DOCSLIB.ORG

Issues Associated with Galilean Invariance on a Moving Solid Boundary in the Lattice Boltzmann Method

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Issues Associated with Galilean Invariance on a Moving Solid Boundary in the Lattice Boltzmann Method Issues associated with Galilean invariance on a moving solid boundary in the lattice Boltzmann method Cheng Peng, Nicholas Geneva Department of Mechanical Engineering, University of Delaware, Newark, DE, 19716-3140, USA Zhaoli Guo State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan, P.R. China Lian-Ping Wang Department of Mechanical Engineering, University of Delaware, Newark, DE, 19716-3140, USA and State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan, P.R. China In the lattice Boltzmann method (LBM), it is well known that the hydrodynamic force evaluation based on the mesoscopic momentum exchange method often fails to satisfy the Galilean invariance. While multiple methods have been proposed in the past to restore Galilean invariance of the evalu- ated hydrodynamic force acting on the solid boundary, the related Galilean-invariance property of the fluid particle bounce-back distributions near the solid boundary has so far been largely ignored or unnoticed. In this work, we first illustrate this problem through a simple physical analysis, then demonstrate through numerical simulations that this problem can severely contaminate LBM re- sults in some cases. Several LBM simulations are presented to show how this problem can render a simulation inaccurate when the entire flow is subjected to a moving reference frame. This reveals the incompleteness of previous modifications on the momentum exchange method. To address this issue, a new bounce-back scheme based on coordinate transformation is proposed. Numerical tests in both laminar and turbulent flows show that the new scheme can effectively eliminate the errors associated with the usual bounce-back implementation on a no-slip solid boundary and maintain an accurate momentum exchange calculation with minimal computational overhead. PACS numbers: 01.55.+b, 01.65.+g 2 I. INTRODUCTION In the past forty years, the lattice Boltzmann method (LBM) has been developed into an viable alternative for solving the Navier-Stokes (N-S) equations governing viscous fluid flows. An important reason for its popularity is its apparent simplicity for handling the no-slip boundary condition within complex geometries, which makes it extremely suitable in simulating particle-laden flows [1{4] and flows through porous media [5{7]. In such simulations, the no-slip boundary condition on moving or fixed (typically curved) solid surfaces is usually realized by a bounce-back scheme, which was first proposed in a simple direct bounce back form [1] and later modified to achieve at least a second-order accuracy on curved boundaries [5, 6, 8, 9]. Also inherited from the gas kinetic theory, the hydrodynamic force acting on a solid body in a LBM simulation can be evaluated by summing the net momentum change between the incident and reflected fluid particles at the boundary nodes during the bounce-back process. This mesoscopic treatment is known as the momentum exchange method (MEM) [1, 10], which is computationally more efficient than its macroscopic counterpart, namely, the stress integration method (SIM), as demonstrated in [10, 11]. While bounce-back schemes together with MEM allow different complex geometries to be implemented in LBM, the existence of potential violation to the Galilean invariance (VGI) in these methods has been exposed both theoretically and numerically [12{16]. By the VGI error, we mean that the physical result of the flow, such as the hydrodynamic force, could change if a constant translational velocity is added to the entire system. Although it is well known that the Galilean-invariant N-S equations can be derived from the LBM scheme via different methods, e.g., Chapman- Enskog analysis [17], asymptotic expansion [18, 19], and linear analysis [20], these derivations often do not consider the presence of a solid boundary. In special cases such as the laminar channel flow, the numerical errors due to a bounce-back scheme can be analyzed theoretically [6, 21]. In general it is very difficult to theoretically study the numerical errors due to a bounce-back scheme alone. Although numerical errors for laminar benchmark flows such as a channel flow or cavity flow are routinely gathered, these errors could contain both the boundary-implementation error and the inherent discretization error within LBM. In order to correct the VGI error associated with the treatment of a solid boundary, several different solutions have been proposed to modify the MEM to achieve Galilean invariance (GI) in the calculation of the hydrodynamic force acting on the solid boundary. To our knowledge, Caiazzo & Junk were among the first to realize the lack of Galilean invariance in the MEM and proposed a corrected MEM by subtracting the link-wise VGI error directly on each link [12]. Later, Clausen & Aidun (hereafter CA09) presented a much more straightforward correction that acts in a node-wise manner for an arbitrarily orientated surface [13]. The link-wise correction intends to remove the VGI error on each link; it is a brute-force treatment that ensures the error be eliminated on all boundary links. The main drawback of the link-wise correction is that it is debatable whether such a correction is over constrained, since our goal is to achieve GI on each node, not on each link. The node-wise correction on the other hand is designed to remove the overall VGI error on each boundary node. Compared to the link-wise treatment, it appears to be more physical, but the major issue for such node-wise correction is that its performance may be affected by the boundary link configuration. Another problem for both correction methods is that they require the errors be formulated and calculated beforehand. The specification of the VGI error involves certain assumptions, so these corrections may not be accurate. Chen et al. [14] proposed a corrected MEM (hereafter Chen13) to incorporate the impulsive force [2] when a solid node becomes a fluid node, or vice versa; Chen13 essentially formulated a continuous correction term to replace an impulsive force (see Section 2.3 for further details). Their treatment has later been interpreted as a link-wise correction that essentially achieves the similar node-wise correction in certain ideal cases [22]. More recently, Wen et al. designed a Galilean-invariant momentum exchange method (GIMEM, hereafter Wen14), where the lattice fluid velocity relative to the wall was used to realize node-wise Galilean invariance [15]. Due to the incompleteness of this scheme later discussed in Section II, this method may not work well in all cases. However, since this formulation does not require explicit computation of the VGI error, which is usually obtained based on the assumption that the system is fully relaxed, this method could potentially work better under insufficient grid resolutions. A similar correction was proposed by Krithivasan et al. around the same time [16]. Numerical simulations of both laminar and turbulent particle-laden flows indicate that the above corrected force evaluation schemes of CA09, Chen13, and Wen14 lead to physically correct particle motions [15, 22, 23]. Further details of these methods are discussed in Wen et al.'s review article [24]. The above modified momentum exchange schemes provide reasonable corrections to the VGI error when the hy- drodynamic force acting on the solid boundary or a suspended solid particle is of concern. However, they should not be regarded as a complete fix to the VGI error. The incompleteness is easily seen by simply considering the Newton's Third law. Namely, if there exists a VGI error in the hydrodynamic force acting on the solid phase, there must be a corresponding VGI error in terms of interaction force experienced by the fluid phase, with the latter being equal and opposite to the former. This rather obvious and critical problem remains essentially untouched by all the proposed corrections listed above. As we will discuss in detail later in this paper, for some cases, the VGI error on the fluid may be corrected impulsively through the momentum gain/loss when a given node switches phase, namely, a fluid node 3 becomes solid or vise versa. However, in many cases in which the impulsive fix does not occur, e.g., the case where the wall moves along a fixed line (in 2D) or surface (in 3D) relative to the lattice nodes, the VGI error on the fluid phase is left uncorrected. Furthermore, the one-way correction on the hydrodynamic force can lead to the violation to the Newton's Third Law. We will discuss this issue in detail in Sec. II. The rest of this paper is arranged as follows. In Sec. II, we shall analyze the VGI error on both the solid and fluid phases and compare the corresponding physical mechanisms on the corrections. For the solid phase, three representative modified momentum exchange schemes proposed by CA09, Chen13 and Wen14, will be further analyzed in detail. The VGI error for the fluid phase and the potential remedies that currently exist will then be thoroughly discussed. We will also point out situations in which these remedies could fail, leading to inaccurate physical results. Then in Sec. III, a new boundary scheme is presented to maintain GI for both the fluid and solid phases. The proposed scheme is validated in Sec. IV, using a Poiseuille flow between two inclined walls and a turbulent channel flow. Finally, the contents of this work are highlighted in Sec. V with a summary of key conclusions. II. ANALYSIS OF DEVIATIONS FROM THE GALILEAN INVARIANCE IN LBM In LBM, the mesoscopic fluid-particle distribution function fi (x; t) is governed by fi (x + ei; t + δt) − fi (x; t) = Ωi; (1) where i refers to a discrete particle velocity ei, x and t are discrete node and time, respectively, δt is the time step size.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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).
