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 2017-18 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 mechanics and an appreciation of how symmetries relate to conservation laws

Syllabus
1 

1.  Classical problems.
We will formulate and solve some foundational problems (e.g. The Catenary Problem, The Brachystochrone, Dido''s Problem) in order to motivate the development of Variational Calculus.
We will also introduce and examine the concepts of Geodesics and minimal surfaces.

2. The first variation.
Formal introduction of the first v ariation.
Euler-Lagrange Equation.
Functionals of several variables & higher order derivatives.
Degenerate cases.
Existence of solutions (Time permitting).

3. Isoperimetric problems
Single and multiple Constraints.
Lag range multiplier.

4. Eigenvalues problems.
Sturm-Liouvelle problems (Time permitting).

5. Constraints.
Holonomic constraints.
Lagrange problems (Time permitting).
Problems with variable endpoints.

6. Hamiltonian formulation.
The Legendre Transformation.
Hamilton''s equations.
Hamilton-Jacobi equation & methods of solution.

7. Conservation laws
Variational symmetries.
Noether''s theorem.

8. The second variation.
Conjugate points.
The Legendre & Jacobi conditions.
Convexity.


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

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:
G101 (3,4); G1F7 (3); 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  100  Yes  Standard UoL penalty applies  Written Exam 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.  
CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes