SYSTEMATIC MATHEMATICS

Symplectic Geometry & Hamiltonian Dynamics

Use ninety visible bilingual chapters in twelve content-sized units to build symplectic geometry from alternating linear algebra through symplectic manifolds, Darboux–Moser theory, Hamiltonian vector fields, Poisson brackets, cotangent mechanics, moment maps and regular reduction, Lagrangian and contact geometry, concrete integrable and perturbed dynamics, capacities and non-squeezing, bounded Floer foundations, pseudoholomorphic curves, and seven original reproducible dossiers. Every chapter starts in ordinary language, defines every symbol and object, states a theorem or deliberately bounded claim, exposes the complete proof route at the length the argument requires, names the nearest failure boundary, solves an exact model step by step, supplies matched practice and a complete solution, and ends with a diagnostic misconception check. Read every formula from left to right: say what space each object lives in, declare the form and contraction signs, identify every derivative, pullback, bracket, quotient, index, integral, action, energy, and auxiliary choice, then test dimensions, scaling, regularity, compactness, transversality, equality, and a negative control. Local Darboux coordinates are never promoted to global standardization; volume is never confused with symplectic preservation; a regular quotient is never reused at a singular value; one plotted orbit, intersection count, discretized complex, or small residual is never called an invariant. The course gives research preparation rather than a complete construction of KAM theory, singular reduction, general contact topology, virtual transversality, all Hamiltonian or Lagrangian Floer theories, Gromov–Witten theory, quantum cohomology, symplectic field theory, or every postdoctoral frontier.

Before this course: Completed Differential Geometry & Manifolds; Differential Topology, Vector Bundles & Morse Theory; Differential Forms, de Rham Cohomology & Hodge Theory; Algebraic Topology; Calculus of Variations; Differential Equations & Dynamical Systems; Sobolev Spaces, Distributions & Weak PDE; and Nonlinear Elliptic & Parabolic Regularity. No prior complete course in symplectic or contact geometry, Hamiltonian reduction, KAM theory, Floer homology, pseudoholomorphic curves, Gromov–Witten theory, or quantum cohomology is assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

A symplectic form is alternating and nondegenerate

Objective: Does ω(v,v)=0 imply ω is nondegenerate?

A symplectic form measures oriented pairs of directions rather than lengths. Nondegeneracy means every nonzero vector has some partner with nonzero pairing. Before calculating, declare the manifold or vector space, dimension, coefficient field, symplectic-form sign, orientation, regularity, compactness, boundary, time interval, and whether the conclusion is linear, local, global, topological, variational, or only valid after choosing auxiliary data.

Read every object in words: On a real vector space V, a bilinear form ω is alternating when ω(v,v)=0 for every v, equivalently ω(u,v)=−ω(v,u). It is nondegenerate when the map ω♭:V→V*, v↦ω(v,·), has zero kernel. The exact statement used here is: A finite-dimensional real vector space carrying a nondegenerate alternating form has even dimension. The pair (V,ω) is then a symplectic vector space. Separate an algebraic identity from a theorem that needs closedness, nondegeneracy, compactness, transversality, regularity, convexity, monotonicity, or a compactness theorem.

The complete proof or honestly bounded proof route is: 1. Choose nonzero v and use nondegeneracy to find w with ω(v,w)≠0. 2. Normalize the pair and split off their two-dimensional span using the symplectic orthogonal. 3. Induct on the remaining nondegenerate complement, removing two dimensions each time. Record every sign, pullback, primitive, isotopy, boundary term, quotient, regular value, gauge, perturbation, orientation, compactification, and equality case instead of hiding it behind a theorem name.

Reconstruct the model “For ω((q,p),(Q,P))=qP−pQ on R², find ω♭(q,p) and test nondegeneracy.”. The checked result is ω♭(q,p)=(−p,q) in dual coordinates, and its kernel is {0}; hence ω is nondegenerate. The nearest failure boundary is: A skew-symmetric matrix in odd dimension has determinant zero, so an odd-dimensional alternating form cannot be nondegenerate. A phase portrait, finite orbit sample, truncated complex, numerical action, mesh, or small residual is evidence only for the recorded model and tolerance; it is not a proof of global integrability, persistence, compactness, transversality, non-squeezing, or a Floer-theoretic invariant.

A finite-dimensional real vector space carrying a nondegenerate alternating form has even dimension. The pair (V,ω) is then a symplectic vector space.

Choose nonzero v and use nondegeneracy to find w with ω(v,w)≠0.

Normalize the pair and split off their two-dimensional span using the symplectic orthogonal.

Induct on the remaining nondegenerate complement, removing two dimensions each time.

For ω((q,p),(Q,P))=qP−pQ on R², find ω♭(q,p) and test nondegeneracy.

  1. Fix (q,p) and read the functional of (Q,P) as −pQ+qP.
  2. Its coordinate row in the dual basis is (−p,q).
  3. This row vanishes only when q=p=0.

Result: ω♭(q,p)=(−p,q) in dual coordinates, and its kernel is {0}; hence ω is nondegenerate.