Canonical Transformations | Hamiltonian Changes of Variables | Symplectomorphisms | Lecture 5

Lecture 5, course on Hamiltonian and nonlinear dynamics. Change of variables in Hamiltonian systems, called symplectic transformations or canonical transformations. The only allowed transformations are those that preserve the form of Hamilton's equations. Thus, the condition is that the equations of motion in the new variables must still be Hamilton's canonical equations.

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

► Class notes in PDF form:
https://is.gd/AdvancedDynamicsNotes

► in OneNote form:
https://1drv.ms/u/s!ApKh50Sn6rEDiRgCY...

►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:
Numerical Hamiltonian Problems by Sanz-Serna & Calvo
Analytical Dynamics by Hand & Finch
A Student’s Guide to Lagrangians and Hamiltonians by Hamill
Classical Mechanics with Calculus of Variations & Optimal Control: An Intuitive Introduction by Levi
Advanced Dynamics by Greenwood

Additional math texts that may also be useful are:
Nonlinear Differential Equations & Dynamical Systems by Verhulst
Introduction to Applied Nonlinear Dynamical Systems & Chaos by Wiggins
Differential Equations, Dynamical Systems, & Linear Algebra by Hirsch & Smale
Introduction to Mechanics & Symmetry by Marsden & Ratiu

Ross Dynamics Lab: http://chaotician.com​

Lecture 2021-07-01

action angle cyclic variables in classical mechanics statistical physics thermal physics thermodynamics general relativity quasiperiodic online course Hamilton Hamilton-Jacobi theory three-body problem orbital mechanics incompressibility integral invariants of Poincare and Cartan Symplectic Geometry topology

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