Symplectic geometry studies a kind of structure that measures oriented two-dimensional interaction rather than length. It is the natural geometry of Hamiltonian phase space, and contact geometry appears on the odd-dimensional boundaries and energy levels that surround it.
A Riemannian metric answers questions such as how long a vector is and how far apart two points are. A symplectic form answers a different question: how do two tangent directions pair as an oriented area element, and how does that pairing constrain dynamics? The form is skew-symmetric rather than positive definite. It does not assign lengths, yet it can control motion with remarkable rigidity.
This final R23 guide connects the preceding Atlas cells. Differential Forms and Integration supplies closed 2-forms and pullbacks. Riemannian Geometry and Geodesics supplies a contrasting metric geometry. Bundles, Characteristic Structures and Global Analysis supplies cotangent bundles, characteristic structures and global-analysis context.
Level: advanced undergraduate to graduate enrichment. Prerequisites are smooth manifolds, tangent/cotangent bundles, differential forms, vector fields and ordinary differential equations. Classical mechanics examples are mathematical models, not claims that every physical system is exactly Hamiltonian.
Conventions: on a cotangent bundle with coordinates (qⁱ,pᵢ), the canonical 1-form is λ=Σpᵢdqⁱ and the symplectic form is ω=−dλ=Σdqⁱ∧dpᵢ. Hamiltonian vector fields satisfy ι_{X_H}ω=dH. With these choices, Hamilton’s equations are q̇ⁱ=∂H/∂pᵢ and ṗᵢ=−∂H/∂qⁱ. Other texts may reverse one or both signs; translate the whole convention consistently.
Reading route: symplectic forms → cotangent phase space → Hamiltonian flows → Poisson brackets → Lagrangians → symmetry and reduction → contact geometry → geometric applications → rigidity and frontier ideas → practice → solutions.
1. A symplectic form is closed and nondegenerate
A symplectic manifold is a smooth manifold M equipped with a differential 2-form ω satisfying two conditions: dω=0 and ωₚ is nondegenerate at every point.
Nondegenerate means that if a tangent vector v satisfies ω(v,w)=0 for every tangent vector w at the same point, then v=0. Equivalently, the map v↦ι_vω from TₚM to Tₚ*M is an isomorphism.
This is not positive definiteness. Because ω is alternating, ω(v,v)=0 for every v. A symplectic form therefore cannot serve as a squared-length rule.
Closedness is a differential condition. It ensures that Hamiltonian vector fields preserve the symplectic form and underlies local normal-form and reduction theorems.
The standard graduate curriculum at UC Berkeley Math 242 lists symplectic linear algebra, symplectic manifolds, Darboux’s theorem, cotangent bundles, Hamiltonian systems, Lagrangian submanifolds, Poisson brackets and momentum mappings as core topics. Michael Hutchings’s 2024 course page also connects symplectic geometry to contact structures, Reeb fields and holomorphic-curve methods.
2. Nondegenerate skew forms force even dimension
Choose a basis of a finite-dimensional tangent space and represent ω by a skew-symmetric matrix A. Nondegeneracy means det A≠0.
If the dimension n were odd, then det A=det Aᵀ=det(−A)=(−1)ⁿdet A=−det A, forcing det A=0. Therefore a symplectic vector space has even dimension 2n.
The top wedge power ωⁿ is a nowhere-zero 2n-form. Hence every symplectic manifold is orientable, with symplectic orientation determined by ωⁿ.
The normalised form ωⁿ/n! is often called the symplectic volume form. It supplies an oriented volume, but again it does not supply a metric length.
3. The standard symplectic plane already contains Hamiltonian geometry
On R² with coordinates (q,p), take ω=dq∧dp. For tangent vectors v=(a,b) and w=(c,d), ω(v,w)=ad−bc.
This is signed area in the ordered (q,p) coordinates. The form is nondegenerate because a nonzero vector always has another vector with nonzero determinant pairing.
It is closed because its coefficients are constant: d(dq∧dp)=0. Thus the ordinary plane, viewed with this 2-form, is the smallest symplectic manifold.
