Aims

The aim of this course is to outline, through a number of concrete examples, certain mathematical structures which are essential in physics, and to exploit them systematically in these examples. The topics covered are restricted to linear theories and to classical physics.

Main themes

. Mathematical tools : vectorial and differential geometry, Fourier series, Fourier integrals, Lie groups and Lie algebras (basic notions), tensor calculus
. Special Relativity
. Classical field theory and conservation laws
. Theory of the electromagnetic field, including the covariant formalism
. Linear partial differential equations of classical physics: derivation, classification, solution
. Rotation group SO(3), Lie algebra, representations

Content and teaching methods

Part I : Symmetries and Invariance in Relativistic Mechanics
1. Classical mechanics, a reminder
2. Theory of Special Relativity
Part II : Field Theory and Partial Differential Equations
3. Notions of vectorial and differential geometry
4. Field theory and conservation laws
5. Theory of the electromagnetic field
6. Linear partial differential equations of classical physics
Part III: Solution of Linear Partial Differential Equations
7. Fourier series and Fourier integrals
8. Classification of the equations. Uniqueness of solutions
9. Solution of the equations
Part IV: Group Theory and Applications
10. Lie groups and Lie algebras. Representations. Tensors
11. Representations of the groups SO(3) and SU(2)

Other credits in programs