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.

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► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
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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

action angle cyclic variables in classical mechanics statistical physics quasiperiodic online course Hamilton Hamilton-Jacobi theory three-body problem orbital mechanics Symplectic Geometry topology

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