Advanced Topics in Mathematical Physics 2

The course aims to address some topics related to partial differential equations in mathematical physics. It will begin by presenting some results on the Schroedinger equation, the equation that constitutes the paradigm of quantum mechanics. In particular, it will be shown how it leads to the quantization of the energy of the harmonic oscillator and its relevance to the study of light. In the second part, the Hamiltonian formalism for partial differential equations will be introduced, starting with the wave equation. Subsequently, the Euler equations of fluids will be introduced, and the equations for surface waves in fluids will be derived. In the final part of the course, the Hamiltonian structure for the equations of surface waves in fluids will be introduced. This structure has played a fundamental role in modern studies of water waves.

Schroedinger equation.

1.1 Crisis of classical mechanics and introduction of the Schroedinger equation as a paradigm of quantum mechanics.

1.2 The eigenvalue problem and its physical interpretation.

1.3 Quantum harmonic oscillator: quantization of energy.

Hamiltonian formalism for partial differential equations: the example of the vibrating string equation (wave equation). Lagrangian formulation and transition to the Hamiltonian formulation.
Equations of fluid dynamics.

3.1 Derivation of the Euler equations of fluids from fundamental principles.

3.2 Sound propagation equation.

3.3 Bernoulli principle, surface wave equation. Wave propagation, tsunamis.

Hamiltonian structure of the surface wave problem.

4.1 Lagrangian of surface waves.

4.2 Transition to the Hamiltonian formalism and Dirichlet-Neumann operator.

4.3 Computation of Hamilton's equations.

4.4 Long-wave approximation and effective equations.