Study-Unit Description

Study-Unit Description


CODE PHY2190

 
TITLE Classical and Relativistic Mechanics

 
UM LEVEL 02 - Years 2, 3 in Modular Undergraduate Course

 
MQF LEVEL 5

 
ECTS CREDITS 4

 
DEPARTMENT Physics

 
DESCRIPTION This study unit constitutes an advanced course in classical mechanics. It comprehends the Lagrangian and Hamiltonian formulation of classical mechanics and the extension of classical mechanics to special relativity.

Study-Unit Aims:

The study of the Lagrangian and Hamiltonian formulation of classical mechanics will provide:

- a review of the vector formulation of classical mechanics;
- an introduction of the notion of virtual work;
- an introduction to calculus of variation;
- an introduction to the principle of stationary and least action;
- the derivation osf the Lagrangian and the Euler-Lagrange equations together with examples of how to use them to solve problems in classical mechanics;
- a detailed description of the phase space and the generalised coordinates and the corresponding momentum;
- a description of the Legendre transformation;
- the derivation of the Hamiltonian together with examples on how to use it to solve problems in classical mechanics;
- the details of how to construct the Lagrangian and the Hamiltonian for a system and the corresponding equations of motion;
- an explanation of how to use the symmetries of the system to find conserved quantities which will lead to the derivation of Noether’s (first) theorem;
- the definition of the Poisson bracket and its relation to the canonical coordinates.

The study of relativistic mechanics will provide:

- an in depth analysis of frames of reference and the principle of relativity;
- a historical overview of the limitations of classical mechanics that led to the formulation of special relativity;
- the basics assumptions of special relativity together with a review of their experimental verification;
- an explanation of the notion of simultaneity in an inertial frame as applicable to special relativity;
- an introduction to space-time diagrams and their use to solve problems;
- the derivation of Lorentz transformations for coordinates, velocity and acceleration together with examples on how to use them to solve problems;
- a detailed analysis on physical measurements from the perspective of special relativity, including in particular the notions of length contraction, time dilation and the relativistic Doppler shift;
- a detailed description of the notion of causality and the limitations that it imposes on events;
- a discussion on the distinction between looking and measuring in relativistic mechanics;
- the derivation of the relativistic principles of conservation of momentum and energy and the their use to solve problems in special relativity with particular reference to relativistic collisions as well as particle production.

Learning Outcomes:

1. Knowledge & Understanding:

By the end of the study-unit the student will be able to:

- define virtual work;
- explain the notion of calculus of variation and its relation to the principle of stationary and least action;
- explain what the phase space, the generalised coordinates and the corresponding momentum are;
- explain what the Legendre transformation is;
- identify the property that distinguishes between inertial and non-inertial reference frames and use it to determine if a given reference frame is inertial or non-inertial;
- explain how the Michelson-Morley experiment was carried out together with the results, and its implication on the speed of light;
- list at least one experiment apart from the Michelson-Morley experiment that led to the formulation of special relativity;
- state the postulates of special relativity and list at least two experiments that verify them;
- explain the notion of simultaneity as applicable to special relativity;
- explain the notion of causality, its links with the past and future events, the "elsewhere" and the space time invariant;
- explain the difference between looking and measuring in relativistic mechanics;
- explain what is meant by the twin paradox.

2. Skills:

By the end of the study-unit the student will be able to:

- determine the Lagrangian and Hamiltonian of a system from the dynamic variables and the coordinates;
- determine the generalised momenta from the generalised coordinates;
- determine the equations of motion for the Lagrangian and Hamiltonian systems;
- use the Lagrangian, Euler-Lagrange and Hamiltonian to solve simple problems of classical mechanics;
- identify the symmetries of the given system and determine the corresponding conserved quantities;
- use the Poisson bracket to determine if a set of coordinates is canonical;
- draw and interpret space-time diagrams as well as use it to solve problems simple problems;
- use the Lorentz transformations for coordinates, velocity and acceleration to solve simple problems in special relativity;
- be able to reproduce the derivations of the length contraction, time dilation and relativistic Doppler shift from the Lorentz transformations;
- use length contraction, time dilation and relativistic Doppler shift to solve simple problems;
- calculate the space-time invariant between two point events and determine if the interval between events is space-like or time-like;
- use the space-time invariant to solve simple problems in special relativity;
- use the relativistic principles of conservation of momentum and energy to solve simple problems in special relativity.

Main Text/s and any supplementary readings:

Lagrangian and Hamiltonian formulation of classical mechanics:

Textbook:

Goldstein, H., Poole, P.C. and Safko, J.L., Classical mechanics, third edition, Addison Wesley

Suggested reading:

- Gignoux, C. and Silvestre-Brac, B, Solved Problems in Lagrangian and Hamiltonian Mechanics, Springer.
- Kamal, A.A., 1000 Solved Problems in Classical Physics: An Exercise Book, Springer.

Special relativity:

Textbooks:

- French, A. P., Special relativity, Van Nostrand Reinhold.

Suggested reading:

- Lightman, A. and R.H. Price, Problem Book in Relativity and Gravitation, Princeton University Press.

The use of other books with similar content would be equivalently good.

 
ADDITIONAL NOTES Pre-Requisites: Basic knowledge of Mathematics including calculus and geometry as well as the vectorial formulation of classical mechanics.

This unit follows from PHY1190 or any course describing the vectorial formulation of classical mechanics. Knowledge of advanced calculus, vector calculus, matrices and geometry at the level of MAT1091, MAT1511 and MAT2512, or equivalent is expected.

 
STUDY-UNIT TYPE Lecture

 
METHOD OF ASSESSMENT
Assessment Component/s Sept. Asst Session Weighting
Examination (2 Hours) Yes 100%

 
LECTURER/S Jackson Said

 

 
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 2025/6. It may be subject to change in subsequent years.

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