This course aims to provide students with advanced theoretical and semi-analytical tools for analyzing dynamical systems, particularly mechanical systems. The learning outcomes include understanding Hamilton-Jacobi theory for finding the best canonical transformation to solve Hamilton's equations, interpreting solutions in original coordinates mapping to equilibria in Hamilton-Jacobi coordinates, and recognizing S as the action integral. The course teaches skills such as deriving Hamilton-Jacobi equations, solving for Hamilton's principal function, and applying the theory to examples like the simple harmonic oscillator and Kepler's 2-body problem. The teaching method involves lectures with examples and geometric interpretations. The intended audience for this course includes individuals interested in advanced dynamics, Hamiltonian systems, nonlinear dynamics, and mechanical systems analysis.
Hamilton-Jacobi Theory - Finding the Best Canonical Transformation + Examples
Overview
Syllabus
Hamilton-Jacobi theory introduction .
Every point in phase space is an equilibrium point.
Derivation of Hamilton-Jacobi equation.
Example: Hamilton-Jacobi for simple harmonic oscillator.
Simplification: if Hamiltonian is time-independent.
Hamilton's Principal function S is the action integral.
Example: Hamilton-Jacobi for Kepler problem.
Simplification: if Hamiltonian is separable.
Taught by
Ross Dynamics Lab
