Mathematics / advanced
Symplectic geometry and Hamiltonian systems
Develop phase-space geometry through differential forms, Hamiltonian flows, canonical transformations, and reduction.
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Module 01 advanced
Symplectic geometry and Hamiltonian systems
Develop phase-space geometry through differential forms, Hamiltonian flows, canonical transformations, and reduction.
- 01
Symplectic forms and Darboux coordinates
Distinguish nondegeneracy from closedness and understand what local canonical coordinates do and do not remove.
29 min - 02
Hamiltonian vector fields and Poisson brackets
Derive Hamilton’s equations from a fixed contraction convention and identify conserved quantities through the bracket.
30 min - 03
Cotangent bundles and the canonical form
Construct the tautological one-form intrinsically and derive the momentum transformation forced by a change of configuration coordinates.
29 min - 04
Symplectomorphisms and generating functions
Verify canonical maps by pullback and derive a local transformation from a generating function with a regular projection.
30 min - 05
Lagrangian submanifolds and action–angle coordinates
Recognize half-dimensional zero-pairing geometry and build action–angle coordinates for a single regular oscillator.
31 min - 06
Momentum maps and Noether conservation
Convert infinitesimal symmetries into Hamiltonian generators and track exactly why invariant energies conserve their momentum components.
32 min - 07
Symplectic reduction and Marsden–Weinstein theory
Restrict to a regular momentum level, identify its null orbit directions and descend the form to a smooth quotient under explicit hypotheses.
34 min - 08
Integrable systems and Liouville tori
Use independent commuting integrals near a compact connected regular fiber to obtain torus geometry and linear angle motion.
34 min - 09
Contact geometry and Reeb flows
Study an odd-dimensional maximally nonintegrable distribution and compute the dynamics selected by a particular contact form.
33 min - 10
Symplectic capacity and nonsqueezing
Compare symplectic size with volume and apply the canonical-cylinder obstruction without overstating what it forbids.
33 min