Canonical Transformations | Hamiltonian Changes of Variables | Symplectomorphisms | Lecture 5 скачать в хорошем качестве

Canonical Transformations | Hamiltonian Changes of Variables | Symplectomorphisms | Lecture 5

Lecture 5, course on Hamiltonian and nonlinear dynamics. Change of variables in Hamiltonian systems, called symplectic transformations or canonical transformations. The only allowed transformations are those that preserve the form of Hamilton's equations. Thus, the condition is that the equations of motion in the new variables must still be Hamilton's canonical equations. ► Next: Principle of least action and Lagrange's equations of mechanics | basics of calculus of variations    • Principle of Least Action, Lagrange's Equa...   ► Previous, Hamiltonian flow Poincare integral invariants| ignorable/cyclic coordinates in Hamiltonian formalism    • Hamiltonian Flow Poincare Integral Invaria...   ► Dr. Shane Ross, Virginia Tech professor (Caltech PhD) Instructor intro    • Professor Shane Ross Introduction   ► New lectures posted regularly Subscribe https://is.gd/RossLabSubscribe​ ► Follow me on Twitter   / rossdynamicslab   ► See the entire playlist for this online course: 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. ► 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, 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 ► A simple introductory course on Nonlinear Dynamics and 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 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 2021-07-01 action angle cyclic variables in classical mechanics statistical physics thermal physics thermodynamics general relativity quasiperiodic online course Hamilton Hamilton-Jacobi theory three-body problem orbital mechanics incompressibility integral invariants of Poincare and Cartan Symplectic Geometry topology #Hamiltonian #CanonicalTransformation #SymplecticTransformation #CyclicCoordinates #DynamicalSystems #IgnorableCoordinate #SymplecticGeometry #IntegralInvariant #PoincareCartan #CyclicVariable #Uncertainty #Streamfunction #Legendre #LegendreTransformation #NonlinearDynamics #DynamicalSystems #Lagrangian #mathematics #Dynamics #Chaos #ChaoticDynamics #Canonical #Poisson #OptimalControl #Poincare #Lindstedt #Mathieu #ChaosTheory #Heisenberg #Lagrange #Hamilton #Jacobi #HamiltonJacobi #ThreeBody #PeriodDoubling #Bifurcation #DifferenceEquation #PoincareMap #chaos #ChaosTheory #Lyapunov #Oscillators #HopfBifurcation #NonlinearOscillators #LimitCycle #Oscillations #VectorFields #topology #geometry #IndexTheory #EnergyConservation #Streamfunction #Streamlines #Vortex #SkewGradient #FixedPoint #DifferentialEquations #SaddleNode #Eigenvalues #HyperbolicPoints #NonHyperbolicPoint #CriticalPoint #PitchforkBifurcation #StructuralStability #DifferentialEquations #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Strogatz #Lorenz #VectorField #GraphicalMethod #FixedPoints #EquilibriumPoints #Stability #StablePoint #UnstablePoint #Stability #LinearStability #LinearStabilityAnalysis #StabilityAnalysis #VectorField #TwoDimensional #Functions #GradientSystem #GradientVectorField #Cylinder #Pendulum #Newton #LawOfMotion #dynamics ​#mathematicians #maths #mathstudents #mathematician #mathfacts #mathskills #mathtricks #KAMtori #Arnold