| CODE | PHY2195 | ||||||||
| TITLE | Classical and Relativistic Mechanics | ||||||||
| UM LEVEL | 02 - Years 2, 3 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 5 | ||||||||
| ECTS CREDITS | 6 | ||||||||
| DEPARTMENT | Physics | ||||||||
| DESCRIPTION | Relativity forms the basis of many fields in modern physics while most modern theories are formulated with classical mechanics as the starting point. This study-unit will cover many of the basic concepts and mechanics of special relativity. The basis and formalism of classical mechanics will also be covered within this context. The emphasis of the study-unit will be the use of special relativity in performing mechanical calculations. The study-unit will conclude with an introduction to some applications of the formalism in several modern fields of physics. Study-Unit Aims: The study-unit segment on Lagrangian and Hamiltonian mechanics will aim to provide a/an: - review of Newtonian mechanics (forces and rotation problems) and various coordinate system settings; - explanation of the calculus of variations with physical examples; - introduction to constrained variations and Hamilton's principle of least action; - in-depth analysis of the Lagrangian and derivation of the Euler Lagrange equations (with examples); - introduction to Lagrangian dynamics as applied to physical systems; - in-depth analysis of generalized coordinates and the Legendre transformations; - explanation of the Hamiltonian derivation with examples; - details of phase space; - explanation of how to setup mechanical problems using the Lagrangian and Hamiltonian analysis as well as setting up their corresponding equations of motion; - introduction to symmetry and conservation laws in physical systems; - introduction to generators of transformations and Poisson brackets; - explanation of Noether's theorem with examples;- introduction to the Euler-Lagrange equations and their derivation; - applications of the principle of least action. The study-unit special relativity segment's aim is to familiarize the student with the basic elements that go into modern mechanics of relativity theory. This includes providing a/an: - an in depth understanding of Galilean relativity and an understanding of the limits that led to the necessity of special relativity; - familiarization with the concept of a reference frame being used to make measurements in an experiment; - basic of the principles of special relativity and an understanding of the limit where is takes over from Galilean relativity; - explanation of the foundational experiments that led to and test special relativity even up to today; - an introduction to the relativity of simultaneity and causality; - introduction and analysis of space-time diagrams for stationary, moving and accelerating systems; - in-depth analysis of the Lorentz transformations including the Lorentz invariant; - introduction to the idea of measurement theory in relativity where length contraction, time dilation and Doppler shifts will all be investigated; - in-depth analysis of relativistic conservation laws with their Lorentz transformation analogues; - introduction to relativistic interactions (four-vectors, four-momentum, types of collisions, scattering); - familiarization with standard relativity `paradoxes' Twin and Pole-barn runner experiments; - explanation of Euclidean and Minkowski metrics; - familiarization with acceleration (constant acceleration, synchronization, limits of special relativity); - analysis of particle dynamics in special relativity. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - explain the notion of calculus of variations; - state Hamilton's principle of least action; - identify the Lagrangian of a system and explain the how the Euler-Lagrange equations evolve for this system; - explain generalized coordinates and Legendre transformations; - identity the Hamiltonian of a system; - explain the concept of phase space; - state how to set-up problems using the Lagrangian and Hamilton mechanics approach; - identify symmetries in a system; - explain the Euler-Lagrange equations of motion in various settings; - explain how Noether's theorem can be used to derive conservation laws; - explain how to use generators of transformations and Poisson brackets; - explain how the Lagrangian and Hamiltonian of a charged particle can be used to determine the dynamics of a system; - identify the distinction between inertial and non-inertial frames of reference; - state the basic principles of special relativity; - identify key experiments that need special relativity to be explained; - explain the concept of relativity of simultaneity and how causality is refined in special relativity; - identify spacetime diagrams and how they can be used to describe systems in motion; - explain the Lorentz transformation and the Lorentz invariant; - explain the distinction between a measurement and rest frame quantities; - state the relativistic conservation laws; - explain scattering in special relativity by using four-vectors; - identify and explain the standard relativity `paradoxes'; - explain relativistic dynamics as applied to particle collisions; - identify and explain the Euclidean and Minkowski metrics; - explain the role of acceleration within relativity and the limits of special relativity as a whole. 2. Skills: By the end of the study-unit the student will be able to: - use calculus of variations to investigate different classical mechanics set-ups; - determine the constrained systems using Hamilton's principle of least action; - determine the Lagrangian of a system and it's Euler-Lagrange evolution; - calculate the output of a system using Lagrangian mechanics; - perform Legendre transformations; - calculate the Hamiltonian of a system; - determine the generalized momenta from the generalized coordinates for a system; - determine phase space behavior; - determine the equations of motion using the Lagrangian and Hamiltonian approach; - use symmetries to identify conservation laws; - use the principle of least action in various scenarios; - determine the Poisson bracket of two functions; - utilize Noether's theorem to derive symmetry relations; - perform calculations on a charged particle using the Lagrangian and Hamiltonian approach;- calculate transformations between inertial frames; - perform calculations on the standard experiments in special relativity; - determine the causal structure of a system; - draw and interpret spacetime diagrams for stationary, moving and accelerating systems; - perform calculations using the Lorentz transformations; - use relativistic analogues of conservations laws; - determine outcomes from scattering experiments in relativistic scenarios; - show consistency in so-called `paradoxes'; - perform calculations on rotational systems; - use Euclidean and Minkowski metrics; - perform calculations in accelerating systems. Main Text/s and any supplementary readings: Main Texts: - Tsamparlis, M., `Special Relativity: An Introduction with 200 Problems and Solutions', first edition, Springer-Verlag Berlin Heidelberg. - French, A.P., ` Special Relativity (M.I.T. Introductory Physics)’, W. W. Norton & Company. Supplementary Readings: - Woodhouse, N.M.J., `Special Relativity', first edition, Springer-Verlag London. |
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| STUDY-UNIT TYPE | Lecture | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Jackson Said |
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2024/5. It may be subject to change in subsequent years. |
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