PHY2032 
Analytical and Chaotic Dynamics 
202324 

Prof. J. Bertolotti 


Delivery Weeks: 
T1:0111 

Level: 
5 (NQF) 

Credits: 
15 NICATS / 7.5 ECTS 

Enrolment: 
33 students (approx) 

Description
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 HamiltonJacobi 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).
Module Aims
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)
A student who has passed this module should be able to:

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

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

Personal and Key Transferable / Employment Skills and Knowledge:
 use mathematics to solve problems.
Syllabus Plan

Generalized coordinates. Holonomic and nonholonomic constraints

Virtual displacement. D'Alembert's principle

The Lagrangian formulation

The Hamiltonian formulation
 Configuration space; generalized (canonical or
conjugate) momentum
 Phase space
 Legendre transformation
 Hamiltonian; Hamilton's equations
 Cyclic coordinates and conservation theorems
 Liouville's theorem

Calculus of variations

Poisson brackets
 Lagrange brackets
 Poisson brackets

HamiltonJacobi equations and actionangle variables

The transition to quantum mechanics

Nonlinear Dynamical Systems
 Chaos and its relevance to mechanics
 The stability of nonlinear equations
 The nonlinear oscillator
 PhaseSpace Methods
 The pendulum revisited
 Mappings
 Characterisation of chaotic systems
Learning and Teaching
Learning Activities and Teaching Methods
Description 
Study time 
KIS type 
20×1hour lectures 
20 hours

SLT 
2×1hour problems/revision classes 
2 hours

SLT 
5×6hour selfstudy packages 
30 hours

GIS 
4×4hour problems sets 
16 hours

GIS 
Reading, private study and revision 
82 hours

GIS 
Assessment
Weight 
Form 
Size 
When 
ILOS assessed 
Feedback 
0% 
Guided selfstudy 
5×6hour packages 
Fortnightly 
17 
Discussion in class 
0% 
4 × Problems sets 
4 hours per set 
Fortnightly 
17 
Solutions discussed in problems classes. 
100% 
Final Examination 
120 minutes 
January assessment period 
17 
Mark via MyExeter, collective feedback via ELE and solutions. 
Resources
The following list is offered as an indication of the type & level of information that
students are expected to consult. Further guidance will be provided by the Module Instructor(s).
Core text:
Supplementary texts:

Calkin M.G. (1996), Lagrangian and Hamiltonian Mechanics, World Scientific, ISBN 9810226721

Englert B.G. (2015), Lectures on Classical Mechanics, World Scientific, ISBN 9789814678445

Fowles G.R. and Cassiday G.L. (2004), Analytical Mechanics (7^{th} edition), Brooks Cole, ISBN 0534494927

Hand L.N. and Finch J.D. (1999), Analytical Mechanics, Cambridge University Press, ISBN 0521575729

Kaplan D. and Glass L. (1995), Understanding Nonlinear Dynamics, SpringerVerlag, ISBN 0387944400

Kibble T.W.B. (1985), Classical Mechanics (3^{rd} edition), Longman, ISBN 0582450233

Landau L.D. and Lifshitz E.M. (1976), Mechanics (Vol. 1) (3^{rd} edition), ButterworthHeinemann, ISBN 9780750628969

Leech J.W. (1965), Classical Mechanics, Methuen and Co., ISBN 0412200708

Symon K.R. (1971), Mechanics (3^{rd} edition), Addison Wesley, ISBN 0201073927

ter Haar D. (1971), Elements of Hamiltonian Mechanics, Pergamon Press

Thornton S.T. and Marion J.B. (2003), Classical Dynamics of Particles and Systems (5^{th} edition), Tomson, ISBN 0534408966
ELE:
Further Information
Prior Knowledge Requirements
Prerequisite Modules 
Vector Mechanics (PHY1021) and Mathematics for Physicists (PHY1026) 
Corequisite Modules 
none 
Reassessment
Reassessment is not available except when required by referral or deferral.
Original form of assessment 
Form of reassessment 
ILOs reassessed 
Time scale for reassessment 
Whole module 
Written examination (100%) 
17 
August/September assessment period 
Notes: See Physics Assessment Conventions.
KIS Data Summary
Learning activities and teaching methods 
SLT  scheduled learning & teaching activities 
22 hrs 
GIS  guided independent study 
128 hrs 
PLS  placement/study abroad 
0 hrs 
Total 
150 hrs 


Summative assessment 
Coursework 
0% 
Written exams 
100% 
Practical exams 
0% 
Total 
100% 

Miscellaneous
IoP Accreditation Checklist 
 Not applicable, this is an optional module.

Availability 
unrestricted 
Distance learning 
NO 
Keywords 
Physics; Equations; Hamilton; Brackets; Dynamical; Poisson; Formulations; Chaos; Hamiltonian; Jacobi; Mechanics. 
Created 
01Feb12 
Revised 
06Aug20 