Undergraduate Course: Variational Calculus (MATH11179)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 11 (Year 5 Undergraduate)
||Availability||Available to all students
|Summary||NB. This course is delivered *biennially* with the next instance being in 2022-23. It is anticipated that it would then be delivered every other session thereafter.
This is a course on the calculus of variations and explores a number of variational principles, such as Hamilton's Principle of Least Action and Shannon's Principle of Maximum Entropy. The approach taken in this course lies at the interface of two disciplines: Geometry and Mathematical Physics. In Geometry you will learn about geodesics, minimal surfaces, etc. In Physics you will learn to elevate Newton's laws to a mathematical principle and discuss lagrangian and hamiltonian formulations. A running theme will be the relationship between symmetries and conservation laws, as illustrated by a celebrated theorem of Emmy Noether's. We will not assume, however, any background in either Physics or Geometry. All the necessary vocabulary and concepts will be introduced in the course.
- Calculus of variations: Euler-Lagrange equations, general variations
- Newtonian mechanics and conservation laws
- Hamilton's principle of least action
- Noether's theorem
- Hamiltonian formalism
- Isoperimetric problems
- Holonomic and nonholonomic constraints
- Variational PDEs
- Noether's theorem revisited
- Classical field theory
Information for Visiting Students
|Pre-requisites||Visiting students are advised to check that they have studied the material covered in the syllabus of any pre-requisite course listed above before enrolling.
|High Demand Course?
Course Delivery Information
|Not being delivered|
On completion of this course, the student will be able to:
- Derive the Euler-Lagrange equations for variational problems, including the case of general variations.
- Derive conserved quantities from symmetries, and use them to solve the Euler-Lagrange equations.
- Solve variational problems with constraints: both algebraic and isoperimetric.
- Calculate effectively using Poisson brackets.
|Lecture notes will be provided, which contain ample bibliography with other sources.|
|Graduate Attributes and Skills
|Course organiser||Dr Jelle Hartong
|Course secretary||Mr Martin Delaney
Tel: (0131 6)50 6427