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

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:

► in OneNote form:

►This course builds on prior knowledge of Lagrangian systems, which have their own lecture series, 'Analytical Dynamics'

► Continuation of this course on a related topic
Center manifolds, normal forms, and bifurcations

► A simple introductory course on Nonlinear Dynamics and Chaos:

► 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

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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[goli]

I am enjoying the course a lot. I just want to make a comment, the equivalence between MJM^T=J and M^T J M=J does not come only from the fact that the set of canonical transformations is a group, it also uses the fact that J is invertible. If one considers then the property MJM^T=J defines a group called presymplectic group and for this group of transformations these two properties are not equivalent. The property M^T J M=J is actually what is required for \phi to be Poisson. Consider the Poisson structure J^{-1}=-J.

[JohnVKaravitis]

Goldstein notes in his textbook "Classical Mechanics" (sec. 8.1) that Considerable ingenuity has been exercised in devising nomenclature schemes that result in entirely symmetric equations, or combine the two sets into one. Most of these schemes have only curiosity value, but one has proved to be an elegant and powerful tool for manipulating the canonical equations and allied expressions." He's referring to the symplectic method. BY ANY CHANCE, do you know what other "schemes" have been attempted?

[brendawilliams]

I really enjoyed your lecture. Thankyou

[Eta_Carinae__]

Forgive me, since I'm going through these one by one, if you cover it later, but I expanded the condition for a coordinate transform. Isn't this the same as anticommutivity? As in like the Poisson bracket {Q, P} = 1?