University College Dublin an Coláiste Ollscoile, Baileátha Cliath School Of
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University College Dublin An Col´aisteOllscoile, Baile Atha´ Cliath School of Mathematical Sciences Scoil na nEola´ıochta´ıMatamaitice Classical Mechanics and Special Relativity (ACM 20050) Dr Lennon O´ N´araigh Lecture notes in Classical Mechanics and Special Relativity, September 2019 Classical Mechanics and Special Relativity (ACM20050) This course starts by developing the principles of Newtonian Mechanics for particles, systems of particles and rigid bodies. It focuses on applications of Newton's laws with an emphasis on problem solving. It then introduces the fundamental concept of Einstein's theory of Special Relativity and the relativistic extension of Newtonian concepts such as energy relativity including Einstein's famous formula E = mc2. [Newton's laws of motion] Revision of Newton's laws of motion and the concept of kinetic and potential energy. Conservative force fields and conservation of energy. Equilibrium of a particle and stability of equilibrium. [Central forces and planetary and satellite motion]: Motion in a plane is discussed with a focus on central forces. Newton's law of gravitation. Determination of the orbit from the central force. Kepler's laws of planetary motion and the motion of satellites. [Moving coordinate systems]: Non-inertial coordinate systems. Rotating coordinate systems. Ve- locity and acceleration in a moving frame. Coriolis and centripetal acceleration. [Systems of Particles]: Centre of mass. Momentum of a system of particles. Motion of the centre of mass. Angular momentum of a system of particles. Total external moment acting on a system and it relation to the angular momentum. [Einstein's Special Theory of Relativity]: Einstein's basic postulates of special relativity. The Lorentz transformation and the concept of simultaneity. Length contraction. Time dilation. Relative velocity. The concepts of inertial mass, energy and momentum in special relativity including E = mc2. What will I learn? On completion of this module students should be able to 1. Analyse the stability of equilibrium positions of a particle; 2. Solve orbit problems in mathematical terms; 3. Explain the concepts of planetary and satellite orbits including Kepler's laws; 4. Derive expressions for the velocity and acceleration in a rotating frame; 5. Explain the concepts of angular velocity, angular momentum for a system of particle; 6. Understand relation to the angular momentum and total external moment acting on a system; 7. Describe Einstein's postulates of special relativity and derive the Lorentz transformation; 8. Explain the geometrical interpretation of the Lorentz transformations in terms of space-time diagrams and the associated concept of 4-vector. 9. Derive the results of special relativity on simultaneity, length contraction, time dilation and relative velocity. 10. Explain the equivalence of mass and energy. i ii Editions This edition: September 2019 iii iv Contents Abstract i 1 Introduction 1 2 Conservative forces in one dimension 5 3 Worked example: Conservative forces in one dimension 13 4 Motion in a plane 17 5 Angular momentum and central forces 25 6 Central forces reduce to one-dimensional motion 32 7 Kepler's Laws 38 8 Kepler's First Law 44 9 Kepler's Second and Third Laws 54 10 Applications of Kepler's Laws in planetary orbits 61 11 Inertial frames of reference 68 12 Rotating Frames 76 13 Particle Moving in a Rotating Frame of Reference 86 14 Introduction to Einstein's theory of Special Relativity 92 15 The Lorentz Transformations 96 v 16 Length contraction and time dilation 104 17 The Geometry of Space-Time 110 18 Relativistic momentum and energy 120 19 Relativisitic Billiard Balls 127 20 Special topics in Special Relativity 139 A Physical units 146 vi Chapter 1 Introduction 1.1 Module Outline Here is an executive summary of the aims of this course. If you cannot remember the more detailed outline that follows, at least keep the following in mind: This course contains two topics: 1. In the first part, we show that Newton's law, Force=mass×acceleration implies Kepler's Laws, or rather, everything we know about the motion of planets around the sun. 2. In the second part, we show that Newtons laws need to be modified at speeds close to the speed of light. These modifications enable us to solve lots of problems involving particle accelerators, cosmic rays, and radioactive decay. Or, in more detail, 1. Advanced Newtonian mechanics: We will formulate Newton's equation in polar co-ordinates and discuss central forces. Using these two ideas, we will write down, and solve, the equations of motion for two bodies interacting through gravity. This enables us to prove Kepler's empirical laws of planetary motion from first principles. As part of this study, we will be motivated to look at the theoretical underpinnings of New- tonian physics, namely the theory of inertial frames and Galilean invariance. This will also motivate us to look at physics in rotating frames of reference. 