In R²ⁿ, the standard form is ω₀=Σᵢdqⁱ∧dpᵢ. Each qⁱ is paired with its conjugate pᵢ. Hamiltonian mechanics interprets q variables as configurations and p variables as momenta after an appropriate modelling step.
4. Symplectic coordinates have no preferred Euclidean scale
Consider the linear change Q=aq and P=p/a for a≠0. Then dQ∧dP=(a dq)∧(dp/a)=dq∧dp. The transformation preserves ω.
Yet it stretches one coordinate direction and compresses the other. It is generally not an isometry of the Euclidean metric dq²+dp².
This illustrates the difference between metric and symplectic invariance. Symplectic geometry preserves conjugate-area structure, not ordinary lengths.
For a linear map represented by S, preservation of the standard form means SᵀJS=J, where J is the standard skew matrix. Such matrices form the symplectic group Sp(2n,R). Their determinants equal one, but determinant one alone is not enough to be symplectic when n>1.
5. Darboux’s theorem removes local symplectic variation
Darboux’s theorem says that around every point of a symplectic 2n-manifold there are local coordinates (q¹,…,qⁿ,p₁,…,pₙ) in which ω=Σdqⁱ∧dpᵢ.
This is strikingly different from Riemannian geometry. A Riemannian metric can be made Euclidean at one point, but curvature remains as a local invariant of its second-order variation. A symplectic form has no analogous local curvature invariant: all symplectic manifolds look locally the same as symplectic manifolds.
The geometry therefore becomes strongly global. Topology, embedded submanifolds, Hamiltonian dynamics and rigidity distinguish symplectic manifolds even though the form has a universal local model.
Hutchings’s Berkeley course explicitly uses the Moser method to prove Darboux and neighbourhood theorems. The proof depends crucially on closedness and nondegeneracy, not merely on linear algebra.
6. Every cotangent bundle carries a canonical symplectic form
Let Q be a smooth n-manifold and π:T*Q→Q its cotangent bundle. At a point (q,p), where p is a covector at q, define the canonical 1-form λ by
λ_{(q,p)}(V)=p(dπ(V)).
This definition requires no coordinates, metric or connection. In local coordinates qⁱ with cotangent components pᵢ, it becomes λ=Σpᵢdqⁱ.
Define ω=−dλ. Then ω=Σdqⁱ∧dpᵢ in the induced coordinates, and it is symplectic.
This construction makes T*Q the natural phase-space geometry associated with any configuration manifold Q. The base can be curved or topologically complicated; its cotangent bundle still carries this canonical exact symplectic structure.
The word “canonical” matters: no Riemannian metric on Q was selected. A metric may later identify velocities with momenta, but the symplectic structure exists before that identification.
7. Cotangent lifts preserve the canonical form
If f:Q→Q is a diffeomorphism, it induces a cotangent lift on T*Q by sending a covector at q to the corresponding pulled-back covector at f(q), with inverse arranged so that the fibre remains over the transformed base point.
The cotangent lift preserves λ and therefore preserves ω. Thus every change of configuration coordinates naturally becomes a symplectomorphism of phase space.
This is one reason momentum must transform as a covector rather than as another copy of the position coordinate. The canonical 1-form pairs momentum with base velocity, and that scalar pairing must survive coordinate changes.
The result connects directly to the earlier tensor guide: vectors push forward, covectors pull back, and the phase-space form is built from that duality.
8. A Hamiltonian function determines a vector field
Let H:M→R be smooth on a symplectic manifold (M,ω). Because ω is nondegenerate, there is a unique vector field X_H satisfying
ι_{X_H}ω=dH.
The integral curves of X_H are Hamiltonian trajectories. The function H is the Hamiltonian. In applications it may represent energy, but mathematically it is any smooth function generating a flow through this equation.
In canonical coordinates, write X_H=aⁱ∂_{qⁱ}+bᵢ∂_{pᵢ}. Contracting with ω gives aⁱdpᵢ−bᵢdqⁱ. Matching dH=(∂H/∂qⁱ)dqⁱ+(∂H/∂pᵢ)dpᵢ yields
q̇ⁱ=∂H/∂pᵢ, ṗᵢ=−∂H/∂qⁱ.
The equations are therefore the coordinate form of one intrinsic identity.
