Hamiltonian Systems Introduction- Why Study Them? | Lecture 1 of a Course on Hamilton's Equations

Lecture 1 of a course on Hamiltonian and nonlinear dynamics. The Hamiltonian formalism is introduced, one of the two great pillars of mechanics, along with the Lagrangian formalism. “What’s the point? Why study Hamiltonian systems?” We answer that, discussing the advantages of the Hamiltonian point of view, and introduce the idea of getting a Hamiltonian system from a Lagrangian system via the Legendre transformation.

► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Introduction to instructor    • Professor Shane Ross Introduction  

► Next, Legendre transformation | going from Lagrangian to Hamiltonian | spring mass worked example
   • Legendre Transformation | Get Hamilto...  

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► A shorter introduction to Hamiltonian systems in 2D
   • Hamiltonian Systems - Nonlinear Syste...  

► 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 & 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.

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

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

►This course builds on prior knowledge of Lagrangian systems
https://is.gd/AnalyticalDynamics

► Continuation of this course:
Center manifolds, normal forms, & bifurcations
https://is.gd/CenterManifolds

► Want a simple introductory course on Nonlinear Dynamics & Chaos?
https://is.gd/NonlinearDynamics

► Courses & Playlists by Dr. Ross

📚Kalman Filter Mini-Course
http://tinyurl.com/kalmanfilters

📚Nonlinear Dynamics & Chaos
https://is.gd/NonlinearDynamics

📚Attitude Dynamics & Control
https://is.gd/SpaceVehicleDynamics

📚Lagrangian & 3D Rigid Body Dynamics
https://is.gd/AnalyticalDynamics

📚Center Manifolds, Normal Forms, & Bifurcations
https://is.gd/CenterManifolds

📚3-Body Problem Orbital Dynamics
https://is.gd/3BodyProblem

📚Space Manifolds
https://is.gd/SpaceManifolds

► Chapters
0:00 Lagrangian and Hamiltonian formalism of mechanics compared
15:14 Advantages of the Hamiltonian formalism
27:48 Hamilton's equations from Lagrange's equations
37:18 Generalized momentum
44:03 Hamiltonian function definition
46:51 Hamilton's canonical equations and advantages
1:00:29 Hamilton's canonical equations do not permit attractors

► References
The class will largely be based on instructor’s notes.
A PDF book available for FREE at
http://shaneross.com/books
Koon, Lo, Marsden, Ross (2011) Dynamical Systems, the Three-Body Problem and Space Mission Design. ISBN 978-0-615-24095-4.

In addition, references are:
Advanced Dynamics by Greenwood
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
Classical Dynamics: A Contemporary Approach by José & Saletan
Classical Mechanics, 3rd Edition by Goldstein, Poole, & Safko

Additional texts that may be useful:
Nonlinear Differential Equations and Dynamical Systems by Verhulst
Introduction to Applied Nonlinear Dynamical Systems and 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 2021-06-17

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