Canonical Transformations & the Symplectic Condition: Hamiltonian Variables | ESM/AOE 6314 L5

Changes of variables in Hamiltonian systems — called canonical transformations or symplectic transformations — are the only allowable coordinate changes: those that preserve the form of Hamilton's equations in the new variables. This lecture builds from motivation (action-angle variables, tori, integrable systems) through the formal symplectic condition (MJMᵀ = J) to worked examples. The strategy: find a change of variables (Q, P) = φ(q, p) such that the equations of motion in the new coordinates are still Hamilton's canonical equations with some new Hamiltonian K(Q, P). The condition on φ is purely geometric — it must be a symplectomorphism — independent of the specific Hamiltonian. This leads to point transformations, swapping q and p, and the harmonic oscillator as worked examples. Hamilton-Jacobi theory and generating functions are previewed as the systematic machinery for finding such transformations. This is Lecture 5 of Advanced Dynamics (ESM/AOE 6314): Hamiltonian & Nonlinear Dynamics, Virginia Tech (graduate-level). ▶️ Chapters: 0:00 Intro: The strategy of canonical transformations 1:54 Motivation: Action-angle variables and integrable systems 10:08 Quasiperiodic orbits on tori in phase space 14:15 Numerical example: Poincaré sections and action-angle coordinates 21:56 Three-body problem and cislunar dynamics in action-angle context 25:52 Beyond action-angle: Hamilton-Jacobi theory overview 31:38 Generating functions for canonical transformations (preview) 34:23 Formal definition: Canonical (symplectic) transformation 38:38 Point transformations as a special class of canonical transformations 47:20 Condition: New variables must satisfy Hamilton's equations 53:48 The symplectic matrix J and the symplectic condition MJMᵀ = J 1:01:04 Symplectic condition is Hamiltonian-independent 1:03:08 Example: Swapping q and p is a valid canonical transformation 1:06:17 Example: Canonical transformation for the harmonic oscillator 1:11:44 Verification: Checking MᵀJM = J 1:12:11 Preview: Generating functions and the least action principle 📘 What you'll learn: Understand why canonical transformations must preserve Hamilton's equations Define and interpret action-angle variables and quasiperiodic tori State the symplectic condition MJMᵀ = J and its geometric meaning Recognize point transformations as a subclass of canonical transformations Verify whether a given transformation is canonical Understand the connection to Hamilton-Jacobi theory and generating functions 🎓 Course: Advanced Dynamics (ESM/AOE 6314): Hamiltonian & Nonlinear Dynamics (Virginia Tech, graduate-level) 🔗 Full course playlist:    • Hamiltonian Mechanics: Full Graduate Cours...   📄 Lecture notes (PDF): https://drive.google.com/drive/folder... 📖 References: Sanz-Serna & Calvo, Numerical Hamiltonian Problems Hand & Finch, Analytical Dynamics Hamill, A Student's Guide to Lagrangians and Hamiltonians Levi, Classical Mechanics with Calculus of Variations and Optimal Control Greenwood, Advanced Dynamics Marsden & Ratiu, Introduction to Mechanics and Symmetry 👨‍🏫 Instructor: Dr. Shane Ross, Virginia Tech (Caltech PhD) Instructor intro:    • Professor Shane Ross Introduction   Research: https://ross.aoe.vt.edu Follow on X: https://x.com/RossDynamicsLab Subscribe: https://www.youtube.com/c/RossDynamic... ▶️ Previous: Hamiltonian Flow, Poincaré Integral Invariants & Cyclic Coordinates (Lecture 4)    • Hamiltonian Flow, Poincaré Integral Invari...   ▶️ Next: Principle of Least Action & Lagrange's Equations | Calculus of Variations (Lecture 6)    • Principle of Least Action & Lagrange's Equ...   ▶️ This course builds on Lagrangian systems:    • Lagrangian Mechanics & 3D Rigid Body Dynam...   ▶️ Continuation: Center Manifolds, Normal Forms & Bifurcations    • Local Bifurcation Theory: Center Manifolds...   🔗 Related courses: Lagrangian & Rigid Body Dynamics:    • Lagrangian Mechanics & 3D Rigid Body Dynam...   Hamiltonian Dynamics:    • Hamiltonian Mechanics: Full Graduate Cours...   Local Bifurcation Theory:    • Local Bifurcation Theory: Center Manifolds...   Three-Body Problem:    • Three-Body Problem: Trajectory Design & Lo...   Nonlinear Dynamics & Chaos:    • Nonlinear Dynamics & Chaos — Full Course F...   Recorded: July 2021 #Hamiltonian #CanonicalTransformation #SymplecticTransformation #Symplectomorphism #ActionAngle #CyclicCoordinates #SymplecticGeometry #HamiltonJacobi #DynamicalSystems #PoincareSections #Tori #IntegrableSystems #NonlinearDynamics #GraduateLevel