Generating Function of a Canonical Transformation | Examples and the Big Picture | Lecture 7

Lecture 7, course on Hamiltonian and nonlinear dynamics. Canonical transformations are a category of change of variables which are central to study of Hamiltonian systems. We discuss the approach of constructing canonical transformations from generating functions, which is related to gauge invariance in the action integral (the Lagrangian function is unique only up to a total derivative of the variables). Examples including for the harmonic oscillator and near-identity transformations related to the Hamiltonian flow map (solution map) are given. ► Next: Hamiltonian Flow is a Canonical Transformation | Strange Non-Intuitive Momentum Definitions    • Hamiltonian Flow is a Canonical Transforma...   ► Previous, Principle of least action and Lagrange's equations of mechanics | basics of calculus of variations    • Principle of Least Action & Lagrange's Equ...   ► Dr. Shane Ross, Virginia Tech professor (Caltech PhD) Instructor intro    • Professor Shane Ross Introduction   ► New lectures posted regularly Subscribe https://is.gd/RossLabSubscribe​ ► Follow me on Twitter   / rossdynamicslab   Chapters 0:00 Summary so far 0:49 Hamilton's canonical equations from the principal of least action 7:28 Generating function approach to canonical transformations 27:17 Harmonic oscillator example 33:31 Aside: photon energy and momentum looks like harmonic oscillator in quantum mechanics 34:28 Different kinds of generating functions 42:51 Near-identity transformations and flow map of Hamilton's equations 52:17 Summary / big picture of canonical transformations ► Class notes in PDF form https://is.gd/AdvancedDynamicsNotes ► in OneNote form https://1drv.ms/u/s!ApKh50Sn6rEDiRgCY... ► See the entire playlist for this online course: Advanced Dynamics - Hamiltonian Systems and Nonlinear Dynamics https://is.gd/AdvancedDynamics This course gives the student advanced theoretical and semi-analytical tools for analysis of dynamical systems, particularly mechanical systems (e.g., particles, rigid bodies, continuum systems). We discuss methods for writing equations of motion and the mathematical structure they represent at a more sophisticated level than previous engineering dynamics courses. We consider the sets of possible motion of mechanical systems (trajectories in phase space), which leads to topics of Hamiltonian systems (canonical and non-canonical), nonlinear dynamics, periodic & quasi-periodic orbits, driven nonlinear oscillators, resonance, stability / instability, invariant manifolds, energy surfaces, chaos, Poisson brackets, basins of attraction, etc. ► This course builds on prior knowledge of Lagrangian systems, which have their own lecture series, 'Analytical Dynamics' https://is.gd/AnalyticalDynamics ► Continuation of this course on a related topic Center manifolds, normal forms, and bifurcations https://is.gd/CenterManifolds ► A simple introductory course on Nonlinear Dynamics and Chaos https://is.gd/NonlinearDynamics ► References The class will largely be based on the instructor’s notes. In addition, references are: A Student’s Guide to Lagrangians and Hamiltonians by Hamill Numerical Hamiltonian Problems by Sanz-Serna & Calvo Analytical Dynamics by Hand & Finch Classical Mechanics with Calculus of Variations & Optimal Control: An Intuitive Introduction by Levi Ross Dynamics Lab: http://chaotician.com​ Lecture 2021-07-08 #Hamiltonian #CanonicalTransformation #GeneratingFunction #EulerLagrange #PrincipleLeastAction #LeastAction #Brachistochrone #HamiltonsPrinciple #CanonicalTransformation #CyclicCoordinates