.1.1 Kinetic energy and the CM 21.2 NII for system of particles - rotational motion 21.2.1 Angular momentum and the CM 21.3 Introduction to Moment of Inertia 21.3.1 Extend the example : J not parallel to ! 21.3.2 Moment of inertia : mass not distributed in a plane 21.3.3 Generalize for rigid bodies 22.1 Moment of inertia tensor 2 22.1.1 Rotation about a principal axis 22.2 Moment of inertia & energy of rotation 22.3 Calculation of moments of inertia 22.3.1 MoI of a thin rectangular plate 22.3.2 MoI of a thin disk perpendicular to plane of disk 22.3.3 MoI of a solid sphere 23.1 Parallel axis theorem 23.1.1 Example : compound pendulum 23.2 Perpendicular axis theorem 23.2.1 Perpendicular axis theorem : example 23.3 Example 1 : solid ball rolling down slope 23.4 Example 2 : where to hit a ball with a cricket bat 23.5 Example 3 : an aircraft landing 24.1 Lagrangian mechanics : Introduction 24.2 Introductory example : the energy method for the E of M 24.3 Becoming familiar with the jargon 24.3.1 Generalised coordinates 24.3.2 Degrees of Freedom 24.3.3 Constraints 3 24.3.4 Configuration -
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Vol. 62 Bollettino Vol. 62 - SUPPLEMENT 1 pp. 327 di Geofisica An International teorica ed applicata Journal of Earth Sciences IMDIS 2021 International Conference on Marine Data and Information Systems 12-14 April, 2021 Online Book of Abstracts SUPPLEMENT 1 Guest Editors: Michèle Fichaut, Vanessa Tosello, Alessandra Giorgetti BOLLETTINO DI GEOFISICA teorica ed applicata 210109 - OGS.Supp.Vol62.cover_08dorso19.indd 3 03/05/21 10:54 EDITOR-IN-CHIEF D. Slejko; Trieste, Italy EDITORIAL COUNCIL SUBSCRIPTIONS 2021 A. Camerlenghi, N. Casagli, F. Coren, P. Del Negro, F. Ferraccioli, S. Parolai, G. Rossi, C. Solidoro; Trieste, Italy ASSOCIATE EDITORS A. SOLID EaRTH GeOPHYsICs N. Abu-Zeid; Ferrara, Italy J. Ba; Nanjing, China R. Barzaghi; Milano, Italy J. Boaga; Padova, Italy C. Braitenberg; Trieste, Italy A. Casas; Barcelona, Spain G. Cassiani; Padova, Italy F. Cavallini; Trieste, Italy A. Del Ben; Trieste, Italy P. dell’Aversana; San Donato Milanese, Italy C. Doglioni; Roma, Italy F. Ferrucci, Vibo Valentia, Italy E. Forte; Trieste, Italy M.-J. Jimenez; Madrid, Spain C. Layland-Bachmann, Berkeley, U.S.A. Bollettino di Geofisica Teorica ed Applicata G. Li; Zhoushan, China c/o Istituto Nazionale di Oceanografia P. Paganini; Trieste, Italy e di Geofisica Sperimentale V. Paoletti, Naples, Italy Borgo Grotta Gigante, 42/c E. Papadimitriou; Thessaloniki, Greece 34010 Sgonico, Trieste, Italy R. Petrini; Pisa, Italy e-mail: [email protected] M. Pipan; Trieste, Italy G. Seriani; Trieste, Italy http-server: bgta.eu A. Shogenova; Tallin, Estonia E. Stucchi; Milano, Italy S. Trevisani; Venezia, Italy M. Vellico; Trieste, Italy A. Vesnaver; Trieste, Italy V. Volpi; Trieste, Italy A.