Nonhyperbolic Equilibria, Hamiltonian & Gradient Systems, Lyapunov Function, Ideal & Damped Pendulum

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Bill Kinney's Differential Equations and Linear Algebra Course, Lecture 33A.

(a.k.a. Differential Equations with Linear Algebra, Lecture 33A, a.k.a. Continuous and Discrete Dynamical Systems, Lecture 33A).

#hamiltoniansystems #lyapunovfunction #pendulum

(0:00) Lecture plans
(1:20) What can happen at non-hyperbolic equilibrium points?
(1:58) One-parameter family of nonlinear (cubic) ODEs where the polar coordinate “r” is relevant
(4:01) Use Mathematica to see what happens
(9:00) Transformed differential equation for dr/dt (differential equations in the polar coordinate r)
(12:45) Transformed differential equation for dθ/dt (differential equations in the polar coordinate θ)
(15:53) General Hamiltonian systems, Hamiltonian functions, and their properties
(18:16) General gradient systems, potential functions, and their properties
(21:20) Lyapunov function for a gradient system is the negative of the potential function
(22:36) What does it mean to be a Lyapunov function?
(25:23) Lyapunov functions and trapping regions for gradient systems (Lyapunov functions can be used to define trapping regions)
(28:34) Theorem about closed and bounded trapping region (there is at least one solution curve inside the trapping region for all time t)
(32:51) Ideal Pendulum (Undamped) Nonlinear and Unforced ODE
(36:37) This is a Hamiltonian system with Hamiltonian function representing total mechanical energy
(39:25) Physical interpretation of solutions (including separatrices)
(40:54) Damped Pendulum (include friction)
(46:33) Visualize the pendulum motion on Mathematica

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Professor Kinney thank you for taking the time to summarize the Hamiltonian Systems, Gradient Systems and the Potential Functions and their contributions to Ordinary Differential Equations. The classical pendulum which is described mathematically as a Nonlinear/Linear Differential Equation is an excellent example of a Lyapunov Functions and Hamiltonian Systems for the damped case. These topics are not covered in Introductory Differential Equations.

georgesadler