Module Details

The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
Title VARIATIONAL CALCULUS AND ITS APPLICATIONS
Code MATH430
Coordinator Dr DJ Colquitt
Mathematical Sciences
D.Colquitt@liverpool.ac.uk
Year CATS Level Semester CATS Value
Session 2018-19 Level 7 FHEQ First Semester 15

Aims

This module provides a comprehensive introduction to the theory of the calculus of variations, providing illuminating applications and examples along the way.


Learning Outcomes

(LO1) Students will posses a solid understanding of the fundamentals of variational calculus

(LO2) Students will be confident in their ability to apply the calculus of variations to range of physical problems

(LO3) Students will also have the ability to solve a wide class of non-physical problems using variational methods

(LO4) Students will develop an understanding of Hamiltonian formalism and have the ability to apply this framework to solve physical and non-physical problems

(LO5) Students will be confident in their ability to analyse variational symmetries and generate the associated conservation laws

(S1) Problem solving skills

(S2) Numeracy


Syllabus

 

1. Some preliminary results in functional analysis 1.1 Revision of some results from classical analysis: Picard's Theorem, Taylor's Theorem, Implicit function theorem, etc. 1.2 Function spaces: Linear spaces, Normed linear spaces, Continuity, C0, C1 2. The first variation 2.1 The fundamental lemma of variational calculus 2.2 Formal introduction of the first variation 2.3 The Euler-Lagrange Equation 2.4 Functionals of several variables & higher order derivatives 2.5 Degenerate cases 3. Isoperimetric problems  3.1 Revision of Lagrange Multipliers in finite dimensional space 3.2 Lagrange Multipliers in infinite dimensional spaces 3.3 Multiple constraints and dependent variables 4. Sturm-Liouville problems 4.1 Sturm-Liuoville problems as constrained isoperimetric problems 4.2 The first eigenvalue 4.3 The Rayleigh quotient and bounds on the first eigenvalue 5. Constraints   5.1 Holonomic constraints 5.2 Lagrange problems 5.3 Problems with variable endpoints 5.4 Co nstrained endpoints & transversality 6. The Hamiltonian formulation 6.1 The Legendre transformation 6.2 Hamilton's equations 6.3 Sympletic transformations 6.4 The Hamilton-Jacobi equation & the method of additive separation 6.5 Hamilton's principle 7. Conservation laws  7.1 Variational symmetries & infinitesimal generators 7.2 A necessary & sufficient condition for the existence of symmetries 7.3 Noether's theorem 8. The second variation 8.1 Formal introduction of the second variation & the definition of local extrema 8.2 The Legendre condition 8.3 The Jacobi accessory equation & conjugate points 8.4 The Jacobi necessary condition 8.5 Weak local extrema 8.6 A sufficient condition for a weak local extremum


Recommended Texts

Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.

Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

 

Co-requisite modules:

 

Modules for which this module is a pre-requisite:

 

Programme(s) (including Year of Study) to which this module is available on a required basis:

 

Programme(s) (including Year of Study) to which this module is available on an optional basis:

 

Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Written Exam There is a resit opportunity. Standard UoL penalty applies for late submission. This is an anonymous assessment. Assessment Schedule (When) :Semester One  150 minutes.    90       
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Regular assignments distributed throughout the semester There is a resit opportunity. Standard UoL penalty applies for late submission. This is not an anonymous assessment. Assessment Schedule   Continuous    10