9. Hamiltonian flows preserve the symplectic form
Use Cartan’s formula for the Lie derivative:
L_{X_H}ω=d(ι_{X_H}ω)+ι_{X_H}dω=d(dH)+0=0.
Thus the local flow φ_t of X_H satisfies φ_t*ω=ω wherever it is defined. Hamiltonian evolution is symplectic.
Consequently it preserves the symplectic volume ωⁿ/n!. This is Liouville volume preservation in Hamiltonian phase space.
Volume preservation alone is weaker than symplectic preservation. In dimensions at least four, many volume-preserving maps fail to preserve the symplectic form.
The Hamiltonian itself is conserved along its own flow: dH(X_H)=ω(X_H,X_H)=0 by antisymmetry. This statement assumes H has no explicit time parameter; time-dependent Hamiltonians require the corresponding extended formulation.
10. Worked model: the harmonic oscillator
Take one degree of freedom with H(q,p)=p²/(2m)+(k/2)q², where m,k>0. Hamilton’s equations are q̇=p/m and ṗ=−kq.
Differentiating the first equation gives q̈=ṗ/m=−(k/m)q. Thus q satisfies q̈+ω₀²q=0 with ω₀=√(k/m).
The energy level H=E is the ellipse p²/(2mE)+kq²/(2E)=1. Hamiltonian trajectories run along these ellipses with orientation determined by X_H.
Directly, dH/dt=H_q q̇+H_p ṗ=kq(p/m)+(p/m)(−kq)=0. The coordinate check agrees with the intrinsic antisymmetry argument.
If m=k=1, the equations are q̇=p,ṗ=−q, so the phase-space vector field rotates clockwise in the (q,p) plane under the chosen convention.
11. Hamiltonian dynamics on a Riemannian manifold recovers geodesic flow
Let (Q,g) be Riemannian. The inverse metric turns a covector p into a squared norm |p|_g²=g^{ij}pᵢpⱼ. On T*Q define the kinetic Hamiltonian H(q,p)=(1/2)g^{ij}(q)pᵢpⱼ.
Hamilton’s equations for this H project to geodesics on Q, with momentum related to velocity by pᵢ=g_{ij}q̇ʲ.
This gives a powerful bridge between the two geometries. Riemannian geodesic motion on Q becomes Hamiltonian flow on the symplectic cotangent bundle T*Q.
The metric determines the Hamiltonian but not the canonical symplectic form. Separating those roles makes it clear which parts of the phase-space geometry are universal and which come from the chosen dynamics.
12. The Poisson bracket measures Hamiltonian interaction
For smooth functions f and g, define the Poisson bracket by
{f,g}=ω(X_f,X_g).
With our convention this becomes {f,g}=Σ(∂f/∂qⁱ ∂g/∂pᵢ−∂f/∂pᵢ ∂g/∂qⁱ).
The bracket is bilinear, antisymmetric, satisfies a Leibniz rule in each argument and obeys the Jacobi identity. Smooth functions therefore form a Poisson algebra.
Along Hamiltonian motion generated by H, an observable f changes according to
df/dt=X_H(f)={f,H}.
Thus {f,H}=0 means f is conserved along the H-flow. The bracket gives a coordinate-independent test of conservation once the symplectic structure is fixed.
13. Worked Poisson calculations
In one canonical pair, {q,p}=1, {q,q}=0 and {p,p}=0. For H=p²/(2m)+V(q), we obtain {q,H}=p/m and {p,H}=−V′(q), reproducing Hamilton’s equations.
Take f=qp. Then f_q=p and f_p=q. For the unit oscillator H=(p²+q²)/2,
{f,H}=p·p−q·q=p²−q².
So qp is not conserved. By contrast {H,H}=0 automatically. A proposed conserved quantity should be tested by its bracket with H, not by whether its formula looks symmetric.
For angular momentum in rotationally symmetric systems, vanishing Poisson brackets with the Hamiltonian encode the familiar conservation law. The deeper reason is symmetry, which leads to moment maps.
14. Lagrangian submanifolds are maximally isotropic
In a symplectic 2n-manifold, a submanifold L is Lagrangian when dim L=n and ω restricts to zero on L.
