Course 
MMAT5260 
Title 
Calculus of Variations 
Units 
3 
Lecturer 
Dr. Liu Chun Lung Kelvin
Office : 
Rm 202A, LSB 
Office number : 
3943 7969 
Email : 
clliu@math.cuhk.edu.hk 

Teaching assistant 
Fung Tsz Hin Antony
Office : 
Rm 232, LSB 
Office number : 
3943 5294 
Email : 
thfung@math.cuhk.edu.hk 

Prerequisite 
The prerequisite of this course is calculus and advanced calculus. Specifically, differentiation, Taylor’s expansion, integration, partial differentiation, Lagrange multipliers and some basic techniques of solving ordinary differential equations are required. 
Course Description 
In calculus of variations, extremal problems for integral functionals on spaces of functions are studied. Motivated by applications in geometry and physics, this branch of analysis was cultivated by the old masters including Fermat, Newton, Euler and Lagrange and it has been studied intensively ever since. In calculus, differentiation theory is applied to study extremal problems in the Euclidean space which is of finite dimension. In calculus of variations, a parallel study on infinite dimensional space–the space of functions–will be carried out. 
Textbook (Lecture Notes) 
The Calculus of Variations, by Prof. KaiSeng Chou, 2014 
Major Topics 
 The EulerLagrange equation
 Generalizations of the EulerLagrange equation
 Isoperimetric problems
 Generalizations of the isoperimetric problems
 Variable endpoints: general variation formulas
 The Legendre condition

Additional Topics 
If time permits, we will also cover some of the following topics :
 Holonomic constraints
 Nonholonomic constraints
 Jacobi's necessary condition
 The Hamiltonian formulation
 Noether's Theorem

Reading 
Reference books are :
 Variational Calculus and Optimal Control, 2nd edition by John L. Troutman, Springer, New York, 1995.
 The Calculus of Variations, by Bruce van Brunt, Springer, New York 2006.
 Calculus of Variations, Dover edition, by I. M. Gelfand and S. V. Fomin, Dover, New York 2000.
 Introduction to Real Analysis, 3rd edition by Robert G. Bartle and Donald R. Sherbert, JohnWiley and Sons, Inc., New York 2000.
 Advanced Calculus: A Course in Mathematical Analysis, 2nd edition, by P. Fitzpatrick, PWS Publishing Co. 1996.
 Differential Equations : Matrices and Models, by Paul Bugl, Prentice Hall, ISBN : 0133209954

Assessment 
Homeworks 
20% 
Midterm 
30% 
Final 
50% 

Homeworks 
Homeworks will be posted on the course homework page. 
Examinations 
There will be one midterm exam and one final exam given during the classes; please see the calendar page for the dates of the exams. No notes (or books) will be allowed during the exams. 
Honesty in Academic Work 
The Chinese University of Hong Kong places very high importance on honesty in academic work submitted by students, and adopts a policy of zero tolerance on cheating and plagiarism. Any related offence will lead to disciplinary action including termination of studies at the University.
Although cases of cheating or plagiarism are rare at the University, everyone should make himself/herself familiar with the content of the following website :
http://www.cuhk.edu.hk/policy/academichonesty/
and thereby help avoid any practice that would not be acceptable. 