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

(LO1) Students will possess 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

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 constrai nts
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