Principle of Least Action, Lagrange's Equations of Mechanics | Calculus of Variations | Lecture 6

Lecture 6, course on Hamiltonian and nonlinear dynamics. Variational principles of mechanics, namely the Principle of Least Action (also called principle of stationary action, principle of critical action, and Hamilton's principle), which is a global principle equivalent to the local principle of Lagrange's equations. We first give a handwavy physicsy way to derive to get Newton's equations, then consider some techniques in the calculus of variations, getting Euler-Lagrange equations.

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

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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.

Chapters
0:00 Canonical transformations come from generating functions via variational principles
2:21 Principal of least action
8:21 Initial approach to understanding how principle of least action leads to Newton's equations
30:32 Euler-Lagrange equations: More general, calculus of variations approach to principle of critical action, leading to Euler-Lagrange equations (Lagrange's equations in mechanics context)
49:00 Euler-Lagrange equations, example uses
49:59 Brachistochrone problem
54:29 Cubic spline curves (data fitting)

► Interactive action integral calculator (courtesy Cleon Teunissen)
http://cleonis.nl/physics/phys256/ene...

► 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
https://is.gd/AnalyticalDynamics

► Center manifolds, normal forms, and bifurcations
https://is.gd/CenterManifolds

► Introductory course on Nonlinear Dynamics & 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

Video clip from ‪@Vsauce‬'s The Brachistochrone
   • The Brachistochrone  

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

Lecture 2021-07-06

action angle in classical mechanics physics quasiperiodic online course Hamilton Hamilton's stationary action critical action Hamilton-Jacobi theory three-body problem orbital mechanics Symplectic Geometry topology

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