Symplectic geometry

Symplectic geometry studies smooth manifolds equipped with a closed nondegenerate differential two-form.

A presymplectic form is a closed two-form, commonly assumed to have constant rank, that need not be nondegenerate. A symplectic form is its nondegenerate special case. Its kernel distribution is integrable: for kernel fields $X,Y$, the identity $d\omega =0$ gives ${\iota }_{[X,Y]}\omega =0$. If the characteristic leaf quotient is smooth and the form descends, that quotient has a symplectic form. A regular Hamiltonian energy hypersurface carries such a form with kernel spanned by its Hamiltonian vector field.

A symplectic capacity is monotone under symplectic embeddings, scales by $c(X,a\omega )=ac(X,\omega )$ for $a>0$, and has positive unit-ball capacity and finite unit-cylinder capacity. It is invariant under symplectomorphisms. A normalized symplectic capacity additionally assigns $\pi$ to both unit domains.

A normalized symplectic capacity has the displayed unit-ball and unit-cylinder values. Dilation gives $\pi R^2$ for both radius-$R$ domains. Monotonicity then implies the sharp Gromov non-squeezing theorem.

A locally uniform limit of symplectomorphisms that is a smooth diffeomorphism is symplectic. By capacity preservation under uniform limits and linear capacity rigidity, its derivative is symplectic or anti-symplectic. Repeat this argument after taking the product of every map with the identity on a symplectic plane. An anti-symplectic derivative on the original factor and a symplectic derivative on the added factor would give neither sign on the product, a contradiction. The argument uses symplectic capacities in the original and stabilized dimensions.

An invertible real linear map preserving symplectic capacities of all solid ellipsoids is symplectic or anti-symplectic. For a pair with $\omega (u,v)=1$, if $0<|\omega (T^{T}u,T^{T}v)|<1$, complete this pair and its normalized transformed pair to symplectic bases. The resulting transpose matrix sends a unit ball into a symplectic cylinder of radius less than one, contradicting the capacity axioms after iterating the resulting linear map: its first coordinate pair contracts geometrically, while the image ellipsoid retains positive capacity. Perturbation handles a zero pairing; using $T^{-1}$ handles absolute value greater than one. Thus the absolute bilinear pairings agree. Their squares agree as polynomials, and the polynomial ring being an integral domain forces one global sign.

Suppose diffeomorphisms preserving capacities of solid ellipsoids converge uniformly on compact sets to a diffeomorphism $\varphi$. For $0<a<1$, uniform closeness and degree of a continuous mapping give ${\varphi }_j(aE)\subseteq \varphi (E)\subseteq {\varphi }_j(a^{-1}E)$. The capacity axioms squeeze $c(\varphi (E))$ between $a^{2}c(E)$ and $a^{-2}c(E)$. Letting $a$ tend to one proves equality. Rescaling a smooth $\varphi$ near a point and repeating the argument also proves capacity preservation by its derivative.

Geometric quantization associates quantum states and observables with a symplectic manifold. A first step uses a complex line bundle with a connection on a vector bundle whose curvature is proportional to the symplectic form divided by $\hslash$; its existence imposes an integrality condition. A real polarization is an integrable half-dimensional distribution with Lagrangian submanifolds as leaves. Requiring sections to be covariantly constant along the leaves reduces the variables to a configuration-space description. Restricting the curvature to a Lagrangian leaf gives zero, permitting local flat sections; global sections can require trivial holonomy. Exact Lagrangian graphs also describe local semiclassical phases through their generating functions.

A polarization is an involutive complex rank-half distribution on which the symplectic form vanishes. Polarized prequantum sections satisfy ${\nabla }_{X}s=0$ for $X\in \mathcal{F}$. A real polarization is tangent to a Lagrangian foliation; a complex Kähler polarization uses the antiholomorphic tangent spaces. Only observables whose Hamiltonian flows preserve the chosen distribution act directly on the polarized space. Singular leaves, holonomy and changes of polarization are genuine quantization difficulties.