The zero condition says every pair of tangent directions inside L has zero symplectic pairing. The dimension condition makes L as large as possible with that property.
The zero section Q⊂T*Q is Lagrangian. Tangent vectors to it have no momentum component, so every dqⁱ∧dpᵢ term vanishes on pairs of such vectors.
If f:Q→R is smooth, the graph of df inside T*Q is also Lagrangian. Pulling the canonical 1-form λ to the graph gives df, so the pullback of ω=−dλ is −d²f=0.
Such a graph is called exact Lagrangian because the restricted canonical 1-form is exact. A graph of a closed 1-form is Lagrangian; it is exact precisely when that 1-form is exact.
15. Generating functions convert Lagrangian geometry into scalar data
The graph-of-df example shows how a scalar function can generate a Lagrangian submanifold. More general generating-function constructions encode canonical transformations and variational relations.
The key point is not that every Lagrangian globally comes from one function on the base. Topology can obstruct a global exact primitive, and a Lagrangian may not project diffeomorphically to Q.
Locally, Darboux and Lagrangian neighbourhood theorems give strong normal forms. Globally, periods, intersection patterns and Floer-theoretic invariants distinguish Lagrangians that look identical in small charts.
This local-versus-global gap is a recurring theme across topology, bundle theory and symplectic geometry.
16. Symplectomorphisms preserve the 2-form, not necessarily a Hamiltonian
A diffeomorphism φ:(M,ω)→(N,η) is symplectic when φ*η=ω. When source and target are the same, it is a symplectomorphism.
A Hamiltonian flow is a special path of symplectomorphisms. Not every symplectic isotopy is necessarily Hamiltonian on every manifold; global cohomological obstructions can intervene.
This distinction matters because “preserves ω” is weaker than “generated by a globally defined Hamiltonian function”. Locally, closed 1-forms are exact, but globally they need not be.
The difference is measured by structures such as flux in more advanced symplectic topology. It is another instance where de Rham cohomology records the failure of a local potential to exist globally.
17. Moment maps turn continuous symmetries into conserved quantities
Suppose a Lie group G acts on (M,ω) by symplectomorphisms. Each Lie algebra element ξ produces an infinitesimal vector field X_ξ on M.
A moment map, under one standard convention compatible with this article, is a map μ:M→g* such that
d⟨μ,ξ⟩=ι_{X_ξ}ω
for each ξ, together with an appropriate equivariance condition when required. Thus each symmetry direction is Hamiltonian, generated by the corresponding component of μ.
If the Hamiltonian H is invariant under the group action, those moment-map components Poisson-commute with H and are conserved. This is the symplectic form of the Noether principle.
Not every symplectic action admits a global moment map. Again, closed 1-forms ι_{X_ξ}ω may fail to be exact.
18. Worked symmetry: rotations in the plane
Take M=R² with coordinates (q,p) and ω=dq∧dp. The infinitesimal generator of anticlockwise rotations is X=−p∂_q+q∂_p.
Contracting gives ι_Xω=(−p)dp−q dq=−d[(q²+p²)/2]. Under our moment-map convention dμ=ι_Xω, one may take μ=−(q²+p²)/2.
The sign is convention-dependent. If the opposite Lie-algebra generator or moment-map sign convention is chosen, μ changes sign. The invariant content is that the radial quadratic generates rotation up to the chosen orientation convention.
For the harmonic oscillator H=(q²+p²)/2, the same radial quantity is conserved. Rotational symmetry and conservation are two views of the same structure.
19. Symplectic reduction removes symmetry and constraints together
Given a Hamiltonian G-action with moment map μ, choose a suitable value c∈g*. Under regularity, freeness and properness hypotheses, the quotient μ⁻¹(c)/G_c can inherit a symplectic form. This is the Marsden–Weinstein reduction mechanism.
For a free action of a d-dimensional group at a regular central value, the dimension typically drops by 2d: d dimensions from the moment-map constraints and d from quotienting the symmetry orbits.
The hypotheses matter. If the value is singular or the action has stabilisers, the quotient may have singular strata rather than being an ordinary smooth symplectic manifold.
