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University College Dublin an Coláiste Ollscoile, Baileátha Cliath School of Mathematical Sciences Scoil Na Neola´Iochta´I University College Dublin An Col¶aisteOllscoile, Baile Atha¶ Cliath School of Mathematical Sciences Scoil na nEola¶³ochta¶³Matamaitice Mechanics and Special Relativity (ACM 10030) Dr. Lennon O¶ N¶araigh Lecture notes in Mechanics and Special Relativity, January 2011 Mechanics and Special Relativity (ACM10030) ² Subject: Applied and Computational Maths ² School: Mathematical Sciences ² Module coordinator: Prof. Adrian Ottewill ² Credits: 5 ² Level: 1 ² Semester: Second This course develops the theory of planetary and satellite motion. It discusses the work of Kepler and Newton that described the elliptic orbits of planets around the earth and which can be applied to the elliptic motion of satellites around the earth. We examine the dynamics of spacecraft. Einstein's Special Theory of Relativity is then introduced. His two basic postulates of relativity are discussed and we show how space and time appear to two observers moving relative to each other. We derive, and discuss the meaning of, Einstein's famous formula E = mc2. What will I learn? On completion of this module students should (be able to) 1. Explain the concepts of planetary and satellite orbits, Kepler's laws and how to boost an earth satellite from one orbit to another; 2. Solve orbit problems in mathematical term; 3. Describe Einstein's postulates and derive the results of special relativity on simultaneity, length contraction, time dilation and relative velocity; 4. Describe Einstein's postulates and derive the results of special relativity on simultaneity, length contraction, time dilation and relative velocity; 5. Solve problems in special relativity in mathematical terms. i ii Editions First edition: January 2010 This edition: January 2011 iii iv Contents Abstract i 1 Introduction 1 2 Vector operations 10 3 Inertial frames of reference 18 4 Worked examples: Vectors and Galilean velocity addition 25 5 Conservative forces in one dimension 33 6 Worked example: Conservative forces in one dimension 42 7 Motion in a plane 46 8 The theory of partial derivatives 55 9 Angular momentum and central forces 60 10 Central forces reduce to one-dimensional motion 67 11 Interlude: Physical units 73 12 Interlude: Energy revisited 78 13 Kepler's Laws 85 14 Kepler's First Law 91 v 15 Kepler's Second and Third Laws 102 16 Applications of Kepler's Laws in planetary orbits 109 17 Mathematical analysis of orbits 116 18 Planning a trip to the moon 119 19 Introduction to Einstein's theory of Special Relativity 125 20 The Lorentz Transformations 129 21 Length contraction and time dilation 137 22 Relativistic momentum and energy 143 23 Special topics in Special Relativity 150 vi Chapter 1 Introduction 1.1 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 ¯rst 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 Newton's laws need to be modi¯ed at speeds close to the speed of light. These modifcations 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 ¯rst principles. Using the same ideas, we will examine the dynamics of spacecraft, showing how they can be boosted from one orbit to another. The twin notions of inertial frames and Galilean invariance are at the heart of this study. 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 2 Chapter 1. Introduction Lorentz transformations1. As a consequence, we will discuss length contraction, time dilation, and relative velocity. We will also discuss the equation E = mc2. Some de¯nitions: ² We call the problem of determining the trajectories of two bodies interacting via Newtonian gravity the orbit problem. ² 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 Learning and Assessment Learning: ² Thirty six classes, three per week. ² In some classes, we will solve problems together or look at supplementary topics. ² One of the main goals of a mathematics degree is to solve problems autonomously. This will be accomplished through homework exercises, independent study, and