The pairing compares state spaces belonging to two polarizations using their prequantum sections and half-forms. When it defines a suitable identification or projection, a classical Hamiltonian flow can transport a state to its transported polarization and the pairing can return it to the original state space. Differentiating this construction supplies operators for some observables that do not directly preserve the original polarization. Existence, domains and unitarity require additional hypotheses; it is not a universal canonical equivalence of all polarizations.

Tensor the prequantum bundle with a square root of the canonical determinant bundle of a polarization, when such a square root exists. The half-form contributes to the state pairing and to the observable action. It improves coordinate invariance and accounts for familiar Maslov and oscillator zero-point shifts. Its existence is additional global data, not automatic from prequantum integrality.

Prequantization realizes a symplectic manifold using a Hermitian line bundle with connection whose curvature is $i\omega /\hslash$ in the convention ${\iota }_{X_f}\omega =df$. Its existence requires integral symplectic periods. On sections, the Kostant-Souriau prequantum operator represents every classical observable and its Poisson bracket, but the resulting states depend on the full phase space. A polarization in geometric quantization removes this excess of variables.

For ${\iota }_{X_f}\omega =df$ and $F_{\nabla }=i\omega /\hslash$, the curvature commutator and $[X_f,X_g]=-X_{\{f,g\}}$ give $[\hat{f},\hat{g}]=i\hslash \widehat{\{f,g\}}$. In cotangent coordinates, $\theta =pdq$ gives $\hat{f}=-i\hslash X_f+f-\theta (X_f)$. Restriction to a vertical polarization yields $\hat{q}=q$ and $\hat{p}=-i\hslash {\partial }_q$.

A prequantum line bundle is a Hermitian line bundle over a symplectic manifold with unitary connection on a vector bundle of the displayed curvature. With $\omega =-d\theta$, a local connection is $\nabla =d-i\theta /\hslash$. Reversing the Hamiltonian-vector-field convention or passing to the dual line bundle reverses the corresponding curvature convention. Integrality of $\omega /(2\pi \hslash )$ on closed two-cycles is the topological existence condition.

A Hamiltonian group action is a Lie group action on a symplectic manifold with an equivariant moment map $\mu :M\to {\mathfrak{g}}^{\ast }$. In the sign convention ${\iota }_{X_H}\omega =dH$, each component satisfies $d⟨\mu ,X⟩={\iota }_{X_M}\omega$, where $X_M$ is the fundamental vector field. The opposite Hamiltonian vector field convention reverses the sign.

A symplectic Lie group action is weakly Hamiltonian when every fundamental vector field is a Hamiltonian vector field. A Hamiltonian group action additionally supplies Hamiltonians as components of an equivariant moment map. Constants in the component Hamiltonians do not affect their vector fields, but the ability to choose them with the required equivariance is an additional global condition.

For a symplectic left Lie group action, use actual fundamental fields ${\xi }_P$ and ${\iota }_{X_f}\omega =df$. They satisfy $[{\xi }_P,{\eta }_P]=-[\xi ,\eta {]}_P$. Cartan's magic formula then gives ${\iota }_{[\xi ,\eta {]}_P}\omega =d[\omega ({\xi }_P,{\eta }_P)]$. Thus every commutator generator has a globally exact contraction. A semisimple Lie algebra is perfect, so every generator has a global Hamiltonian. The separate semisimple moment-map equivariance result removes the constant two-cocycle obstruction and gives an equivariant moment map for a connected group.

For a Hamiltonian group action, a regular value level preserved by a free proper group action has a smooth quotient. The restricted symplectic form descends to a symplectic form on that quotient when the kernel is exactly the orbit directions; this is symplectic reduction.

At a regular free circle group level of a moment map, the inverse image of a Lagrangian submanifold of the reduced space is a Lagrangian submanifold of the original space. Its restricted symplectic form is the pullback of the reduced form and hence vanishes. Its dimension is one more than that of the reduced Lagrangian, exactly half the ambient dimension. It is a circle bundle over the original Lagrangian.