This is why a constrained mechanical system cannot always be handled by “number of variables minus number of equations”. One must distinguish independent constraints, gauge/symmetry directions and the geometry of the quotient.
Berkeley symplectic courses place Hamiltonian group actions and symplectic quotients after the core theory, reflecting this need for prior manifold, group-action and moment-map control.
20. Coisotropic submanifolds are the natural home of many constraints
For a subspace W of a symplectic vector space, define W^ω={v:ω(v,w)=0 for all w∈W}. A submanifold C is coisotropic when (TₚC)^ω⊂TₚC at every point.
The subspaces (TC)^ω form the characteristic distribution under suitable regularity. Its integral leaves represent directions along which the restricted 2-form is degenerate.
Quotienting by those characteristic leaves, when the quotient is smooth and well behaved, can produce a reduced symplectic space.
This language clarifies constrained phase spaces: degeneracy on the constraint surface is not necessarily an error. It can identify the redundant directions that must be quotiented out.
Regularity is again essential. Leaf spaces can be non-Hausdorff or singular, so a formal quotient symbol is not proof that a smooth reduced phase space exists.
21. Contact geometry is an odd-dimensional nonintegrability structure
On a (2n+1)-dimensional manifold Y, a 1-form α is a contact form when
α∧(dα)ⁿ
is nowhere zero. The hyperplane field ξ=ker α is then a contact structure.
Multiplying α by a nowhere-zero function leaves the same hyperplane field ξ, though it generally changes the distinguished Reeb vector field. Thus the contact distribution and a chosen contact form are related but distinct data.
The nonvanishing condition is the opposite of integrability. Frobenius theory says a codimension-one distribution locally tangent to hypersurfaces satisfies an integrability condition; a contact distribution is maximally nonintegrable.
Hutchings’s Math 242 course describes contact geometry as an odd-dimensional partner to symplectic geometry and includes contact structures, Reeb vector fields and contact-type hypersurfaces.
22. Worked contact form on R³
Take coordinates (x,y,z) and α=dz−y dx. Then dα=dx∧dy.
Compute α∧dα=(dz−y dx)∧dx∧dy=dz∧dx∧dy. The second term vanishes because dx repeats. This 3-form is nowhere zero, so α is a contact form.
The contact plane ξ at a point consists of vectors V=a∂_x+b∂_y+c∂_z satisfying c−ya=0. It is spanned, for example, by ∂_y and ∂_x+y∂_z.
These planes twist as y changes. They cannot be the tangent planes to a family of ordinary surfaces, because the contact condition forbids such local integrability.
23. The Reeb vector field is determined by two equations
For a contact form α, the Reeb vector field R_α is uniquely defined by
α(R_α)=1, ι_{R_α}dα=0.
For α=dz−y dx on R³, dα=dx∧dy. The condition ι_R dα=0 forces R to have no ∂_x or ∂_y component. Then α(R)=1 gives R=∂_z.
Changing α to fα for a positive nonconstant function generally changes the Reeb vector field even though ker(fα)=ker α. Reeb dynamics belong to the chosen contact form, not only to the underlying hyperplane field.
Closed Reeb orbits are central objects in contact and symplectic topology. The Weinstein conjecture asserts existence in broad contact settings and has been proved in many important cases, including all closed contact 3-manifolds; the general high-dimensional landscape requires hypotheses and remains an active area.
24. Contact hypersurfaces arise naturally from symplectic manifolds
Suppose a symplectic form is exact, ω=dβ, near a hypersurface Y, and the associated Liouville vector field points transversely through Y. Then the restriction α=β|_Y is a contact form under the standard contact-type hypothesis.
Regular energy levels of Hamiltonian systems often carry contact-type structures when appropriate transversality conditions hold. Reeb flow on the hypersurface can then be related to the Hamiltonian characteristic flow up to reparametrisation.
This does not mean every energy surface is automatically contact type. The Liouville/transversality condition must be verified.
Conversely, a contact manifold (Y,α) has a symplectization R×Y with symplectic form d(eᵗα). The extra real coordinate converts the odd-dimensional contact structure into an even-dimensional exact symplectic structure.