    [Show full text]
  • Newton's Laws
    Newton’s Laws First Law A body moves with constant velocity unless a net force acts on the body. Second Law The rate of change of momentum of a body is equal to the net force applied to the body. Third Law If object A exerts a force on object B, then object B exerts a force on object A. These have equal magnitude but opposite direction. Newton’s second law The second law should be familiar: F = ma where m is the inertial mass (a scalar) and a is the acceleration (a vector). Note that a is the rate of change of velocity, which is in turn the rate of change of displacement. So d d2 a = v = r dt dt2 which, in simplied notation is a = v_ = r¨ The principle of relativity The principle of relativity The laws of nature are identical in all inertial frames of reference An inertial frame of reference is one in which a freely moving body proceeds with uniform velocity. The Galilean transformation - In Newtonian mechanics, the concepts of space and time are completely separable. - Time is considered an absolute quantity which is independent of the frame of reference: t0 = t. - The laws of mechanics are invariant under a transformation of the coordinate system. u y y 0 S S 0 x x0 Consider two inertial reference frames S and S0. The frame S0 moves at a velocity u relative to the frame S along the x axis. The trans- formation of the coordinates of a point is, therefore x0 = x − ut y 0 = y z0 = z The above equations constitute a Galilean transformation u y y 0 S S 0 x x0 These equations are linear (as we would hope), so we can write the same equations for changes in each of the coordinates: ∆x0 = ∆x − u∆t ∆y 0 = ∆y ∆z0 = ∆z u y y 0 S S 0 x x0 For moving particles, we need to know how to transform velocity, v, and acceleration, a, between frames.
    [Show full text]
  • Teaching and Learning Special Relativity Theory in Secondary and Lower Undergraduate Education: a Literature Review
    PHYSICAL REVIEW PHYSICS EDUCATION RESEARCH 17, 023101 (2021) Teaching and learning special relativity theory in secondary and lower undergraduate education: A literature review Paul Alstein ,* Kim Krijtenburg-Lewerissa , and Wouter R. van Joolingen Freudenthal Institute, Utrecht University, P.O. Box 85170, 3508 AD Utrecht, Netherlands (Received 20 November 2020; accepted 7 June 2021; published 12 July 2021) This review presents an overview and analysis of the body of research on special relativity theory (SRT) education at the secondary and lower undergraduate level. There is currently a growing international interest in implementing SRT in pre-university education as an introduction to modern physics. For this reason, insights into learning opportunities and challenges in SRT education are needed. The field of research in SRT education is still at an early stage, especially at the level of secondary education, and there is a shortage of empirical evaluation of learning outcomes. In order to guide future research directions, there is a need for an overview and synthesis of the results reported so far. We have selected 40 articles and categorized them according to reported learning difficulties, teaching approaches, and research tools. Analysis shows that students at all educational levels experience learning difficulties with the use of frames of reference, the postulates of SRT, and relativistic effects. In the reported teaching sequences, instructional materials, and learning activities, these difficulties are approached from different angles. Some teaching approaches focus on thought experiments to express conceptual features of SRT, while others use virtual environments to provide realistic visualization of relativistic effects. From the reported teaching approaches, three learning objectives can be identified: to foster conceptual understanding, to foster understanding of the history and philosophy of science, and to gain motivation and confidence toward SRT and physics in general.