For a planar cotangent bundle, rotation has moment map $\ell =x{p}_y-y{p}_x$. At $\ell \ne 0$ its level is regular and the circle action is free. Polar canonical coordinates give $\omega =dr\wedge d{p}_r+d\theta \wedge d\ell$. Fixing $\ell$ and quotienting $\theta$ leaves $T^{\ast }{\mathbb{R}}_{>0}$ with the displayed form. The unit-mass free Hamiltonian becomes $p_r^2/2+{\ell }^2/(2r^2)$.

For $H={\sum }_j(q_j^2+p_j^2)/2$ and $E>0$, the oscillator flow is a free circle action on the sphere of radius $\sqrt{2E}$. Its quotient is Complex projective space. For two modes, the reduced form in $w=z_2/z_1$ is $iEdw\wedge d\bar{w}/(1+|w{|}^2{)}^2$, of total area $2\pi E$. Reduction at zero energy is singular instead of a smooth quotient of the asserted dimension.

For a Hamiltonian group action, let $c$ be a regular value of the moment map fixed by the coadjoint action, and assume the action on ${\mu }^{-1}(c)$ is free and proper. The quotient is a symplectic manifold of dimension $\dim M-2\dim G$. Its unique reduced symplectic form is characterized by ${\pi }^{\ast }{\omega }_c={\iota }^{\ast }\omega$, where $\iota$ includes the level set and $\pi$ is the quotient projection. The kernel of the restricted two-form consists precisely of the orbit directions. More general noncentral levels use their stabilizer group; the stated central-level version suffices for circle group reduction.

Restriction to a Lie subgroup with Lie algebra inclusion $i:\mathfrak{h}\to \mathfrak{g}$ has moment map $i^{\ast }\mu$. The dual map $i^{\ast }$ composes with the original moment map.

A moment map assigns to each element of a Lie algebra a Hamiltonian function for its fundamental vector field, and is equivariant for the coadjoint action. For an abelian Lie group, constant shifts are allowed; the numerical scale also depends on the chosen Lie algebra generators and the normalization of the symplectic form.

On the canonical cotangent bundle $T^{\ast }{\mathbb{R}}^n$, $\omega =i\sum d{z}_j\wedge d{\bar{z}}_j$ and $H=z^{†}z$. The unitary group acts by $z\mapsto Uz$, preserving both. For a Hermitian $A$, the generator $-iA$ has Hamiltonian ${\mu }_A=z^{†}Az$. The Poisson bracket satisfies $\{{\mu }_A,{\mu }_B\}={\mu }_{-i[A,B]}$, with no central term. The central generator has moment map $H$; imaginary antisymmetric matrices give ordinary angular momentum, while real symmetric matrices give quadratic observables mixing position and momentum.

For a weakly Hamiltonian left Lie group action on a connected symplectic manifold, choose ${\iota }_{X_f}\omega =df$. Fundamental vector fields and Hamiltonian vector fields are both antihomomorphisms, so the displayed difference is constant. The Jacobi identity for the Poisson bracket makes it a two-cocycle. Changing ${\mu }_X$ by $b(X)$ changes the difference by $-b([X,Y])$. Thus its class in Lie algebra cohomology obstructs choosing an equivariant moment map. For a connected group, vanishing of this class gives infinitesimal and group equivariance.

Suppose a symplectic Lie group action on a connected symplectic manifold has global component Hamiltonians and a common fixed point $p_0$. Choose their constants linearly so ${\mu }_{\xi }(p_0)=0$. The difference $c(\xi ,\eta )=\{{\mu }_{\xi },{\mu }_{\eta }\}-{\mu }_{[\xi ,\eta ]}$ has zero Hamiltonian vector field because fundamental fields and Hamiltonian fields have compatible Lie-bracket conventions; it is therefore constant. At $p_0$, all fundamental fields vanish, so both terms in $c$ vanish. Thus $c=0$ everywhere and the Poisson brackets of the components reproduce the group Lie algebra. For a connected group this also yields coadjoint equivariance. A linear unitary action on a vector space has the origin as such a fixed point.

