Unit information: Mechanics 2 in 2021/22

Unit name Mechanics 2 MATH21900 20 I/5 Teaching Block 2 (weeks 13 - 24) Dr. Mike Blake Not open MATH10012 ODEs, Curves and Dynamics (or MATH11007 Calculus 1 and PHYS10006 Core Physics 1) and MATH11005 Linear Algebra and Geometry None School of Mathematics Faculty of Science

Description

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.

Relation to Other Units

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.

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.

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 details

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 Details

90% 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.

Recommended

• Douglas R. Gregory, Classical Mechanics: An Undergraduate Text, Cambridge University Press, 2006

Further

• Richard P. Feynman, Robert B. Leighton and Matthew L. Sands, The Feynman Lectures on Physics Volume 2, Addiston-Wesley, 1964
• Grant R. Fowles, Analytical Mechanics, Thomson Brooks/Cole, 2005
• Herbert Goldstein, Charles P. Poole, and John L. Safko, Classical Mechanics, Addison-Wesley, 2002
• T. W. B. Kibble and F. H. Berkshire, Classical Mechanics, Imperial College Press 2004
• Cornelius Lanczos, The Variational Principles of Mechanics, Dover, 1986
• Wolfgang Yourgrau and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Theory, Dover, 1979