Diff Eqs Lec #30, Hyperbolic Equilibria, Hamiltonian Systems, Pendulum

Differential Equations, Lecture 30. (0:00) Exam 3 is graded. (0:30) Project directions. (6:53) Extra reading assignment. (7:23) Hyperbolic versus nonhyperbolic equilibrium points. Definition. Relationship to Hartman-Grobman Theorem (when Linearization is helpful). (12:48) The unforced, undamped harmonic oscillator as a Hamiltonian System. (15:20) Conservation of mechanical energy (potential plus kinetic), verified "along a solution curve" with a multivariable version of the chain rule. (23:34) Confirm with DSolveValue on Mathematica and then simplification of the formula for the energy along an arbitrary solution curve. (26:40) Elaboration on why its a Hamiltonian system (find a Hamiltonian function), and a discussion of the form of 2-dimensional Hamiltonian systems in general. (31:41) Confirmation that the Hamiltonian function is a conserved quantity in general, and therefore solution curves of the system must lie on top of the level curves of the Hamiltonian function. (34:46) The (nonlinear) undamped and unforced pendulum. (43:46) Use Mathematica (ultimately NDSolve) to graph solution curves in the phase plane (for the nonlinear pendulum). (49:22) Find Hamiltonian function (mistake made, but noted on video).

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