25. Legendrian submanifolds are tangent to the contact distribution
In a (2n+1)-dimensional contact manifold, a Legendrian submanifold has dimension n and its tangent spaces lie inside ξ=ker α.
In contact R³, Legendrian curves satisfy α(γ′)=0. For α=dz−y dx, a parametrised curve (x(t),y(t),z(t)) is Legendrian when ż=y ẋ.
Thus if x=t and y=t², a compatible z is z=t³/3+C. The contact condition converts a geometric tangency requirement into an ordinary differential relation.
Legendrian and Lagrangian objects are related but live in different dimensions and structures. Lagrangians belong to symplectic manifolds; Legendrians belong to contact manifolds.
26. Hamiltonian mechanics is one application, not the whole subject
Classical conservative mechanics motivates much of the language: configuration space Q, phase space T*Q, Hamiltonian H, trajectories from X_H, symmetries from group actions and conserved quantities from moment maps.
But symplectic geometry has developed far beyond a bookkeeping language for mechanics. It studies embeddings, Lagrangian intersections, Hamiltonian diffeomorphism groups, pseudoholomorphic curves, Floer homology and rigidity phenomena with no direct analogue in ordinary volume geometry.
The Berkeley graduate-course descriptions reflect this breadth, moving from Hamiltonian systems into Darboux theory, symplectic quotients, holomorphic curves and global invariants.
Applications to celestial mechanics, optics and dynamical systems must still be interpreted through their modelling assumptions. The mathematical phase-space structure does not by itself prove that omitted forces, dissipation or measurement uncertainty are negligible.
27. Symplectic integrators preserve structure rather than only local accuracy
Numerical methods for Hamiltonian systems can be designed so that the discrete update map is symplectic. Such a method preserves a discrete analogue of the phase-space 2-form.
Symplecticity does not mean the numerical method conserves the exact Hamiltonian at every step. Instead, backward-error analysis often shows that over suitable regimes the numerical trajectory follows the exact flow of a nearby modified Hamiltonian for long times.
This can give superior long-time qualitative behaviour compared with a method having similar local truncation error but no geometric preservation.
A numerical claim therefore needs at least three separate checks: order of accuracy, stability/step-size assumptions and the geometric structure the algorithm actually preserves.
28. Optics and wavefront geometry lead naturally toward contact structures
In geometric optics, rays can be described through Hamiltonian systems on cotangent bundles, while spaces of oriented contact elements and wavefronts naturally carry contact structures.
The mathematics separates a point in physical space from a covector encoding a propagation direction or phase gradient. That is the same tangent/cotangent distinction used throughout the differential-geometry route.
Caustics occur when a projection from a smooth Lagrangian or Legendrian object develops singularities. The object in phase/contact space may remain smooth even while its visible projection folds.
This provides a geometric lesson useful well beyond optics: a singularity in a chosen representation need not be a singularity of the underlying higher-dimensional object.
29. Statistical and quantum bridges use related geometry with different meanings
Symplectic structures also appear in geometric formulations of classical statistical mechanics and in the geometry underlying quantum phase spaces, projective Hilbert spaces and semiclassical analysis.
These bridges require care. A classical symplectic phase point is not a quantum state vector, and a Poisson bracket is not simply the same operation as a quantum commutator. Correspondence principles relate structures under specified limits and quantisation procedures.
BTT’s Quantum Mathematics route owns the finite-dimensional Hilbert-space entry. This article provides geometric background rather than replacing that route.
Keeping ownership clear avoids a common error in advanced exposition: similar-looking algebra does not imply identical physical interpretation.
30. Gromov non-squeezing shows that symplectic geometry is more rigid than volume
In R²ⁿ with n≥2, a ball of radius R cannot be symplectically embedded into a cylinder B²(r)×R²ⁿ⁻² when R>r, even though the cylinder has infinite ordinary volume.
This is Gromov’s non-squeezing theorem. It says that preserving symplectic structure imposes a two-dimensional capacity constraint invisible to volume alone.
A volume-preserving transformation might reshape a region dramatically while preserving total 2n-dimensional measure. A symplectic map must respect additional conjugate-plane structure.
Symplectic capacities abstract this phenomenon by assigning sizes with monotonicity, conformality and normalisation properties. Different capacities can reveal different global symplectic constraints.
