Legendre Transformation | Get Hamiltonian from Lagrangian | Spring Mass, Harmonic Oscillator, Lect 2

Lecture 2 of a course on Hamiltonian and nonlinear dynamics. The Legendre transformation is a general mathematical technique for transforming variables of a scalar function of multiple variables, using the partial derivatives as the new variables. We give a geometric interpretation. Then we use the Legendre transformation to derive Hamilton's canonical equations. Finally we consider a 1 degree of freedom system, the spring-mass example, a version of the harmonic oscillator.

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► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
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► A shorter, gentler introduction to Hamiltonian systems in 2D
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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.

► The entire class notes are in PDF form:
https://is.gd/AdvancedDynamicsNotes

► and 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

► If you want a simple introductory course on Nonlinear Dynamics and Chaos, see:
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 2021-06-22

action angle variables in classical mechanics quantum mechanics statistical physics thermal physics thermodynamics general relativity Jerrold Marsden Gibbs free energy quasiperiodic online course principle of least action cosmology universe quarks William Rowan Hamilton Hamilton-Jacobi theory three-body problem orbital mechanics incompressibility integral invariants of Poincare streamfunction fluids

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