### 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

Students will posses a solid understanding of the fundamentals of variational calculus

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

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

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

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

### 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

< div xmlns="http://www.w3.org/1999/xhtml">5. Constraints
5.1 Holonomic constraints
5.2 Lagrange problems
5.3 Problems with variable endpoints
5.4 Constrained 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

Explanation of Reading List:

B. Dacorogna, Introduction to the Calculus of Variations, Imperial College Press, 2004.
I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, 1963.
B. van Brunt, The Calculus of Variations, Spinger, 2004.

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

M201/224; MATH101; MATH102; MATH103 Some knowledge of MATH225 would be useful, but not essential.

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

G101 (3,4); F344 (3,4); FGH1 (3,4); MMAS (1)

### Assessment

EXAM Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Unseen Written Exam  150  Semester One  90  Yes  Standard UoL penalty applies  Written Exam
CONTINUOUS Duration Timing
(Semester)
% of
final
mark
Resit/resubmission
opportunity
Penalty for late
submission
Notes
Coursework  Continuous  Semester One  10  Yes  Standard UoL penalty applies  Regular assignments distributed throughout the semester Notes (applying to all assessments) Full marks will be awarded for complete answers to FOUR questions. Only the best FOUR answers will be taken into account.