Mathematical Methods of Classical Mechanics

This is a second semester course for M1 of the Master Program in Mathematical Physics at the IMB. The lectures will be in English.

Description

This course focuses on the study of classical mechanics from a modern mathematical point of view.

Outline

1. Introduction. Newtonian mechanics.
2. Lagrangian and Hamiltonian mechanics in $\R^n$.
3. Oscillations.
4. Rigid bodies.
5. Lagrangian and Hamiltonian mechanics on manifolds.
6. Symplectic and Poisson manifolds.

Course material

The course material will be posted on the Teams group of the course.

Lectures

• Basic experimental principles of classical mechanics
• Galilean space-time and Galilean transformations
• One-dimensional systems, phase diagrams
• System of particles interacting in space, motion of the center of mass, conservation laws

• Euler-Lagrange equations for a system of particles in a conservative field
• Invariance of Lagrange equations under lift of coordinate transformations
• Lagrangians dependent on velocities, system of particles in the electomagnetic field
• Calculus of variations, Hamilton's principle
• Euler-Lagrange equations for systems with holonomic constraints

• D'Alembert principle
• Conditional variations and holonomic constraints
• Noether theorem

• $SO(3)$ invariance and conservation of angular momentum
• Hamilton equations
• Legendre transform
• Similarity

• Equivalence of Euler-Lagrange and Hamiltonian equations
• Poisson brackets
• Variational principle for Hamilton equations
• Cyclic coordinates

• Symplectic and Hamiltonian matrices
• Hamiltonian vector fields
• Generating functions for canonical transformations

• Equivalent definitions of canonical transformations
• Canonical transformations preserve the canonical structure of Hamilton equations
• Point transformations are canonical
• Generating functions give canonical transformations

• Lie condition
• Hamiltonian flow is canonical
• Hamilton-Jacobi equation

• Hamilton-Jacobi, examples.
• Separation of variables
• Lagrangian mechanics on the tangent bundle
• Euler-Lagrange vector field
• Variational formulation
• Fibre derivative

• Energy functional and Hamiltonian function
• Poisson manifolds and Hamiltonian vector fields
• Canonical transformations
• Symplectic structure on the cotangent bundle
• Tautological $1$-form
• Symplectic manifolds and symplectomorphisms
• Relation between symplectic and Poisson manifolds
• Liouville theorem

• Poincaré recurrence
• Darboux theorem
• Poisson structure on the dual of a Lie algebra
• Completely integrable systems
• Liouville-Arnold theorem
• Action-angle variables