For a semisimple Lie algebra, its nondegenerate Killing form turns a two-cocycle into an operator $D$ through $B(DX,Y)=\kappa (X,Y)$. The cocycle identity makes $D$ a derivation. Choose $Z$ by $B(Z,X)=\mathrm{tr}(D\mathrm{ad}X)$. Using $[D,\mathrm{ad}X]=\mathrm{ad}(DX)$ and cyclicity of trace gives $B([Z,X],Y)=B(DX,Y)$, so $D=\mathrm{ad}Z$. Consequently $\kappa (X,Y)=B(Z,[X,Y])$ is a coboundary. This proves that component Hamiltonians can be normalized to a Lie algebra homomorphism.

For the symplectic form $\omega =idz\wedge d\bar{z}/(1+|z{|}^2{)}^2$, the circle group rotates $z$ by $e^{i\theta }$. Its real generator is $K=iz{\partial }_z-i\bar{z}{\partial }_{\bar{z}}$. The contraction is ${\iota }_K\omega =-dH$, so $H$ is a Hamiltonian function in this sign convention. In the other chart $w=1/z$, $H=1/(1+|w{|}^2)$, proving smoothness at infinity. Its image is $[0,1]$ with the endpoints at the fixed poles. The symplectic area is $2\pi$; rescaling to the round area $4\pi$ would double this moment function.

An anti-symplectic map between symplectic manifolds is a diffeomorphism $f$ satisfying $f^{\ast }{\omega }^{\prime }=-\omega$. For symplectic vector spaces, the analogous definition uses an invertible linear map.

An anti-symplectic involution is an anti-symplectic map whose square is the identity. Its nonempty fixed-point set is a Lagrangian submanifold; the linear version has complementary $+1$ and $-1$ Lagrangian subspaces.

A nonempty fixed-point set of an anti-symplectic involution is a Lagrangian submanifold. Local linearization identifies its tangent space with the $+1$ eigenspace of the derivative.

The real symplectic group consists of matrices $A$ satisfying $A^{T}JA=J$. It is an embedded matrix Lie group of dimension $2n^2+n$ in size $2n$.

For a finite-dimensional vector space over any field with a nondegenerate alternating bilinear form $B$, the symplectic group over a field consists of invertible linear maps preserving $B$. In a symplectic basis its matrices satisfy $g^{\mathsf{T}}Jg=J$. Matching two symplectic bases gives a unique such linear map, and a symplectic basis exists by successively splitting off paired vectors. This includes arbitrary-characteristic and infinite-field versions, beyond the real Lie group and the symplectic group over a finite field.

An invertible real matrix preserving the standard symplectic form. With $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$, it satisfies $A^{T}JA=J$, equivalently $AJ{A}^T=J$. The first identity describes preservation of the form; the second describes preservation of coordinate Poisson brackets.

The stabilizer of a totally isotropic $k$-space in a finite symplectic space also stabilizes its orthogonal complement. Its block decomposition has a unipotent normal subgroup $Q$ and a block-diagonal complement $L\cong {\mathrm{GL}}_k(q)\times {\mathrm{Sp}}_{2m-2k}(q)$. The paired $k$-space carries the contragredient action. A row-vector convention gives upper-triangular blocks; a column-vector convention transposes the arrangement. The unipotent radical count for a symplectic parabolic subgroup yields its order.

With a Lagrangian subspace first in a paired symplectic basis, its stabilizer consists of $\left(\begin{array}{cc}A& AS\\ 0& A^{-\mathsf{T}}\end{array}\right)$ with $S$ symmetric. The additive symmetric-matrix unipotent radical has order $q^{m(m+1)/2}$; the Levi subgroup is $G{L}_m(q)$, acting by $S\mapsto AS{A}^{\mathsf{T}}$. Its index in $S{p}_{2m}(q)$ is ${\prod }_{i=1}^m(q^i+1)$, the number of Lagrangian subspaces.

