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Mechanics 2 (MATH 21900)

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Administrative Information

  1. Unit number and title: MATH 21900 Mechanics 2
  2. Level: I/5
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 1999
  6. Unit Organiser: Sebastian Muller
  7. Lecturer: Dr Sebastian Muller
  8. Teaching block: 1
  9. Prerequisites: MATH 11007 Calculus 1 and MATH 11005 Linear Algebra & Geometry and either MATH 11009 Mechanics 1 or Core Physics A

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.

General Description of the Unit

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.  There is a level 3 version, Mechanics 23.  Students may NOT take both Mechanics 2 and Mechanics 23.

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.

Teaching Methods

Lectures supported by problem classes and problem and solution sheets. 

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

Assessment Methods

The unit mark for Mechanics 2 is calculated from one 2 ½ -hour examination in April consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted.

Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

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)

Texts

Lecture notes will be provided. (See http://www.maths.bris.ac.uk/~maxsm/mechnotes.pdf for last year's version.)

Also the later chapters of

  • Classical Mechanics, R. Douglas Gregory, Cambridge University Press (2006)

are especially recommended.

Further literature:   

  • Classical Mechanics,  B. Kibble & Frank H. Berkshire, Imperial College Press (2004)
  • Analytical Mechanics, G.R. Fowles & G.L. Cassiday, Saunders College Publishing (1993)
  • Richard Feynman's lecture on the principle of least action in The Feynman Lectures on Physics, Vol II, Ch 19, R.P. Feynman, R.B. Leighton, and M Sands, Addison-Wesley Publishing (1964) 
  • Mechanics, 3 ed, L.D. Landau & E.M Lifschitz, Pergamon (1976)
  • Mathematical Methods of Classical Mechanics, V.I. Arnold, Springer-Verlag (1978)
  • Classical Mechanics,  2 ed., H. Goldstein,  Addison-Wesley (1980)
  • Variational Principles in Dynamics and Quantum Theory, W. Yourgrau and S. Mandelstam, Dover Publications (1968)
  • The Variational Principles of Mechanics, 4 ed., C. Lanzcos, Dover Publications (1986)

Syllabus

Weeks per topic is approximate at three lectures per week

0. Introduction

1. Calculus of variations [2 weeks] 

  • Euler-Lagrange equations in one and more dimensions. Alternative form. Examples: brachistochrone, Fermat's principle.

1. Lagrangian mechanics [3 weeks] 

  • Principle of least action and Lagrange's equations. Generalised coordinates. Constraints. Derivation of Lagrange's equations from Newton's laws. Conserved quantitities (generalised energy, generalised momenta, Noether's theorem). Examples, including spherical pendulum.

3. Small oscillations [2 weeks]

  • Normal modes. Stability of equilibria. Examples.  
       

4. Rigid bodies [1.5 weeks]

  • Angular velocity. Inertia tensors. Euler's equations.   

5. Hamiltonian mechanics [2.5 weeks] 

  • Hamilton's equations. Phase space. Conservation laws and Poisson brackets. Canonical transformations. Action-angle variables. Chaos.

There may be minor changes to this syllabus.