III.88 Symplectic Manifolds

Gabriel P. Paternain

Symplectic geometry is the geometry that governs classical physics, and more generally plays an important role in helping us to understand the actions of groups on manifolds. It shares some features with Riemannian geometry and complex geometry, and there is an important special class of manifolds, the Kähler manifolds, in which all three geometric structures are unified.

1 Symplectic Linear Algebra

Just as RIEMANNIAN GEOMETRY [I.3 §6.10] is based on EUCLIDEAN GEOMETRY [I.3 §6.2], symplectic geometry is based on the geometry of the so-called linear symplectic space (²ⁿ, ω₀).

Given two vectors v = (q, p) and v′ = (q′ , p′) in the plane ², the signed area ω₀(v , v′) of the parallelogram spanned by v and v′ is given by the formula

It can also be written using matrices and inner products as ω₀(v, v′) = v′ · Jv, where J is the 2 x 2 matrix

If a linear transformation A : ² → ² is area preserving and orientation preserving, then ω₀(Aυ,Aυ′) = ωo(υ, υ′) for every υ and υ′.

Symplectic geometry studies two-dimensional signed area measurements like this, as well as transformations that preserve these measurements, but it applies to general spaces of dimension 2n rather than just to the plane.

If we split ²ⁿ up as ⁿ x ⁿ, then we can write a vector υ in ²ⁿ as v = (q, p), where q and p each belong to ⁿ. The standard symplectic form ω₀ : ²ⁿ x ²ⁿ → is defined by the formula

ω₀(υ,υ′) = p · q′ - q · p′,

where “·” denotes the usual inner product in ⁿ. Geometrically, ω₀(υ, υ′) can be interpreted as the sum of the signed areas of the parallelograms spanned by the projections of υ and υ′ to the q_ip_i-planes. In terms of matrices, we can write

where J is the 2n × 2n matrix

and I is the n × n identity matrix.

A linear map A : ²ⁿ → ²ⁿ that preserves the product ω₀ of any two vectors (that is, ω₀(Av,Av′) = ω₀(v, v′) for all v, v′ ∈ ²ⁿ) is called a symplectic linear transformation; equivalently, a 2n × 2n matrix A is symplectic if and only if A^TJA = J, where A^T is the transpose of A. Symplectic linear transformations are to symplectic geometry as rigid motions are to Euclidean geometry. The set of all symplectic linear transformations of (²ⁿ, ω₀) is one of the classical LIE GROUPS [III.48 §1] and is denoted by Sp(2n). One can show that symplectic matrices A ∈ Sp(2n) always have DETERMINANT [III.15] 1, and are thus volume preserving. However, the converse does not hold when n 2. For instance, if n = 2, the linear map

(q₁, q₂, p₁, p₂) → (aq₁, q₂/a, ap₁, p₂/a)

has determinant 1 for any a ≠ 0, but it is symplectic only if a² = 1.

The standard symplectic form ω₀ has three properties worth noting. First, it is bilinear: the expression ω₀(υ,υ′) varies linearly in υv′ when υ′ is held fixed, and vice versa. Second, it is crntisymmetric: we have ω₀(υ, υ′) = -(υ′, υ) for all υ and υ′, and in particular ω₀(υ, υ) = 0. Finally, it is nondegenerate, which means that for every nonzero υ there is a nonzero υ′ such that ω₀(υ, υ′) ≠ 0. The standard symplectic form ω₀ is not the only form that obeys these three properties; however, it turns out that any form with these three properties can be converted into the standard form ω₀ after an invertible linear change of variables. (This is a special case of Dgrboux’s theorem.) Thus (²ⁿ, ω₀) is essentially the “only” linear symplectic geometry in 2n dimensions. There are no symplectic forms in odd-dimensional spaces.

2 Symplectic Diffeomorphisms of (²ⁿ, ω₀)

In Euclidean geometry, all rigid motions are automatically linear (or affine) transformations. However, in symplectic geometry there are many more symplectic maps than just the symplectic linear transformations. These nonlinear symplectic maps in (²ⁿ, ω₀) are one of the principal objects of study in symplectic geometry.

Let U ⊂ ²ⁿ be an open set. Recall that a map : U → ²ⁿ is called smooth if it has continuous partial derivatives of all orders. A diffeomorphism is a smooth map with smooth inverse.

A smooth nonlinear map : U → ²ⁿ is said to be symplectic if, for every x ∈ U, the Jacobian matrix ′ (x) of first derivatives of is a symplectic linear transformation. Informally, a symplectic map is one that behaves like a symplectic linear transformation at infinitesimally small scales. Since symplectic linear transformations have determinant 1, we can conclude using several-variable calculus that a symplectic map is always locally volume preserving and locally invertible; roughly speaking, this means that the map : A →, (A) is invertible whenever A is a sufficiently small subset of U, and (A) has the same volume as A. However, the converse is not true when n 2; the class of symplectic maps is much more restricted than that of volume-preserving maps. In fact, Gromov’s non-squeezing theorem (see below) shows how striking this difference can be.

Symplectic maps have been around for quite a long time in Hamiltonian mechanics under the name of canonical transformations. We briefly explain this in the next subsection.

