Hamiltonian Systems Introduction- Why Study Them? | Lecture 1 of a Course on Hamilton's Equations

Lecture 1 of a course on Hamiltonian and nonlinear dynamics. The Hamiltonian formalism is introduced, one of the two great pillars of mechanics, along with the Lagrangian formalism. “What’s the point? Why study Hamiltonian systems?” We answer that, discussing the advantages of the Hamiltonian point of view, and introduce the idea of getting a Hamiltonian system from a Lagrangian system via the Legendre transformation. ► Dr. Shane Ross, Virginia Tech professor (Caltech PhD) Introduction to instructor https://youtu.be/iAlNnYJinRs ► Next, Legendre transformation | going from Lagrangian to Hamiltonian | spring mass worked example https://youtu.be/xJGluipIV2k ► Stay informed Subscribe https://is.gd/RossLabSubscribe​ ► Follow me https://twitter.com/RossDynamicsLab ► A shorter introduction to Hamiltonian systems in 2D https://youtu.be/wxRZwLFS6lk ► 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 & 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. ► Entire class notes in PDF form: https://is.gd/AdvancedDynamicsNotes ► OneNote form: https://1drv.ms/u/s!ApKh50Sn6rEDiRgCYBrjscAYJDUk?e=IE5wnQ ►This course builds on prior knowledge of Lagrangian systems https://is.gd/AnalyticalDynamics ► Continuation of this course: Center manifolds, normal forms, & bifurcations https://is.gd/CenterManifolds ► Want a simple introductory course on Nonlinear Dynamics & Chaos? https://is.gd/NonlinearDynamics ► *Courses & Playlists by Dr. Ross* 📚Kalman Filter Mini-Course http://tinyurl.com/kalmanfilters 📚Nonlinear Dynamics & Chaos https://is.gd/NonlinearDynamics 📚Attitude Dynamics & Control https://is.gd/SpaceVehicleDynamics 📚Lagrangian & 3D Rigid Body Dynamics https://is.gd/AnalyticalDynamics 📚Center Manifolds, Normal Forms, & Bifurcations https://is.gd/CenterManifolds 📚3-Body Problem Orbital Dynamics https://is.gd/3BodyProblem 📚Space Manifolds https://is.gd/SpaceManifolds ► Chapters 0:00 Lagrangian and Hamiltonian formalism of mechanics compared 15:14 Advantages of the Hamiltonian formalism 27:48 Hamilton's equations from Lagrange's equations 37:18 Generalized momentum 44:03 Hamiltonian function definition 46:51 Hamilton's canonical equations and advantages 1:00:29 Hamilton's canonical equations do not permit attractors ► References The class will largely be based on instructor’s notes. A PDF book available for FREE at http://shaneross.com/books Koon, Lo, Marsden, Ross (2011) Dynamical Systems, the Three-Body Problem and Space Mission Design. ISBN 978-0-615-24095-4. In addition, references are: Advanced Dynamics by Greenwood 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 Classical Dynamics: A Contemporary Approach by José & Saletan Classical Mechanics, 3rd Edition by Goldstein, Poole, & Safko Additional texts that may be useful: Nonlinear Differential Equations and Dynamical Systems by Verhulst Introduction to Applied Nonlinear Dynamical Systems and 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 2021-06-17 action angle variables in classical mechanics quantum mechanics statistical physics thermal physics thermodynamics general relativity Jerrold Marsden quasiperiodic online course cosmology universe Hamilton-Jacobi theory three-body problem orbital mechanics incompressibility Poincare fluids #Hamiltonian #Lagrangian #NonlinearDynamics #DynamicalSystems #mathematics #Dynamics #Chaos #ChaoticDynamics #Canonical #Poisson #OptimalControl #Poincare #Lindstedt #ChaosTheory #Legendre #Hamilton #Jacobi #Bifurcation #PoincareMap #chaos #Lyapunov #Oscillators #NonlinearOscillators #LimitCycle #Oscillations #VectorFields #topology #geometry #IndexTheory #EnergyConservation #Streamfunction #Vortex #SkewGradient #DifferentialEquations #SaddleNode #Eigenvalues #dimensions #PhaseSpace #PhasePortrait #PhasePlane #Strogatz #FixedPoints #EquilibriumPoints #Stability #StablePoint #UnstablePoint #LinearStability