    [Show full text]
  • 1. Special Relativity (SR) — Introduction Lectures • Introduction
    1. Special Relativity (SR) — Introduction Lectures • Introduction, Invariants • Lorentz transformations • Algebra of the Lorentz group Links • Lecture notes by David Hogg: http://cosmo.nyu.edu/hogg/sr/sr.pdf – or: http://web.vu.lt/ff/t.gajdosik/wop/sr.pdf • Tatsu Takeuchi: http://www.phys.vt.edu/~takeuchi/relativity/notes/ • ”Special Relativity for Particle Physics”: http://web.vu.lt/ff/t.gajdosik/wop/sr4wop.pdf Thomas Gajdosik – World of Particles Special Relativity 1 1. Special Relativity (SR) — Introduction History • 1632: Galileo Galilei describes the principle of relativity: – ”Dialogue concerning the Two Chief World Systems” • 1861: Maxwell’s equations • 1887: Michelson-Morley experiment • 1889 / 1892: Lorentz – Fitzgerald transformation • 1905: Albert Einstein publishes the Theory of Special Relativity: – ”On the Electrodynamics of Moving Bodies” • 1908: Hermann Minkovsky introduces 4D space-time Thomas Gajdosik – World of Particles Special Relativity 2 1. Special Relativity (SR) — Introduction Galilean Invariance: Every physical theory should mathematically look the same to every inertial observer • for Galileo it was the mechanics and kinematics: – water dropping down – throwing a ball or a stone – insects flying – jumping around . Thomas Gajdosik – World of Particles Special Relativity 3 1. Special Relativity (SR) — Introduction Galilean Invariance/ Galilean transformations: t ! t0, ~x ! ~x0 Two inertial observers, O and O0, • measure the same absolute time (i.e.: 1 second = 1 second0) – Time translations: t0 = t + τ, ~x0 = ~x
    [Show full text]
  • CHAPTER 2: Special Theory of Relativity
    The Foucault pendulum in Aggieland What does it show?? Seeing is believing T=10 sec how long is it?? https://sibor.physics.tamu.edu/home/courses/physic s-222-modern-physics/ CHAPTER 2 Special Theory of Relativity ◼ 2.1 The Apparent Need for Ether ◼ 2.2 The Michelson-Morley Experiment ◼ 2.3 Einstein’s Postulates ◼ 2.4 The Lorentz Transformation ◼ 2.5 Time Dilation and Length Contraction ◼ 2.6 Addition of Velocities ◼ 2.7 Experimental Verification ◼ 2.8 Twin Paradox It was found that there was no ◼ 2.9 Space-time displacement of the interference ◼ 2.10 Doppler Effect fringes, so that the result of the experiment was negative and would, ◼ 2.11 Relativistic Momentum therefore, show that there is still a ◼ 2.12 Relativistic Energy difficulty in the theory itself… ◼ 2.13 Computations in Modern Physics - Albert Michelson, 1907 ◼ 2.14 Electromagnetism and Relativity Newtonian (Classical) Relativity Assumption ◼ It is assumed that Newton’s laws of motion must be measured with respect to (relative to) some reference frame. Inertial Reference Frame ◼ A reference frame is called an inertial frame if Newton laws are valid in that frame. ◼ Such a frame is established when a body, not subjected to net external forces, is observed to move in rectilinear (along a straight line) motion at constant velocity. Newtonian Principle of Relativity or Galilean Invariance ◼ If Newton’s laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system. ◼ This is referred to as the Newtonian principle of relativity or Galilean invariance.
    [Show full text]
  • General Relativity
    Institut für Theoretische Physik der Universität Zürich in conjunction with ETH Zürich General Relativity Autumn semester 2016 Prof. Philippe Jetzer Original version by Arnaud Borde Revision: Antoine Klein, Raymond Angélil, Cédric Huwyler Last revision of this version: September 20, 2016 Sources of inspiration for this course include • S. Carroll, Spacetime and Geometry, Pearson, 2003 • S. Weinberg, Gravitation and Cosmology, Wiley, 1972 • N. Straumann, General Relativity with applications to Astrophysics, Springer Verlag, 2004 • C. Misner, K. Thorne and J. Wheeler, Gravitation, Freeman, 1973 • R. Wald, General Relativity, Chicago University Press, 1984 • T. Fliessbach, Allgemeine Relativitätstheorie, Spektrum Verlag, 1995 • B. Schutz, A first course in General Relativity, Cambridge, 1985 • R. Sachs and H. Wu, General Relativity for mathematicians, Springer Verlag, 1977 • J. Hartle, Gravity, An introduction to Einstein’s General Relativity, Addison Wesley, 2002 • H. Stephani, General Relativity, Cambridge University Press, 1990, and • M. Maggiore, Gravitational Waves: Volume 1: Theory and Experiments, Oxford University Press, 2007. • A. Zee, Einstein Gravity in a Nutshell, Princeton University Press, 2013 As well as the lecture notes of • Sean Carroll (http://arxiv.org/abs/gr-qc/9712019), • Matthias Blau (http://www.blau.itp.unibe.ch/lecturesGR.pdf), and • Gian Michele Graf (http://www.itp.phys.ethz.ch/research/mathphys/graf/gr.pdf). 2 CONTENTS Contents I Introduction 5 1 Newton’s theory of gravitation 5 2 Goals of general relativity 6 II Special Relativity 8 3 Lorentz transformations 8 3.1 Galilean invariance . .8 3.2 Lorentz transformations . .9 3.3 Proper time . 11 4 Relativistic mechanics 12 4.1 Equations of motion . 12 4.2 Energy and momentum .
