Hamiltonian Flow is a Canonical Transformation | Strange Momenta | ESM/AOE 6314 Lecture 8

Prove that the flow of Hamilton's equations is itself a canonical transformation, and explore the surprising nature of generalized momentum — which often differs from familiar mechanical momentum (mass × velocity). Examples include a charged particle in an electromagnetic field (where momentum picks up a vector potential term) and motion in a rotating frame, as in the circular restricted three-body problem (CR3BP). The vector field of Hamilton's equations is infinitesimally symplectic, and this constrains the eigenspectrum at equilibrium points: eigenvalues come in symmetric quadruples (±λ, ±λ*). This is why equilibria in Hamiltonian systems are saddle-centers, center-centers, or saddle-saddles — never simple sinks or sources. The L1 and L2 points of the three-body problem are classic saddle-centers. This is Lecture 8 of Advanced Dynamics (ESM/AOE 6314): Hamiltonian & Nonlinear Dynamics, Virginia Tech. ▶️ Chapters: 0:00 Strange momentum definitions: Generalized vs. mechanical momentum 0:53 Charged particle in an electromagnetic field 2:38 The Lorentz force and the magnetic vector potential 5:44 Generalized momentum picks up a vector potential term 8:31 Motion in a rotating frame: The circular restricted three-body problem 10:44 Transformation from inertial to rotating frame 13:10 Generalized momentum in the rotating frame 16:44 Working out Hamilton's equations: Recovering the velocity 21:55 The flow of Hamilton's equations is a canonical transformation 28:58 The infinitesimally symplectic condition 30:48 Conditions on the eigenspectrum at equilibrium points 32:10 Eigenvalue quadruples: ±λ and ±λ* for symplectic systems 34:28 Saddle-centers, center-centers, and saddle-saddles 41:22 3-body problem: L1, L2 as saddle-centers; L4, L5 as center-centers 42:52 The pendulum: Saddle (inverted) and center (hanging) equilibria 45:27 Example Hamiltonian flow maps 56:01 Final example: Falling in uniform gravity 📘 What you'll learn: Distinguish generalized momentum from mechanical momentum Compute generalized momentum for a charged particle and in a rotating frame Prove that Hamiltonian flow is a canonical transformation State the infinitesimally symplectic condition on the Jacobian Understand how the symplectic structure constrains equilibrium eigenvalues Classify equilibria as saddle-centers, center-centers, or saddle-saddles 🎓 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: Hamill, A Student's Guide to Lagrangians and Hamiltonians Sanz-Serna & Calvo, Numerical Hamiltonian Problems Hand & Finch, Analytical Dynamics Levi, Classical Mechanics with Calculus of Variations and Optimal Control 👨‍🏫 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: Generating Functions of a Canonical Transformation — Examples & the Big Picture (Lecture 7)    • Generating Function of a Canonical Transfo...   ▶️ Next: Hamilton-Jacobi Theory — Finding the Best Canonical Transformation (Lecture 9)    • Hamilton-Jacobi Theory | Canonical Transfo...   🔗 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 #HamiltonianMechanics #CanonicalTransformation #GeneralizedMomentum #SymplecticGeometry #ThreeBodyProblem #LorentzForce #SaddleCenter #LagrangePoints #InfinitesimallySymplectic #NonlinearDynamics #GraduateLevel