Integrable & Non-Integrable Hamiltonian Systems, KAM Tori, Poincare Section, Poisson Bracket, Lec 11

► Next: Poisson brackets, non-canonical Hamiltonian systems and Euler's rigid body equations

► Previous, Action-Angle Variables in Hamiltonian Systems | Visualizing Tori and Spheres in N Dimensions

► *Advanced Dynamics - Hamiltonian Systems and Nonlinear Dynamics*

► *Dr. Shane Ross, Virginia Tech professor (Caltech PhD)*

► *Follow me*

► Missed the first lecture, introducing Hamiltonian systems?

► A shorter, gentler introduction to Hamiltonian systems in 2D

► Course lecture notes (PDF)
Lecture 2020-03-26

► Course lecture notes (OneNote):

► Continuation of this course on a related topic
Center manifolds, normal forms, and bifurcations

► Chapters:
0:00 Introduction
0:30 Integrable and Non-Integrable Hamiltonian Systems
22:12 Non-Integrable Hamiltonian Systems
33:46 KAM Theorem and KAM tori
40:19 Poincare section, Poincare map
1:03:23 Poisson brackets and Poisson systems

Part of a graduate level course:
Advanced Dynamics (ESM/AOE 6314)
Spring Semester, 2020

► Courses and Playlists by Dr. Ross

📚Lagrangian and 3D Rigid Body Dynamics

📚Center Manifolds, Normal Forms, and Bifurcations

📚Attitude Dynamics and Control

📚Nonlinear Dynamics and Chaos

📚Hamiltonian Dynamics

📚3-Body Problem Orbital Dynamics

📚Space Manifolds

Charles Conley index theory gradient system autonomous on the plane phase plane are introduced 2D ordinary differential equations 2d ODE vector field topology cylinder bifurcation robustness fragility cusp unfolding perturbations structural stability emergence critical point critical slowing down supercritical bifurcation subcritical bifurcations buckling beam model change of stability nonlinear dynamics dynamical systems differential equations dimensions phase space Poincare Strogatz graphical method Fixed Point Equilibrium Equilibria Stability Stable Point Unstable Point Linear Stability Analysis Vector Field Two-Dimensional 2-dimensional Functions Hamiltonian Hamilton streamlines weather vortex dynamics point vortices pendulum Newton's Second Law Conservation of Energy topology

#Hamiltonian #NonlinearDynamics #DynamicalSystems #Poisson #Kolmogorov #Arnold #Moser #KAM #KAMtorus #Euler #DynamicalSystems #Poincare #PoincareMap #PoincareSection #Torus #geometry #dynamics #ChaoticDynamics #StandardMap #tsunami #transformations #mechanics #DynamicalSystems #NonlinearDynamics #Hamiltonian #dynamics #mechanics #physics #sphere #Nsphere #hypersphere #Tori #torus #topology #QuasiPeriodic #Oscillations #LimitCycles #VectorFields #topology #IndexTheory #EnergyConservation #Hamiltonian #Streamfunction #Streamlines #Vortex #SkewGradient #Gradient #PopulationBiology #FixedPoint #DifferentialEquations #Bifurcation #SaddleNode #Eigenvalues #HyperbolicPoints #NonHyperbolicPoint #CuspBifurcation #CriticalPoint #buckling #PitchforkBifurcation #robust #StructuralStability #DifferentialEquations #dynamics #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Poincare #Strogatz #Wiggins #VectorField #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #NonlinearODEs #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #TwoDimensional #Functions #PopulationGrowth #DynamicalSystems #PopulationDynamics #Population #Logistic #GradientSystem #GradientVectorField #Cylinder #Pendulum #Newton #LawOfMotion #dynamics

[lantonovbg]

Thank you, a complicated matter made understandable

[Nidish]

That visualization from 59.30 is amazing! Do you have more details on the parameterization used here? Great lecture, thanks! :)

[lantonovbg]

Kolmogorov theorem (KAM theory): If the unperturbed system is nondegenerate or isoenergetically nondegenerate, then for a sufficiently small Hamiltonian perturbation most nonresonant invariant tori do not vanish but are only slightly deformed, so that in the phase space of the perturbed system there are invariant tori densely filled with conditionally-periodic phase curves winding around them, with a number of independent frequencies equal to the number of degrees of freedom. These invariant tori form a majority in the sense that the measure of the complement of their union is small when the perturbation is small. In the case of isoenergetic nondegeneracy the invariant tori form a majority on each level manifold of the energy.

[nikkatalnikov]

thank you so much, very enlightening!