Berkeley courses use holomorphic curves as one route into non-squeezing, showing how complex-analytic methods enter global symplectic rigidity.
31. Floer theory turns infinite-dimensional variational problems into topology
Hamiltonian Floer theory studies action functionals whose critical points correspond, in suitable settings, to periodic Hamiltonian trajectories or intersection points.
The differential counts solutions of a nonlinear PDE connecting critical points. Compactness, transversality and orientation issues make the construction substantially deeper than finite-dimensional Morse theory.
The resulting homology can prove existence and intersection results that are invisible to local Darboux coordinates. It is one major tool behind modern forms of Arnold-type fixed-point and Lagrangian-intersection theorems.
This article does not attempt a Floer-homology construction. The correct learning boundary is to understand why local symplectic triviality forces global invariants to carry much of the interesting information.
Hutchings’s Berkeley course materials explicitly place Arnold’s conjecture and Floer homology among advanced topics after the foundational symplectic and contact structures.
32. A dependable symplectic/contact workflow
First identify the manifold dimension and the form. For a proposed symplectic form, check dω=0 and nondegeneracy. For a proposed contact form, check α∧(dα)ⁿ≠0.
State sign conventions before deriving Hamiltonian vector fields, Poisson brackets, canonical forms or moment maps. A single sign reversal can propagate through every later formula while leaving each line internally plausible.
For cotangent-bundle calculations, distinguish base vectors from covectors and identify the canonical 1-form before choosing coordinates. For constraints, test regularity and symmetry before subtracting dimensions.
Keep local and global conclusions separate. Darboux makes every symplectic manifold locally standard, so global topology, periodic orbits, Lagrangian intersections and embedding obstructions require additional evidence.
Finally, return applications to their modelling domain. Preserving a symplectic form is a mathematical property of a model or numerical map; it does not by itself establish the physical adequacy of that model.
33. Independent practice: sixteen questions
- Show that dq∧dp is nondegenerate on R².
- Explain why a nondegenerate alternating form cannot exist on an odd-dimensional vector space.
- For Q=aq,P=p/a, verify that dQ∧dP=dq∧dp.
- Starting from λ=p dq, compute ω=−dλ.
- For H=p²/(2m)+V(q), derive Hamilton’s equations.
- For H=(p²+q²)/2, solve the Hamiltonian system with q(0)=A,p(0)=0.
- Compute {q,p}, {q,H} and {p,H} for the unit harmonic oscillator.
- Prove that H is conserved along its own Hamiltonian flow.
- Show that the graph of df in T*Q is Lagrangian.
- For the rotation vector field X=−p∂_q+q∂_p, compute ι_X(dq∧dp) and find a moment-map component under this article’s convention.
- A free d-dimensional Hamiltonian group action is reduced at a regular central value. What dimension drop should you expect and why?
- For α=dz−y dx on R³, verify the contact condition and compute the Reeb vector field.
- Find z(t) so that γ(t)=(t,t²,z(t)) is Legendrian for α=dz−y dx and z(0)=0.
- Explain why multiplying a contact form by a positive nonconstant function keeps the contact distribution but can change Reeb dynamics.
- State what Gromov non-squeezing says about embedding a ball of radius 2 into a symplectic cylinder whose two-dimensional radius is 1.
- Why does Darboux’s theorem make global invariants especially important in symplectic geometry?
34. Worked solutions and checks
1. Let v=a∂_q+b∂_p. If ω(v,w)=0 for every w, use w=∂_p to get a=0 and w=∂_q to get −b=0. Hence v=0, so the form is nondegenerate.
2. A skew-symmetric n×n matrix A satisfies det A=det(−A)=(−1)ⁿdet A. For odd n this forces det A=−det A, hence det A=0. Nondegeneracy requires nonzero determinant.
3. dQ=a dq and dP=dp/a. Their wedge product is dq∧dp. The map may distort Euclidean lengths while preserving the symplectic form.
4. dλ=dp∧dq=−dq∧dp, so −dλ=dq∧dp.
5. dH=V′(q)dq+(p/m)dp. Writing X_H=a∂_q+b∂_p gives ι_Xω=a dp−b dq. Matching coefficients yields a=p/m and b=−V′(q).
