Stellar Orbits: Integrals of Motion and Phase Space Visualization

This lecture focuses on the concept of integrals of motion in stellar dynamics, emphasizing their significance in understanding stellar orbits within a six-dimensional phase space. The instructor begins by redefining integrals of motion, explaining that they are functions of phase-space coordinates that remain constant along an orbit. Various examples illustrate how these integrals help simplify complex dynamical systems by reducing the number of independent variables. The discussion then transitions to the introduction of surfaces of section, a powerful visualization tool that aids in analyzing the structure of phase space and identifying regular and chaotic orbits. The instructor demonstrates how to create these surfaces by plotting points that cross a defined plane with positive velocity, leading to insights about the organization of orbits. The lecture concludes with examples of orbits in different potentials, highlighting the emergence of new orbit families and the concept of bifurcation, which occurs at higher energies. Overall, the lecture provides a comprehensive overview of the mathematical tools used to study stellar dynamics.