through independently practising the problems in the recommended textbooks. Assessment: ² One end-of-semester exam, 60%. ² Two in-class exams, for a total of 20%. ² Four homework assignments, for a total 20%. Policy on late submission of homework: All uncerti¯ed late submissions will attract a penalty of 50%, that is, if the late assignment receives a grade X, only the grade X=2 will be entered. Textbooks: ² Lecture notes will be put on the web. These are self-contained. ² However, here are some books for extra reading: 1These are referred to as the Fitzgerald{Lorentz transformations in another Dublin university, in honour of George Francis Fitzgerald. 1.3. A modern perspective on classical mechanics 3 { An Introduction To Mechanics, D. Kleppner, R. J. Kolenkow (one copy in library). { For the second part of the course, look at the later chapters in Sears and Zemansky's University Physics With Modern Physics, H. D. Young, R. A. Freedman, T. R. Sandin, A. L. Ford (Many copies in library). { You might also be interested in A. Einstein's book for `the man (and woman) on the street', Relativity: The Special and the General Theory. This is out of copyright and I have placed it on Blackboard for downloading. 1.3 A modern perspective on classical mechanics Before beginning the lecture course, let us discuss a problem in Newtonian mechanics that is the focus of continuing research. The problem also involves orbits. The three-body problem: We shall show in this lecture course that an analytical solution in terms of integrals exists for a system of two particles interacting via gravity. No such solution exists for three particles, and the motion can become very complex and is not yet understood fully. Even the motion of such a system in the plane R2 is complicated, but can be tackled numerically. This is what we examine here. Newton's law of gravity states that the gravitational force on particle i due to particle j Fij is given by µ ¶ xi ¡ xj Fij = ¡Gmimj 3 (1.1) jxi ¡ xjj In the schematic diagram (Fig. 1.1), particle (1) experiences a force from particle (0) and particle (2), to give a net force µ ¶ µ ¶ x1 ¡ x2 x1 ¡ x0 F12 + F10 = ¡Gm1m2 3 ¡ Gm1m0 3 : (1.2) jx1 ¡ x2j jx1 ¡ x0j Newton's law of motion, force=mass£acceleration gives m1xÄ1 = F12 + F10, or, µ ¶ µ ¶ x ¡ x x ¡ x Ä 1 2 1 0 m1x1 = ¡Gm1m2 3 ¡ Gm1m0 3 : (1.3a) jx1 ¡ x2j jx1 ¡ x0j The equations for the other two bodies can be written down immediately by permutations 0 ! 1, 1 ! 2, 2 ! 0: µ ¶ µ ¶ x ¡ x x ¡ x Ä 2 0 2 1 m2x2 = ¡Gm2m0 3 ¡ Gm2m1 3 : (1.3b) jx2 ¡ x0j jx2 ¡ x1j µ ¶ µ ¶ x ¡ x x ¡ x Ä 0 1 0 2 m0x0 = ¡Gm0m1 3 ¡ Gm0m2 3 : (1.3c) jx0 ¡ x1j jx0 ¡ x2j 4 Chapter 1. Introduction Figure 1.1: Schematic diagram of three-body problem. Position vectors and their time derivatives lie in the plane. Radius vectors: q rij = rji = [(xi ¡ xj) ¢ (xi ¡ xj)]: Equations (1.3) can be solved numerically as an ODE problem by de¯ning the vector Y : T Y = [x0; y0; x_ 0; y_0; x1; y1; x_ 1; y_1; x2; y2; x_ 2; y_2] ; (1.4) 1.3. A modern perspective on classical mechanics 5 such that 0 1 x_ 0 B C B C B y_0 C B 3 3 C B ¡Gm1 (x0 ¡ x1) =r ¡ Gm2 (x0 ¡ x2) =r C B 01 02 C B 3 3 C B ¡Gm1 (y0 ¡ y1) =r01 ¡ Gm2 (y0 ¡ y2) =r02 C B C B x_ C B 1 C B C dY B y_1 C = B C : dt B 3 3 C B ¡Gm2 (x1 ¡ x2) =r12 ¡ Gm0 (x1 ¡ x0) =r10 C B C B ¡Gm (y ¡ y ) =r3 ¡ Gm (y ¡ y ) =r3 C B 2 1 2 12 0 1 0 10 C B C B x_ 2 C B C B C B y_2 C B 3 3 C @ ¡Gm0 (x2 ¡ x0) =r20 ¡ Gm1 (x2 ¡ x1) =r21 A 3 3 ¡Gm0 (y2 ¡ y0) =r20 ¡ Gm1 (y2 ¡ y1) =r21 We perform a numerical simulation with the following initial conditions (G = 1): m0 = 1000; m1 = 100; m2 = 0:001; (1.5a) ³ p ´ 1 x0 (t = 0) = (0; 0) ; x1 (t = 0) = (1; 0) ; x2 (t = 0) = 2 1; 3 ; (1.5b) x_ 0 (t = 0) = (0; ¡10) ; x_ 1 (t = 0) = (0; 10) ; x_ 2 (t = 0) = (1; ¡4) : (1.5c) (See Fig. 1.2.) ² The particle orbits are shown in Fig. 1.3. ² An animation of the orbits is on the web. ² A phase portrait (x_ versus x) for m1 is shown in Fig. 1.4. ² The conserved energy X3 Gm m Gm m Gm m E = 1 m x_ 2 ¡ 1 2 ¡ 1 0 ¡ 2 1 2 i i r r r i=1 12 10 21 is shown in Fig.
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