Topics in Nonlinear Dynamics

Overview

Mechanics of a system of particles in vector form. Conservation of linear momentum, energy and angular momentum - Degrees of freedom, generalized coordinates and velocities. Lagrangian, action, and the principle of extremal action. Euler-Lagrange equations. Constraints. Applications of the Lagrangian formalism - Legendre transform, generalized momenta. Hamiltonian and Hamilton's equations of motion. Phase space, phase trajectories. Applications to systems with a small number of degrees of freedom - Scattering in a central potential, Rutherford formula, scattering cross section - Charged particle in an electromagnetic field - Non-inertial frames of reference and pseudo-forces: centrifugal, Coriolis and Euler forces - Rotations in three dimensions, Euler angles. Angular momentum. Elements of rigid-body dynamics. The symmetric top - Small oscillations. Normal mode analysis. Normal modes of simple molecules and harmonic chains - Elementary ideas on general dynamical systems: conservative versus dissipative systems. Hamiltonian systems and Liouville's theorem - Constants of the motion and symmetries in classical mechanics. Noether's Theorem and the connection between symmetry, invariance and conservation principles - Canonical transformations, Poisson brackets. Action-angle variables. Integrable Hamiltonians. Non-integrable systems and elements of chaotic motion - Special relativity: Inertial frames. Principle and postulate of Relativity. Lorentz transformations. Length contraction, time dilation and the Doppler effect. Velocity addition formula. Rapidity - Four-vectors and tensors. Energy-momentum of a free particle. Relativistic invariance of physical laws. The wave equation. Transformation properties of electric and magnetic fields. The homogeneous and inhomogeneous Lorentz groups