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MAS8809 : Variational Methods and Lagrangian Dynamics

  • Offered for Year: 2023/24
  • Module Leader(s): Professor Ian Moss
  • Owning School: Mathematics, Statistics and Physics
  • Teaching Location: Newcastle City Campus
Semester 2 Credit Value: 10
ECTS Credits: 5.0


To present basic ideas and techniques of variational calculus, including relevant applications.

Module Summary

What is the shortest route between two places on the Earth's surface (the answer isn't a straight line, because the Earth isn't flat!)? How to run a utilitarian economy? How do you reformulate dynamical motion in terms of geometry? To answer questions such as these, we need a way to find the path between two points which minimises some quantity (such as length or time). The calculus of variations is a very elegant and powerful way of doing this, and consequently has wide application to real-world problems. The ideas provide the basis for a reformulation of dynamics which underpins modern theoretical physics.

Outline Of Syllabus

Review of standard methods for finding extrema. Definition of, and method for calculating, extremals
(minima/maxima) of functionals. The Euler-Lagrange equation. Classical examples from different disciplines.
Lagrange multipliers. Many dependent variables. The action principle, Lagrangian
and Hamiltonian dynamics with applications to astro- and particle physics.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture51:005:00Problem Classes
Scheduled Learning And Teaching ActivitiesLecture21:002:00Revision Lectures
Scheduled Learning And Teaching ActivitiesLecture201:0020:00Formal Lectures
Guided Independent StudyAssessment preparation and completion151:0015:00Completion of in course assessments
Guided Independent StudyIndependent study581:0058:00Preparation time for lectures, background reading, coursework review
Jointly Taught With
Code Title
PHY3029Variational Methods and Lagrangian Dynamics
MAS3809Variational Methods and Lagrangian Dynamics
Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Description Length Semester When Set Percentage Comment
Written Examination1202A80N/A
Exam Pairings
Module Code Module Title Semester Comment
PHY3029Variational Methods and Lagrangian Dynamics2N/A
MAS3809Variational Methods and Lagrangian Dynamics2N/A
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises2M5Problem-solving exercises assessment
Prob solv exercises2M5Problem-solving exercises assessment
Prob solv exercises2M5Problem-solving exercises assessment
Prob solv exercises2M5Problem-solving exercises assessment
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

Reading Lists