| CODE | PHY2285 | ||||||||||||
| TITLE | Classical Mechanics and Symmetries | ||||||||||||
| UM LEVEL | 02 - Years 2, 3 in Modular Undergraduate Course | ||||||||||||
| MQF LEVEL | 5 | ||||||||||||
| ECTS CREDITS | 6 | ||||||||||||
| DEPARTMENT | Physics | ||||||||||||
| DESCRIPTION | Classical mechanics and symmetries forms the basis of many fields in modern physics since it provides a basis on which to construct physical theories and to explore their properties such as their conservation laws. The vast array of topics that are contained in physics can be related together in classical mechanics where the Lagrangian formulation of physical problems can be used to form equations of motion for a large plethora of physical setups. This study-unit will cover this formalism and extend it to the Hamiltonian formalism which can provide further information about these physical systems. Students will also be introduced to canonical transformations which are an important part of Hamiltonian analysis. In all areas, students will be shown the crucial role that symmetries play in physics. Study-unit Aims: Classical mechanics: - Review of Newtonian mechanics (forces and rotation problems) and coordinate systems; - Explanation of the calculus of variations with 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; - In-depth analysis of generalized coordinates and the Legendre transformation; - Explanation of the Hamiltonian with examples; - Explanation of how to set-up mechanical problems using the Lagrangian and Hamiltonian as well as setting up their corresponding equations of motion; - Introduction to symmetry and conservation laws; - Introduction to generators of transformations and Poisson brackets; - Explanation of Noether's theorem with examples; - Application of canonical transformations. Symmetries: - Introduction to many-body systems; - Applications of the coupled harmonic oscillators model to many-body systems; - Analysis of the role of symmetries in Kepler's problem; - Analysis of the angular momentum as symmetry generator; - Explanation of the relation between symmetry and phases of matter. Learning Outcomes: 1. Knowledge & Understanding By the end of the study-unit the student will be able to: Classical mechanics: - 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; - State how to set-up problems using the Lagrangian and Hamiltonian mechanics approach; - Identify symmetries in a system; - Explain how Noether's theorem can be used to derive conservation laws; - Explain how to use generators of transformations and Poisson brackets; - Explain how canonical transformations can be used with canonical equations. Symmetries: - Distinguish between discrete and continuous symmetries; - Relate parity, time inversion and charge conjugation symmetries to conserved quantities; - Relate rotational symmetry and angular momentum; - Understand the concept of orbits in the two-body central forces problem; - Understand the role of symmetries in solving many body systems problems; - Describe the concept of symmetry breaking mechanism; - Relate phase transitions and spontaneous symmetry breaking. 2. Skills By the end of the study-unit the student will be able to: Classical mechanics: - Use calculus of variations to investigate different classical mechanics set-ups; - Determine the constrained systems using Hamilton's principle of least action; - Determine the 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; - Determine the Poisson bracket of two functions; - Utilize Noether's theorem to derive symmetry relations; - Use canonical transformations within a canonical system. Symmetries: - Integrate the equations of motion in the presence of a central force; - Find the orbits in Kepler’s problem; - Solve coupled harmonic oscillators utilising lattice translational symmetry; - Determine the symmetries of a many-body system and their associated conserved quantities; - Identify different phases of matter on the basis of symmetry arguments. Main Text/s and any supplementary readings: Supplementary Textbook - Landau, L.D. and Lifshitz, E.M., `Mechanics', Third Edition, Pergamon Press. - Woodhouse, N. M. J., `Introduction to Analytical Dynamics', second edition, Springer-Verlag London. |
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| ADDITIONAL NOTES | Pre-Requisite Study-unit: Introduction to Classical Mechanics and Waves | ||||||||||||
| STUDY-UNIT TYPE | Lecture | ||||||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Tony John George Apollaro 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 2023/4. It may be subject to change in subsequent years. |
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