2. The principle of Galilean invariance says that the laws of physics are the same in all inertial frames. Using this idea, together with Einstein's second postulate of relativity, we will write down the basis of Einstein's special theory of relativity. These postulates allow us to derive the 1 Chapter 1. Introduction Lorentz transformations. As a consequence, we will discuss length contraction, time dilation, and relative velocity. We will also discuss the equation E = mc2. Again, we will be motivated to look at the theoretical underpinnings of Special Relativity, namely the theory of four-vectors and Lorentz invariance. Some notation We use the notation x_ and dx=dt interchangeably, to signify the derivative of the quantity x with respect to time. For the second derivative, we employ two dots. 1.2 Assessment and Learning Assessment • One end-of-semester exam, 70%. • Midterm exam, for a total of 20%. • Two graded homework assignments, for a total 10%. Learning • Thirty six classes, three per week. • Optional tutorial. • Graded homework assignments . • Non-graded homework assignments . There will be a weekly homework assignment, however only two of these will be graded, as per the above assessment strategy. You are encouraged to work through all of the homework assignments, whether graded or not, because this is helpful for your learning and also, these problems will feature in the final exam, in some shape or form. Model answers for the homework assignments will be presented in the tutorial class. Moreover, one of the main goals of any degree in the mathematical sciences is to put problems into a quantitative framework and hence to solve them. This will be accomplished through the home- work assignments, and autonomous student learning, or what used to be known less grandiosely as studying. One way of studying is by independently practising the non-graded homework assignments and by subsequently going through the answers in the tutorial classes. Textbooks These lecture notes are self-contained. For extra reading, students may refer to the following recommended textbooks: 2 1.3. UCD policies as applied to this module • An Introduction To Mechanics, D. Kleppner, R. J. Kolenkow. • For the second part of the module, look at the later chapters in Sears and Zemansky's Uni- versity Physics With Modern Physics, H. D. Young, R. A. Freedman, T. R. Sandin, A. L. Ford. • You might also be interested in A. Einstein's book for the popular audience, Relativity: The Special and the General Theory. This book is now in the public domain and is readily available online, e.g. http://www.gutenberg.org/ebooks/30155 1.3 UCD policies as applied to this module Policy on late submission of homework The official UCD policy will be absolutely strictly adhered to1. As such, coursework that is late by up to one week after the due date will have the grade awarded reduced by 10% (e.g. from 87% to 77%); coursework submitted up to two weeks after the due date will have the grade reduced by 20% (e.g. from 87% to 67%). Coursework received more than two weeks after the due date will not be accepted. In addition, it will not be possible for me to accept late coursework via email in a scan or otherwise { it will be the student's responsibility to submit such late homework directly to me in paper form. There are many reasons for this { not least because of incidents in the past where I have had to print down pages and pages of late submissions consisting of jpeg after jpeg of mobile-phone photographs of homework. Extenuating circumstances This module adheres to the policy on extenuating circumstances as outlined in the UCD Science Handbook: • Serious issues (serious illness, hospitalization, etc.) are dealt with through a formal process administered by the College of Science. • Minor issues (e.g. assignment/midterm missed due to minor illness) are dealt with through direct contact with the lecturer { via email, with supporting documentation as necessary. In order to keep track of such extenuating circumstances, please use the phase `minor extenuating circumstances ACM20050' in the subject line in the email. • Retrospective issues (e.g. illness affected my performance in the midterm exam, so I don't want my grade to stand and would prefer extenuating circumstances) will not be entertained by the lecturer but may be referred to the formal process administered by the College of Science. 1https://sisweb.ucd.ie/usis/!W HU MENU.P PUBLISH?p tag=GD-DOCLAND&ID=137 3 Chapter 1. Introduction Plagiarism Plagiarism is a serious academic offence. While plagiarism may be easy to commit unintentionally, it is defined by the act not the intention. • All students are responsible for being familiar with the University's policy statement on pla- giarism2 and are encouraged, if in doubt, to seek guidance from an academic member of staff.