Study information

# Analytical and Chaotic Dynamics - 2023 entry

MODULE TITLE CREDIT VALUE Analytical and Chaotic Dynamics 15 PHY2032 Prof Jacopo Bertolotti (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11
 Number of Students Taking Module (anticipated) 33
DESCRIPTION - summary of the module content

This module introduces some fundamental concepts in analytical dynamics, and illustrates their applications to relevant problems. The module covers the calculus of variations, Lagrangian and Hamiltonian formulations of dynamics, Poisson brackets, canonical transformations, and Hamilton-Jacobi equations. The approach is necessarily mathematical and students are advised to take this optional module only if they have got marks of at least 60% in both PHY1021 Vector Mechanics and PHY1026 Mathematics for Physicists (or in equivalent modules in other departments).

AIMS - intentions of the module

This module will be of interest to students wishing to develop their grasp of theoretical physics. The subject of analytical dynamics provides advanced theoretical developments which prove elegant and versatile in solving dynamical problems.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)
A student who has passed this module should be able to:

Module Specific Skills and Knowledge:
1. solve advanced dynamical problems involving classical particles by applying the Lagrangian and Hamiltonian formulations;
2. explain the calculus of variations and apply it to the solution of problems;
3. state the Hamilton-Jacobi equations and apply them to the solution of problems;
4. describe the relationship between poisson brackets and quantum mechanical commutation relations;
5. describe the basic concepts of chaos theory and explain how chaos theory may be used in different disciplines;

Discipline Specific Skills and Knowledge:
6. formulate mathematical descriptions of physical systems;

Personal and Key Transferable / Employment Skills and Knowledge:
7. use mathematics to solve problems.
SYLLABUS PLAN - summary of the structure and academic content of the module

I. Generalized coordinates. Holonomic and nonholonomic constraints
II. Virtual displacement. D'Alembert's principle
III. The Lagrangian formulation
IV. The Hamiltonian formulation
1. Configuration space; generalized (canonical or conjugate) momentum
2. Phase space
3. Legendre transformation
4. Hamiltonian; Hamilton's equations
5. Cyclic co-ordinates and conservation theorems
6. Liouville's theorem
V. Calculus of variations
VI. Poisson brackets
1. Lagrange brackets
2. Poisson brackets
VII. Hamilton-Jacobi equations and action-angle variables
VIII. The transition to quantum mechanics
IX. Nonlinear Dynamical Systems
1. Chaos and its relevance to mechanics
2. The stability of non-linear equations
3. The non-linear oscillator
4. Phase-Space Methods
5. The pendulum revisited
6. Mappings
7. Characterisation of chaotic systems
LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 22 128
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning & teaching activities 20 hours 20×1-hour lectures Scheduled learning & teaching activities 2 hours 2×1-hour problems/revision classes Guided independent study 30 hours 5×6-hour self-study packages Guided independent study 16 hours 4×4-hour problems sets Guided independent study 82 hours Reading, private study and revision

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Guided self-study (0%) 5×6-hour packages (fortnightly) 1-7 Discussion in class
4 × Problems sets (0%) 4 hours per set (fortnightly) 1-7 Solutions discussed in problems classes.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 0 100
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Final Examination 100% 120 minutes (May/June assessment period) 1-7 Mark via MyExeter, collective feedback via ELE and solutions.

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
Whole module Written examination (100%) 1-7 August/September assessment period

Re-assessment is not available except when required by referral or deferral.

RE-ASSESSMENT NOTES
An original assessment that is based on both examination and coursework, tests, etc., is considered as a single element for the purpose of referral; i.e., the referred mark is based on the referred examination only, discounting all previous marks. In the event that the mark for a referred assessment is lower than that of the original assessment, the original higher mark will be retained.

Physics Modules with PHY Codes
Referred examinations will only be available in PHY3064, PHYM004 and those other modules for which the original assessment includes an examination component - this information is given in individual module descriptors.
RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener
ELE:

Type Author Title Edition Publisher Year ISBN
Set Gregory, R.D. Classical Mechanics Cambridge University Press 2006 0-521-534097
Extended Calkin, M.G. Lagrangian and Hamiltonian Mechanics World Scientific 1996 981-02-2672-1
Extended Englert, B. G. Lectures on Classical Mechanics World Scientific 2015 978-981-4678-44-5
Extended Fowles, G.R. and G. L. Cassiday Analytical Mechanics 7th edition Brooks Cole 2004 0-534-49492-7
Extended Hand, L. N. and J. D. Finch Analytical Mechanics Cambridge University Press 1999 0-521-57572-9
Extended Kaplan, D. and L. Glass Understanding Nonlinear Dynamics Springer-Verlag 1995 0-387-94440-0
Extended Kibble, T.W.B. Classical Mechanics 3rd edition Longman 1985 0-582-45023-3
Extended Landau, L. D. and E. M. Lifshitz Mechanics, Vol. 1 3rd edition Butterworth-Heinemann 1976 978-0-750-62896-9
Extended Leech, J.W. Classical Mechanics Methuen and Co. 1965 0-412-20070-8
Extended Symon, K.R. Mechanics 3rd edition Addison-Wesley 1971 0-201-07392-7
Extended ter Haar, D. Elements of Hamiltonian Mechanics Pergamon Press 1971
Extended Thornton, S.T. and J. B. Marion Classical Dynamics of Particles and Systems 5th edition Tomson 2003 0-534-40896-6
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES PHY1021, PHY1026
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 5 No Monday 6th December 2021 Thursday 26th January 2023
KEY WORDS SEARCH Physics; Equations; Hamilton; Brackets; Dynamical; Poisson; Formulations; Chaos; Hamiltonian; Jacobi; Mechanics.