6. The equations are q̇=p,ṗ=−q. Hence q̈+q=0. With q(0)=A and q̇(0)=p(0)=0, q(t)=A cos t and p(t)=−A sin t.
7. {q,p}=1. For H=(p²+q²)/2, {q,H}=p and {p,H}=−q, reproducing the equations of motion.
8. X_H(H)=dH(X_H)=ω(X_H,X_H)=0 because ω is alternating. Thus H is constant along each trajectory on which the autonomous flow is defined.
9. If ι_f:Q→T*Q is q↦(q,df_q), then ι_f*λ=df. Hence ι_f*ω=−d(df)=0. The graph has dimension dimQ, half the dimension of T*Q, so it is Lagrangian.
10. ι_Xω=−p dp−q dq=−d[(q²+p²)/2]. Thus a compatible moment-map component is μ=−(q²+p²)/2. Changing generator or sign convention changes the sign consistently.
11. The regular moment-map level imposes d independent equations, reducing dimension by d. Quotienting the free d-dimensional group orbits removes another d, for a total reduction of 2d.
12. dα=dx∧dy and α∧dα=dz∧dx∧dy, which never vanishes. For R=a∂_x+b∂_y+c∂_z, ι_Rdα=a dy−b dx, so a=b=0. Then α(R)=c=1, hence R=∂_z.
13. The Legendrian condition is ż−y ẋ=0. Here ẋ=1 and y=t², so ż=t². With z(0)=0, z=t³/3.
14. ker(fα)=kerα when f is nowhere zero, so the same hyperplane distribution remains. But the Reeb equations use the particular form and its derivative d(fα)=df∧α+f dα, so the normalisation and characteristic direction generally change.
15. It is impossible. A symplectic ball of radius R cannot be embedded into a standard symplectic cylinder of two-dimensional radius r when R>r. Here 2>1.
16. Darboux says all symplectic forms have the same local model, so local coordinate coefficients cannot distinguish most interesting symplectic manifolds. Topology, embeddings, dynamics, Lagrangian intersections and global invariants must carry the distinguishing information.
35. Questions that protect the final boundaries
Is every area-preserving map symplectic?
In two dimensions with a fixed area form, preserving that form is the symplectic condition. In higher dimensions, preserving total volume is weaker than preserving the symplectic 2-form.
Does a symplectic form define shortest paths?
No. A symplectic form does not define lengths. A compatible almost complex structure can be used with ω to construct a Riemannian metric, but that is additional structure.
Is every symplectic vector field Hamiltonian?
Locally yes under the usual smooth assumptions, because the associated closed 1-form is locally exact. Globally no: its cohomology class can obstruct a global Hamiltonian potential.
Is contact geometry just symplectic geometry in one fewer dimension?
No. They are tightly related but have different defining objects and invariants. Contact structures are maximally nonintegrable hyperplane fields on odd-dimensional manifolds, while symplectic structures are closed nondegenerate 2-forms on even-dimensional manifolds.
What completes R23?
The R23 route now spans concrete curves and surfaces, smooth manifolds, tangent/cotangent/tensor fields, differential forms, Riemannian geometry, connections and curvature, bundle/global-analysis structures, and symplectic/contact geometry. The Atlas completion test still requires audit, bridges and mathematical review before the region is labelled complete for v1.
Sources and further study
The UC Berkeley Geometry/Topology graduate course listing gives the standard Math 242 scope: symplectic linear algebra, manifolds, Darboux, cotangent bundles, Hamiltonian systems, Lagrangians, Poisson brackets and momentum mappings. Michael Hutchings’s Math 242 course page extends the route through contact geometry, Reeb dynamics, Hamiltonian actions and holomorphic curves. Fraydoun Rezakhanlou’s Berkeley course materials likewise place Hamiltonian systems, cotangent bundles, Darboux, contact manifolds and global symplectic rigidity in one progression.
Return to Bundles, Characteristic Structures and Global Analysis for the bundle/global-analysis bridge, Riemannian Geometry and Geodesics for metric geometry, or the BTT Mathematics Hub for another route.