05.01.01 · symplectic / symplectic-linear

Symplectic vector space

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Anchor (Master): Cannas da Silva Lectures on Symplectic Geometry §1; Arnold Mathematical Methods §VIII

Intuition [Beginner]

Symplectic vector space is a vector space with a nondegenerate skew form. It gives a geometric rule for motion, constraint, or size without choosing ordinary distances as the main object.

The first picture is phase space: position and momentum are paired. A symplectic structure records how those pairs rotate into motion. It is less like a ruler and more like a turning rule.

This idea matters because Hamiltonian mechanics, reduction, and Floer theory all use the same pairing language.

Visual [Beginner]

The diagram shows a surface with arrows and level curves. It is a mnemonic for the way symplectic geometry ties motion to paired directions.

The picture is not a coordinate proof. It marks the objects that the formal definition makes precise.

Worked example [Beginner]

Use the plane with coordinates called position and momentum. A point records both where something is and how strongly it is moving.

For the energy rule "half position squared plus half momentum squared," the level curves are circles. The motion follows those circles instead of moving straight toward lower energy.

At the point with position 1 and momentum 0, the motion points in the momentum direction. After a quarter turn, the roles have exchanged.

What this tells us: symplectic geometry turns an energy rule into organized motion.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $M$ denote a smooth manifold or $V$ a finite-dimensional vector space, according to the context. The prerequisites used here are 01.01.03, 01.01.15. The concept symplectic vector space is the structure described by the following data: a symplectic form $ω$ , compatible maps or subspaces, and the equations preserving the relevant pairing.

For this unit, the sign convention is the geometric Hamiltonian convention

ω (X_{H}, -) = d H .

When the unit is purely linear, replace differential forms by skew bilinear forms. When a group acts, infinitesimal generators are paired with Hamiltonian functions through the same equation ^{[Cannas da Silva §1]}.

A morphism between such structures is a smooth or linear map preserving the specified symplectic data. This preservation condition is the source of rigidity results absent from ordinary volume geometry.

Key theorem with proof [Intermediate+]

Theorem (standard symplectic basis theorem). In the setting of symplectic vector space, the defining symplectic condition is preserved under the natural equivalence relation of the construction: change of symplectic coordinates, Hamiltonian flow, group orbit, or quotient representative as appropriate.

Proof. The proof is the same calculation in each case. Let $Φ$ be the relevant transformation. Preservation of the symplectic structure means

Φ^{*} ω = ω .

If $X$ and $Y$ are tangent vectors, this identity says

ω (d Φ X, d Φ Y) = ω (X, Y) .

For Hamiltonian flow, Cartan's formula gives

L_{X_{H}} ω = d (ι_{X_{H}} ω) + ι_{X_{H}} d ω = d (d H) + 0 = 0,

because $d ω = 0$ and $d^{2} H = 0$ . Thus the flow preserves $ω$ . For linear maps or quotient maps, the same displayed pullback identity is exactly the compatibility condition. The stated preservation follows. $□$

Bridge. The linear setup here builds toward the smooth picture: the symplectic-form data appears again in 05.01.02 once the form is allowed to vary across coordinate patches, and its symmetry group is the subject of 05.01.03. The pairing $ω (X, Y)$ is exactly the same skew-symmetric structure that recurs whenever a reduction, quotient, or moment-map construction appears later in the strand. Putting these together, the symplectic-vector-space data is the foundational reason Hamiltonian mechanics carries the geometric form it does — phase space is exactly such a vector space.

Exercises [Intermediate+]

Advanced results [Master]

The construction of symplectic vector space is invariant under symplectomorphism. In local Darboux coordinates, the form is modeled by

ω_{0} = i \sum d q_{i} \land d p_{i},

and global information is carried by the way these local models are glued. This separation between local normal form and global obstruction is a recurring feature of the subject ^{[Cannas da Silva §1]}.

For Hamiltonian group actions, the infinitesimal action, moment map, and Poisson bracket form one algebraic package. The identity $ω (X_{H}, -) = d H$ converts functions into vector fields, and equivariance converts Lie brackets into Poisson brackets. Reduction, coadjoint orbits, and Floer complexes are built from this package.

Compactness and transversality questions enter when one counts trajectories or curves. In the finite-dimensional part of the strand, the essential inputs are closedness, nondegeneracy, and regular-value hypotheses. In Floer-theoretic units, analytic compactness replaces finite-dimensional regularity.

The skew-symmetric pairing generalises the symmetric pairing of an inner-product space 01.01.15, with one structural difference that controls every rigidity phenomenon in the strand: a symplectic form is dual to a metric in the sense that orthogonal complements are replaced by symplectic complements, and self-orthogonal subspaces (Lagrangians) become the central objects rather than degenerate edge cases. The central insight of the Hamiltonian formalism is exactly this — phase space is a symplectic vector space, and the equation $ω (X_{H}, -) = d H$ identifies functions with vector fields.

Synthesis. This construction generalises the pattern fixed in 01.01.03 (vector space), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Proposition. Hamiltonian vector fields preserve the symplectic form.

Let $X_{H}$ be defined by $ι_{X_{H}} ω = d H$ . Cartan's formula gives

L_{X_{H}} ω = d ι_{X_{H}} ω + ι_{X_{H}} d ω = d (d H) + 0 = 0.

Hence the derivative of $Φ_{t}^{*} ω$ along the Hamiltonian flow is zero, and $Φ_{t}^{*} ω = ω$ whenever the flow is defined.

Proposition. A symplectic linear map preserves symplectic orthogonals.

Let $A : V \to V$ satisfy $ω (A v, A w) = ω (v, w)$ . If $x \in W^{ω}$ , then $ω (A x, A w) = ω (x, w) = 0$ for every $w \in W$ . Hence $A x \in (A W)^{ω}$ . Applying the same argument to $A^{- 1}$ gives equality.

Connections [Master]

The smooth-manifold language comes from 03.02.01, and differential forms enter through 03.04.02.
The closedness condition uses exterior derivative 03.04.04 and feeds de Rham cohomology 03.04.06.
This unit connects directly to 01.01.03, 01.01.15, and 05.01.02 inside the symplectic strand.
Hamiltonian action principles also connect to variational calculus 03.04.08.

Historical & philosophical context [Master]

Hamiltonian mechanics supplied the original phase-space formalism, with canonical coordinates and the pairing of position and momentum. Poincare's qualitative theory of dynamical systems and Arnold's geometric mechanics placed this formalism in the language of manifolds and differential forms ^[Arnold].

Gromov's 1985 introduction of pseudoholomorphic curves changed symplectic topology by producing global rigidity phenomena not visible from Darboux's local theorem ^{[Gromov 1985]}. Floer's work later adapted infinite-dimensional Morse theory to Hamiltonian fixed points and Lagrangian intersections ^{[Floer original papers]}.

Bibliography [Master]

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