    [Show full text]
  • 1.1. Galilean Relativity
    1.1. Galilean Relativity Galileo Galilei 1564 - 1642 Dialogue Concerning the Two Chief World Systems The fundamental laws of physics are the same in all frames of reference moving with constant velocity with respect to one another. Metaphor of Galileo’s Ship Ship traveling at constant speed on a smooth sea. Any observer doing experiments (playing billiard) under deck would not be able to tell if ship was moving or stationary. Today we can make the same Even better: Earth is orbiting observation on a plane. around sun at v 30 km/s ! ≈ 1.2. Frames of Reference Special Relativity is concerned with events in space and time Events are labeled by a time and a position relative to a particular frame of reference (e.g. the sun, the earth, the cabin under deck of Galileo’s ship) E =(t, x, y, z) Pick spatial coordinate frame (origin, coordinate axes, unit length). In the following, we will always use cartesian coordinate systems Introduce clocks to measure time of an event. Imagine a clock at each position in space, all clocks synchronized, define origin of time Rest frame of an object: frame of reference in which the object is not moving Inertial frame of reference: frame of reference in which an isolated object experiencing no force moves on a straight line at constant velocity 1.3. Galilean Transformation Two reference frames ( S and S ) moving with velocity v to each other. If an event has coordinates ( t, x, y, z ) in S , what are its coordinates ( t ,x ,y ,z ) in S ? in the following, we will always assume the “standard configuration”: Axes of S and S parallel v parallel to x-direction Origins coincide at t = t =0 x vt x t = t Time is absolute Galilean x = x vt Transf.
    [Show full text]
  • Generalized Galilean Transformations of Tensors and Cotensors with Application to General Fluid Motion
    GENERALIZED GALILEAN TRANSFORMATIONS OF TENSORS AND COTENSORS WITH APPLICATION TO GENERAL FLUID MOTION VÁN P.1;2;3, CIANCIO, V,4 AND RESTUCCIA, L.4 Abstract. Galilean transformation properties of different physical quantities are investigated from the point of view of four dimensional Galilean relativistic (non- relativistic) space-time. The objectivity of balance equations of general heat con- ducting fluids and of the related physical quantities is treated as an application. 1. Introduction Inertial reference frames play a fundamental role in non-relativistic physics [1, 2, 3]. One expects that the physical content of spatiotemporal field theories are the same for different inertial observers, and also for some noninertial ones. This has several practical, structural consequences for theory construction and development [4, 5, 6, 7]. The distinguished role of inertial frames can be understood considering that they are true reflections of the mathematical structure of the flat, Galilean relativistic space-time [8, 9, 10]. When a particular form of a physical quantity in a given Galilean inertial reference frame is expressed by the components of that quantity given in an other Galilean inertial reference frame, we have a Galilean transformation. In a four dimensional representation particular components of the physical quantity are called as timelike and spacelike parts, concepts that require further explanations. In non-relativistic physics time passes uniformly in any reference frame, it is ab- solute. The basic formula of Galilean transformation is related to both time and position: t^= t; x^ = x − vt: (1) Here, t is the time, x is the position in an inertial reference frame K, x^ is the position in an other inertial reference frame K^ and v is the relative velocity of K arXiv:1608.05819v1 [physics.class-ph] 20 Aug 2016 and K^ (see Figure 1).
