Lecture 8, course on Hamiltonian and nonlinear dynamics. Symplecticness of vector field and flow map for Hamilton's canonical equations of motion. Infinitesimally symplectic matrix. Example systems, showing that their flow map satisfies the condition of a canonical transformation. Strange momentum definitions, that is, generalized momentum vs mechanical momentum. Examples: charged particle in electromagnetic fields, motion viewed in a rotating frame: circular restricted 3-body problem; and a particle in linear restoring force and nonlinear term.
► Next: Hamilton-Jacobi Theory | Finding the Best Canonical Transformation | Examples
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► Previous, Generating Function of a Canonical Transformation | Examples and the Big Picture
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► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Instructor intro [ Ссылка ]
► New lectures posted regularly
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Chapters
0:00 Strange momentum definitions
22:00 Flow of Hamilton's Equations is a Canonical Transformation
30:48 Conditions on eigenspectrum for equilibrium points
45:27 Example Hamiltonian flow maps
► Class notes in PDF form
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► in OneNote form
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► See the entire playlist for this online course:
Advanced Dynamics - Hamiltonian Systems and Nonlinear Dynamics
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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'
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► Continuation of this course on a related topic
Center manifolds, normal forms, and bifurcations
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► A simple introductory course on Nonlinear Dynamics and Chaos
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► 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: [ Ссылка ]
Lecture 2021-07-13
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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