Unit information: Mechanics 23 in 2021/22

Unit name	Mechanics 23
Unit code	MATH31910
Credit points	20
Level of study	H/6
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Dr. Mike Blake
Open unit status	Not open
Pre-requisites	Year 2 Theoretical Physics. OR MATH10012 ODEs, Curves and Dynamics and MATH10015 Linear Algebra Core Physics A can be used in replace of Mechanics 1.
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Unit Aims

• To introduce variational principles in mechanics.
• To introduce Lagrangian and Hamiltonian mechanics and their applications.
• To provide a foundation for further study in mathematical physics.

Unit Description

In Newtonian mechanics, the trajectory of a particle is governed by the second-order differential equation F = ma. An equivalent formulation, due to Maupertuis, Euler and Lagrange, determines the particle's trajectory as that path which minimises (or, more generally, renders stationary) a certain quantity called the action. The mathematics which links these two formulations (which at first seem so strikingly different) is the calculus of variations.

The known fundamental laws of physics (e.g., Maxwell's equations for electricity and magnetism, the equations of special and general relativity, and the laws of quantum mechanics) can be formulated in terms of variational principles, and indeed find their simplest expression in this way. The principle of least action in classical mechanics is conceptually one of the simplest, and historically one of the first such examples.

The course covers the principle of least action, the calculus of variations, Lagrangian mechanics, the relation between symmetry and conservation laws, and the theory of small oscillations. The last part of the course is an introduction to Hamiltonian mechanics, including Poisson brackets, canonical transformations. Hamilton-Jacobi theory and some qualitative results.

Relation to Other Units

This unit is a more advanced version of Mechanics 2. The lectures for Mechanics 2 and Mechanics 23 are the same, but the problem sheets and examination questions for Mechanics 23 are more challenging. Students may NOT take both Mechanics 2 and Mechanics 23.

This unit develops the mechanics met in the first year from a more general and powerful point of view. Lagrangian and Hamiltonian methods are used in many areas of Mathematical Physics. Familiariaty with these concepts is helpful for Quantum Mechanics, Quantum Chaos, Quantum Information Theory, Statistical Mechanics and General Relativity. Variational calculus, which forms part of the unit, is an important mathematical idea in general, and is relevant to Control Theory and to Optimisation.

Intended Learning Outcomes

Learning Objectives

At the end of the unit the student should:

• understand the notions of configuration space, generalised coordinates and phase space in mechanics
• be able to obtain the Euler-Lagrange equations from a variational principle
• understand the relation between Lagrange's equations and Newton's laws
• be able to use Lagrange's equations to solve complex dynamical problems
• be able to calculate the normal modes and characteristic frequencies of linear mechanical systems
• be able to obtain the Hamiltonian formulation of a mechanical system
• understand Poisson brackets
• understand canonical transformations

Transferable Skills

• Use of mathematical methods to describe "real world" systems
• Development of problem-solving and analytical skills, assimilation and use of complex and novel ideas
• Mathematical skills: Knowledge of the calculus of variations; an understanding of the importance of variational principles in physical theory; analysis of complex problems in mechanics; analysis of linear systems (normal modes, characteristic frequencies)

Teaching Information

The unit will be taught through a combination of

• synchronous online and, if subsequently possible, face-to-face lectures
• asynchronous online materials, including narrated presentations and worked examples
• guided asynchronous independent activities such as problem sheets and/or other exercises
• synchronous weekly group problem/example classes, workshops and/or tutorials
• synchronous weekly group tutorials
• synchronous weekly office hours

Assessment Information

90% Timed, open-book examination 10% Coursework

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

Resources

If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.

If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATH31910).

How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours of study to complete. Your total learning time is made up of contact time, directed learning tasks, independent learning and assessment activity.

See the Faculty workload statement relating to this unit for more information.

Assessment
The Board of Examiners will consider all cases where students have failed or not completed the assessments required for credit. The Board considers each student's outcomes across all the units which contribute to each year's programme of study. If you have self-certificated your absence from an assessment, you will normally be required to complete it the next time it runs (this is usually in the next assessment period).
The Board of Examiners will take into account any extenuating circumstances and operates within the Regulations and Code of Practice for Taught Programmes.