    [Show full text]
  • Relativity Chap 1.Pdf
    Chapter 1 Kinematics, Part 1 Special Relativity, For the Enthusiastic Beginner (Draft version, December 2016) Copyright 2016, David Morin, [email protected] TO THE READER: This book is available as both a paperback and an eBook. I have made the first chapter available on the web, but it is possible (based on past experience) that a pirated version of the complete book will eventually appear on file-sharing sites. In the event that you are reading such a version, I have a request: If you don’t find this book useful (in which case you probably would have returned it, if you had bought it), or if you do find it useful but aren’t able to afford it, then no worries; carry on. However, if you do find it useful and are able to afford the Kindle eBook (priced below $10), then please consider purchasing it (available on Amazon). If you don’t already have the Kindle reading app for your computer, you can download it free from Amazon. I chose to self-publish this book so that I could keep the cost low. The resulting eBook price of around $10, which is very inexpensive for a 250-page physics book, is less than a movie and a bag of popcorn, with the added bonus that the book lasts for more than two hours and has zero calories (if used properly!). – David Morin Special relativity is an extremely counterintuitive subject, and in this chapter we will see how its bizarre features come about. We will build up the theory from scratch, starting with the postulates of relativity, of which there are only two.
    [Show full text]
  • Notes on Special Relativity Honours-Level Module in Applied Mathematics
    School of Mathematics and Statistics Te Kura M¯ataiTatauranga MATH 466/467 Special Relativity Module T1/T2 2020 Math 466/467: Notes on special relativity Honours-level module in Applied Mathematics Matt Visser School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand. E-mail: [email protected] URL: http://www.victoria.ac.nz/sms/about/staff/matt-visser Draft form: 3 March 2020; LATEX-ed March 3, 2020; c Matt Visser |VUW| Warning: Semi{Private | Not intended for large-scale dissemination. These notes are provided as a reading module for Math 466/467: Honours-level Applied Mathematics. There are still a lot of rough edges. Semi{Private | Not intended for large-scale dissemination. Math 466/467: Special Relativity Module 2 Contents I Notes: The special relativity 5 1 Minkowski spacetime 5 2 The Lorentz and Poincare groups 9 2.1 Full Lorentz group: 10 2.2 Restricted Lorentz group: 14 2.3 Poincare group: 15 2.4 Explicit Lorentz transformations 16 3 Quaternions (an aside) 21 4 Bi-quaternions (an aside) 24 5 Causal structure 25 6 Lorentzian metric 28 7 Inner product 32 8 Index-based methods for SR 33 9 Relativistic combination of velocities 34 10 Relativistic kinematics 34 11 Relativistic dynamics 38 12 Electromagnetic fields and forces 39 12.1 Lorentz transformations for electromagnetic fields 42 12.1.1 Transformation along the x direction 42 12.1.2 Summary: 44 12.1.3 Transformation along a general direction 44 12.2 Some exercises: 47 12.3 Lorentz force law 48 13 Relativistic scalar potential 54 14
    [Show full text]
  • Transformations and Conservation Laws Galilean Transformation We Mainly Use Inertial Frames in Which a Free Body (No Forces Applied) Moves with a Constant Velocity
    Transformations and conservation laws Galilean transformation We mainly use inertial frames in which a free body (no forces applied) moves with a constant velocity. A frame moving with a constant velocity with respect to an inertial frame is inertial, too. Thus there is an infinite number of inertial frames. The equations of motion are invariant with respect to transformations from one inertial frame to another, and the transformed Lagrange function can differ from the initial one only by an irrelevant full derivative. This is the principle of the Galilean invariance , i.e., invariance with resperct to Galilean transfromations , that is valid in the classical mechanics. K‘ V K Frame K‘ moves with a velocity V = const with respect to frame K O‘ v' = -v V, r' = r − Vt O t' = t Transformation of the Lagrange function of a free particle: m m m m L' = v' 2 = (v − V)2 = v2 − mv • V + V 2 2 2 2 2 ¢ ¥ d £ m 2 - Thus the Lagrange equations are the same in both frames ¡ = L + ¤ − mr • V + V t dt 2 (Prove the same for systems with interaction) 1 Irrelevant full derivative The proof of the Galilean invariance on the previous page (Landau and Lifshitz) is not fully satisfactory since the Largange function in the K‘ frame is expressed as the function of the variables in the K frame. The true check of the Galilean invariance should be that of the covariance of the Lagrange equations, that is, ∂L ∂L' d d = 0 = 0 dt ∂v dt ∂v' under the Galilean transformation with the same functional forms of the Lagrange function.
    [Show full text]