For a line $⟨e⟩$, complete $(e,f)$ to a symplectic basis with complementary symplectic space $W$. The unipotent radical has elements $u(b,t)$ sending $e$ to $e$, $w$ to $w-B(w,b)e$ and $f$ to $f+b+te$. Its multiplication is $(b,t)(b^{\prime },t^{\prime })=(b+b^{\prime },t+t^{\prime }+B(b,b^{\prime }))$, giving order $q^{2m-1}$. Its commutator is central with parameter $2B(b,b^{\prime })$. It is Heisenberg-type in odd characteristic and abelian in characteristic two. The full stabilizer also scales $e,f$ inversely and acts symplectically on $W$.

In row-block order $(W^{\prime },U,W)$ with dual bases, a unit-diagonal symplectic flag stabilizer satisfies $E=J_U{D}^T$ and $F-F^T=D{J}_U{D}^T$. The matrix $D$ is arbitrary. The latter right side is alternating, so each off-diagonal pair of entries of $F$ gives one free choice and each diagonal entry is free. ConsequentlyReversed dual-basis order inserts a reversal matrix but does not change the count. The argument holds in characteristic two without division by two.

Counting successive symplectic pairs givesFor each nonzero first vector there are $q^{2m-1}$ partners with pairing one, after which its nondegenerate plane splits off. The exact defining-characteristic part is $q^{m^2}$, realized by an upper symplectic unitriangular subgroup.

The map $F:M_{2n}(\mathbb{R})\to {\mathrm{Skew}}_{2n}(\mathbb{R})$ given by $F(A)=A^{T}JA$ has $Sp(2n,\mathbb{R})=F^{-1}(J)$. At a symplectic $A$,is surjective, so the regular level set theorem makes the symplectic group an embedded submanifold.

At the identity, the symplectic Lie algebra consists of $X$ satisfying $X^{T}J+JX=0$; tangent spaces elsewhere are its left translates.

A Poisson manifold is a smooth manifold with a bilinear bracket on smooth functions that is antisymmetric, obeys the Jacobi identity, and is a derivation in each argument. Its Poisson bivector may have nonconstant rank, unlike the inverse of a symplectic form.

A uniform magnetic field defines a constant Poisson bivector on momentum space by the displayed components. For the Hamiltonian $E(\mathbf{p})$ and convention $\dot{f}=\{f,E\}$, it gives $\dot{\mathbf{p}}=e\nabla E\times \mathbf{B}$. Constancy makes every term in the coordinate Jacobi identity vanish. If $e\mathbf{B}\ne 0$, the symplectic leaves are the planes $\mathbf{B}\cdot \mathbf{p}=c$, and energy is conserved along the flow. The whole three-dimensional momentum space is not symplectic, and an energy surface need not itself be a Poisson submanifold.

The symplectic foliation of a Poisson manifold consists of maximal connected integral manifolds of its Hamiltonian distribution. Each leaf has a nondegenerate closed two-form inverse to the restricted Poisson structure, with compatible sign conventions. Ranks can differ between leaves, making this a singular foliation. Hamiltonian flows remain within a leaf, and Casimir functions of a Poisson manifold are constant along every leaf; a Casimir level set need not be a single leaf.

A smooth map $r:M\to N$ between Poisson manifolds is a Poisson map when $\{f\circ r,g\circ r{\}}_M=\{f,g{\}}_N\circ r$ for all smooth functions on $N$. When a Lie group acts by Poisson automorphisms and the quotient is smooth, brackets of invariant functions descend and make the quotient projection Poisson. The quotient can be Poisson without being symplectic.

A Poisson tensor is a smooth bivector field defining $\{f,g\}=\pi (df,dg)$ whose bracket obeys the Jacobi identity. In coordinates, ${\pi }^{i\ell }{\partial }_{\ell }{\pi }^{jk}+{\pi }^{j\ell }{\partial }_{\ell }{\pi }^{ki}+{\pi }^{k\ell }{\partial }_{\ell }{\pi }^{ij}=0$. Its Hamiltonian distribution integrates into symplectic leaves, even though its rank can vary across the manifold. The tensor specifies a bracket without requiring a globally nondegenerate symplectic form.

