Study-Unit Description

Study-Unit Description


TITLE Classical and Relativistic Mechanics

LEVEL 02 - Years 2, 3 in Modular Undergraduate Course



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 generalised coordinates and the Legendre transformations;
- explanation of the Hamiltonian derivation with examples;
- details of phase space and Liouville's theorem;
- 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 Hamilton Jacobi Equations;
- application of the Lagrangian and Hamiltonian mechanics to a charged particle;
- application of the Lagrangian and Hamiltonian mechanics to special relativity.

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 Polebarn runner experiments;
- analysis of rotation including Thomas precession;
- explanation of Euclidean and Minkowski metrics;
- familiarization with acceleration (constant acceleration, synchronization, limits of 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 generalised coordinates and Legendre transformations;
- identity the Hamiltonian of a system;
- explain the concept of phase space and how to use Liouville's theorem;
- state how to set-up problems using the Lagrangian and Hamilton 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 the Hamilton-Jacobi equations can be used to solve mechanical problems;
- explain how the Lagrangian and Hamiltonian of a charged particle can be used to determine the dynamics of a system;
- explain how special relativity can tie into this new way of doing mechanics;
- 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 the phenomenon of Thomas precession;
- 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 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 generalised momenta from the generalised 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;
- determine the Hamilton-Jacobi equations for a system;
- perform calculations on a charged particle using the Lagrangian and Hamiltonian approach;
- calculate simple problems in special relativity 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:

Principle Textbook

- Tsamparlis, M., `Special Relativity: An Introduction with 200 Problems and Solutions', first edition, Springer-Verlag Berlin Heidelberg.
- Woodhouse, N. M. J., `Introduction to Analytical Dynamics', second edition, Springer-Verlag London.

Supplementary Textbook

- Woodhouse, N.M.J., `Special Relativity', first edition, Springer-Verlag London.
- Landau, L.D. and Lifshitz, E.M., `Mechanics', Third Edition, Pergamon Press.

ADDITIONAL NOTES Pre-Requisite qualifications: A basic knowledge of mechanics and an understanding of calculus


Assessment Component/s Assessment Due Resit Availability Weighting
Examination (3 Hours) SEM1 Yes 100%

LECTURER/S Jackson Said

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It should be noted that all the information in the study-unit description above applies to the academic year 2019/0, if study-unit is available during this academic year, and may be subject to change in subsequent years.