A Poisson structure is an antisymmetric smooth bivector field whose induced function bracket satisfies the Jacobi identity. In coordinates its condition is the vanishing cyclic sum of ${\omega }^{dc}{\partial }_d{\omega }^{ab}$. Applying Jacobi to coordinate functions proves necessity; expanding arbitrary brackets cancels the second derivatives and proves sufficiency. A function bracket-commuting with every function is a Casimir function of a Poisson manifold.

For field functionals, a Poisson operator $J$ is an operator whose displayed pairing of variational derivatives defines a Poisson bracket. It must be skew under the integral pairing and satisfy the functional Jacobi identity. A constant skew differential operator, such as $J={\partial }_x$ with periodic or vanishing boundary conditions, gives the simplest examples. The Hamiltonian evolution is $u_t=J(\delta H/\delta u)$.

Two Poisson structures are compatible when every linear combination ${\Pi }_1+\lambda {\Pi }_2$ is a Poisson structure. This family is a Poisson pencil. In terms of the Schouten-Nijenhuis bracket, compatibility means $[{\Pi }_1,{\Pi }_2]=0$, in addition to the two individual Jacobi identities. Constant Poisson bivectors are compatible. Over a real manifold, complex parameter values refer to the complexification of its function algebra.

Start from a Casimir function $H_0$ of the first Poisson structure. Whenever possible, choose $H_n$ whose first Hamiltonian vector field is minus the second Hamiltonian vector field of $H_{n-1}$. A sequence extending indefinitely is the coefficient sequence of a formal Casimir function for the Poisson pencil. Its functions pairwise commute for both Poisson brackets. Existence at each stage is an additional solvability requirement; compatibility alone does not guarantee it.

The next step of a Lenard-Magri recursion requires $-{\Pi }_{2}d{H}_{n-1}$ to be a Hamiltonian vector field for ${\Pi }_1$. It must be tangent to the symplectic leaves, and its contraction with the leaf symplectic form must have a global potential, with a compatible smooth extension between leaves. Even the first step may fail: on ${\mathbb{R}}^2$, take ${\Pi }_1=0$, ${\Pi }_2={\partial }_x\wedge {\partial }_y$ and $H_0=x$. Both Poisson structures are compatible, but ${\Pi }_{1}d{H}_1=-{\Pi }_{2}dx$ cannot hold, because its left side is zero and its right side is nonzero.

For a Lenard-Magri recursion, antisymmetry shifts an index: $\{H_i,H_j{\}}_1=\{H_{i-1},H_{j+1}{\}}_1$ when $i\ge 1$. Repeating reaches $\{H_0,H_{i+j}{\}}_1=0$, because $H_0$ is a Casimir function. The recursion then gives $\{H_i,H_j{\}}_2=-\{H_{i+1},H_j{\}}_1=0$. This proof applies to every finite initial portion for the indices that can be shifted inside it, and to all pairs in an infinite sequence.

The dual space ${\mathfrak{g}}^{\ast }$ of a finite-dimensional Lie algebra has this canonical Poisson bracket. For a basis $v^i$, linear coordinates $x^i(\ell )=\ell (v^i)$ give $\{x^i,x^j\}=c_k^{ij}{x}^k$. The Lie algebra's Jacobi identity is precisely the coordinate Jacobi condition for a Poisson bivector. This construction is intrinsic on the dual space; writing the same formula in an arbitrary manifold chart does not by itself provide a compatible global atlas.

A smooth antisymmetric contravariant two-tensor defines a biderivation $\{F,G\}=\omega (dF,dG)$. It is a Poisson bivector when this bracket satisfies the Jacobi identity. Its rank may vary; where it is nondegenerate it inverts to a symplectic form. The coordinate Jacobi condition for a Poisson bivector makes the integrability constraint explicit.

A nondegenerate Poisson bivector inverts to a closed differential form of degree two. Differentiating $S=P^{-1}$ gives ${\partial }_{d}S=-S({\partial }_{d}P)S$. Contracting the coordinate Jacobi condition for a Poisson bivector with three copies of $S$ yields ${\partial }_i{S}_{jk}+{\partial }_j{S}_{ki}+{\partial }_k{S}_{ij}=0$, which is precisely closure of the resulting symplectic form. An overall sign in the inverse convention does not change closure.

The Jacobi identity for a Poisson bivector reduces to the displayed cyclic identity in every local chart. Necessity follows by applying the bracket to coordinate functions. Sufficiency follows because the second-derivative terms in an arbitrary Jacobiator cancel by antisymmetry, leaving this coefficient multiplying first derivatives of the three functions.

A Casimir function $C$ Poisson-commutes with every smooth function, $\{C,f\}=0$. Its regular level sets are unions of symplectic leaves.

A symplectic leaf is a maximal connected immersed submanifold on which the Poisson bivector has constant full rank and therefore inverts to a symplectic form.

The rotational Lie-Poisson bracket on ${\mathbb{R}}^3$ isIts Casimir is $x_1^2+x_2^2+x_3^2$, and every sphere centered at the origin is a two-dimensional symplectic leaf on which $SO(3)$ acts symplectically.

On ${\mathbb{R}}^{2n}$ let $J_0$ be the standard complex structure, let ${\omega }_0$ be the standard symplectic form, and let $g_0(u,v)={\omega }_0(u,J_{0}v)$ be the Euclidean inner product. A linear map preserving any two of $J_0$, ${\omega }_0$, and $g_0$ preserves the third. Equivalently,

A Lagrangian subspace of a $2n$-dimensional symplectic vector space $(V,\omega )$ is an $n$-dimensional subspace $L$ on which $\omega$ vanishes.

A Lagrangian complement to $L$ in a symplectic vector space is a Lagrangian subspace $K$ with $E=L\oplus K$. Such complements exist smoothly for Lagrangian subbundles. Identify any initial complement with $L^{\ast }$ using the symplectic cross-pairing. If its internal pairing is $b(\eta ,\zeta )$, correct its lift by a vector $a\eta \in L$ with $\zeta (a\eta )=-b(\eta ,\zeta )/2$. The corrected complement has zero internal pairing.

The Lagrangian Grassmannian is the space of all Lagrangian subspaces of a symplectic vector space. The unitary group acts transitively on $\mathrm{LGr}({\mathbb{R}}^{2n})$, and the stabilizer of ${\mathbb{R}}^n$ is the orthogonal group $O(n)$, giving

Relative to a fixed Lagrangian subspace $L_0$, the Maslov cycle consists of Lagrangian subspaces $L$ with $L\cap L_0\ne 0$. In a two-dimensional symplectic space all lines are Lagrangian, and the cycle is the single line $L_0$ in the projective line. Fibrewise versions yield sections of a projective-line bundle and record passage through a prescribed vertical direction.

For a surface thermostat, the planes containing $F$ form the projective bundle of $T(SM)/\mathbb{R}F$. The vertical Maslov cycle is the section selecting the image of the vertical vector field of a surface unit tangent bundle. It is a smooth hypersurface. In the local projective chart $\mathbb{R}F+\mathbb{R}(V+rH)$, it is $r=0$; the lifted flow satisfies $\dot{r}=1$ on the cycle, hence crosses it transversely.

The Maslov index assigns an integer to a loop of Lagrangian subspaces. Under $\mathrm{LGr}({\mathbb{R}}^{2n})\cong U(n)/O(n)$, it is the degree of the map induced by $A\mapsto \det (A{)}^2$.

The Maslov map is well defined because every matrix in $O(n)$ has determinant $\pm 1$. The loopmaps to $e^{2\pi it}$ and therefore has Maslov index one.

For a rank-$2n$ symplectic vector bundle $E\to B$, the Lagrangian Grassmannian bundle has fiber $\mathrm{LGr}(E_b)$ over $b$. A symplectic trivialization of $E$ induces a trivialization $\mathcal{L}Gr(E)\cong B\times \mathrm{LGr}({\mathbb{R}}^{2n})$.

If $E\to B$ is an oriented real plane bundle, then $\mathcal{L}Gr(E)=\mathbb{P}(E)$ is an oriented circle bundle. Its transition rotations have twice the angle of those of $E$, so

A symplectic manifold is an even-dimensional smooth manifold $M$ equipped with a closed nondegenerate differential two-form $\omega$.

A vector field on a symplectic manifold is symplectic when its flow preserves the symplectic form. Infinitesimally this is ${\mathcal{L}}_X\omega =0$. By Cartan's magic formula it is equivalent to $d({\iota }_X\omega )=0$. A Hamiltonian vector field has an exact contraction with the form, so is symplectic. On the standard two-dimensional torus, ${\partial }_x$ contracts to the closed but nonexact form $dy$, providing a symplectic non-Hamiltonian example.

A coisotropic submanifold $N$ satisfies $(T_{p}N{)}^{\omega }\subset T_{p}N$, where the superscript denotes the symplectic orthogonal complement. Every hypersurface of a symplectic manifold is coisotropic.

For a hypersurface $N$ of a symplectic manifold, the restriction of the symplectic form has a one-dimensional kernel, its characteristic line field. On a regular value level $N=H^{-1}(c)$, this line is generated by the Hamiltonian vector field of $H$.

An isotropic submanifold has every tangent space an isotropic subspace of a symplectic vector space. Its dimension is at most half the ambient dimension, and equality makes it a Lagrangian submanifold.

The symplectic form on a $2n$-dimensional symplectic manifold determines the positive volume form ${\omega }^n/n!$ and its total integral, the symplectic volume, using the induced orientation.

The top-degree differential form ${\omega }^n/n!$ on a $2n$-dimensional symplectic manifold. Every canonical transformation preserves this form. For $n>1$, preservation of this volume alone does not imply preservation of the symplectic form.

A symplectic form on a $2n$-dimensional smooth manifold determines the orientation in which the nowhere-zero volume form ${\omega }^n/n!$ is positive. Consequently every symplectic manifold is orientable. In dimension two this forbids a symplectic form on the Möbius strip.

An exact symplectic manifold has a symplectic form that is an exact differential form, written here $\omega =-d\lambda$. A positive-dimensional example without boundary cannot be compact, by the Generalized Stokes theorem applied to the top wedge product of differential forms.

For an exact symplectic manifold with $\omega =d\lambda$, the Liouville vector field is defined by ${\iota }_Z\omega =\lambda$. Cartan's magic formula gives ${\mathcal{L}}_Z\omega =\omega$. On a cotangent bundle using $\omega =-d\alpha$, take $\lambda =-\alpha$; then $Z={\sum }_i{p}_i{\partial }_{p_i}$ generates the cotangent fiber-dilation flow.

The vertical Liouville vector field on a cotangent bundle has complete flow $D_s(x,p)=(x,e^{s}p)$. Its pullbacks scale both the Liouville one-form and the canonical symplectic form by $e^s$. As $s\to -\infty$, each orbit tends to its base point on the zero section. This contraction proves rigidity of cotangent Liouville-form preservation.

On a nonempty compact $2n$-dimensional symplectic manifold without boundary, with $n\ge 1$, the class $[\omega {]}^n$ in top-degree de Rham cohomology is nonzero, since its integral in the symplectic orientation is positive. In particular $\omega$ cannot be an exact differential form: if $\omega =d\alpha$, then ${\omega }^n=d(\alpha \wedge {\omega }^{n-1})$ and the Generalized Stokes theorem would make that integral zero. Thus vanishing second de Rham cohomology obstructs a closed positive-dimensional symplectic manifold.

A symplectic form is a closed differential form of degree two whose value on every tangent space is a nondegenerate alternating bilinear form. In canonical position-momentum coordinates it is ${\sum }_{i}d{q}_i\wedge d{p}_i$.

The standard symplectic form on ${\mathbb{R}}^{2n}$ is