2.1 Hamilton’s Equations

How can we produce nonlinear symplectic maps? Let us begin by exploring a familiar example. Consider the motion of a simple pendulum with length l and mass m and let q(t)be the angle it makes with the vertical at time t. The equation of motion is

where g is the acceleration due to gravity. If we define the momentum as = ml² , then we may transform this second-order differential equation into a first-order system in the phase plane ², namely

where the vector field X : ² → ² is given by the formula X(q,p) = (p/ml², -mglsin q). For each (q(0), p(0)) ² there is a unique solution (q(t), p(t)) to (3) with initial condition (q(0), p(0)). Then for any fixed time t we obtain an evolution map (or flow) t : ² → ² given by t(q(0), p(0)) = (q(t), p(t)), which has the remarkable property of being area preserving. This can be deduced from the observation that X is divergence free, or in other words that

In fact, for every time t, _t is a symplectic map on (², ω₀).

More generally, any system in classical mechanics with finitely many degrees of freedom can be reformulated in a similar fashion, so that the evolution maps _t are always symplectic maps; in this context, they are also known as canonical transformations. The Irish mathematician WILLIAM ROWAN HAMILTON [VI.37] showed us how to do this in general more than 170 years ago. Given any smooth function H :_2n → (called the Hamiltonian), the system of first-order differential equations given by

will (under some mild growth assumptions on H, which we ignore here) give rise to evolution operators t : ²ⁿ → ²ⁿ, which are symplectic maps on (²ⁿ, ω₀) for every time t. To see the connection with the symplectic form ω₀, observe that we may rewrite (4) and (5) in the following equivalent form:

where ∇ H is the usual GRADIENT [I.3§5.3] of H and J was defined in (2). From (6), (1), and the antisymmetry property of ω₀, it is then not difficult to verify that _t is symplectic for every t (the main trick is to compute the derivative of ω₀ ((x)v,(x)v′) n t and check that it equals zero).

We have already pointed out that symplectic maps are volume preserving. The preservation of volume by Hamiltonian systems (a result known as Liouville’s theorem) attracted considerable attention in the nineteenth century and it was a driving force in the development of ERGODIC THEORY [V.9], which studies recurrence properties of measure-preserving transformations.

Symplectic maps or canonical transformations play an important role in classical physics, as they allow one to replace a complicated system by an equivalent system that is simpler to analyze.

2.2 Gromov’s Nonsqueezing Theorem

What is the difference between a symplectic map and a volume-preserving map? In order to answer this question, suppose that we have two connected open sets U and V in ²ⁿ and that we wish to embed one into the other using a symplectic map. This means that we are looking for a symplectic map : U → V such that is a homeomorphism onto its image. We know such a must be volume preserving, so we clearly have the restriction that the volume of U should be at most the volume of V, but is this restriction all that matters? Consider the open ball B(R) = {x ²ⁿ : |x| < R}, which has radius R and center at the origin, and clearly has finite volume. It is not hard to embed it symplectically into the infinite-volume cylinder given by

C(r) = {(q, p) ²ⁿ : + < r²},

whatever the values of R and r. Indeed, the linear symplectic map

(q,p) (aq1, aq2, q3, . . . , q_n, p1/a,p2/a,p3, . . . , pn)

will do the trick when a is sufficiently small and positive. However, the situation is radically different if instead we consider the infinite-volume cylinder

Z(r) = {(q, p) ²ⁿ : + <r²}.

We could try with a similar linear map like

(q,p) (aqb₁,q₂/a,q₃, . . . ,q_n,ap₁,p₂/a,p₃, . . . ,p_n).

This map is volume preserving (it has determinant 1) and for a small it embeds B(R) into Z(r). However, it is symplectic only if a = 1, so it will give a symplectic embedding only if R ≤ r. One is tempted to think that if R > r, then there should still be a nonlinear symplectic embedding squeezing B(R) into Z(r), but a remarkable theorem of Gromov from 1985 asserts that it is not possible to find such a map.

In spite of this deep result of Gromov, and other results that followed it, we still do not know much about how sets in ²ⁿ embed into one another.

3 Synrplectic Manifolds

Recall from DIFFERENTIAL TOPOLOGY [IV.7] that a manifold of dimension d is a TOPOLOGICAL SPACE [III.90] such that each point has a neighborhood that is homeomorphic to an open set in Euclidean space ^d. One can think of ^d as a local model for this manifold, in the sense that it describes what the manifold looks like at very small distance scales. Recall also that a smooth manifold is one for which the “transition functions” are smooth. This means that if ψ : U → ^d and φ : V → ^d are coordinate charts, then the transition function ψ φ^-1 between the open sets φ (U ∩ V) and ψ(U ∩ V) is smooth.

A symplectic manifold is defined similarly, but now the local model is the linear symplectic space (²ⁿ, ω₀). More precisely, a symplectic manifold M is a manifold of dimension 2n that can be covered with domains of coordinate charts whose transition functions are symplectic diffenmorphisms of (²ⁿ, ω₀).

Of course, any open set in (²ⁿ, ω₀) is a symplectic manifold. An example of a compact symplectic manifold is the torus ²ⁿ, which is obtained as the quotient of ²ⁿ by the action of ²ⁿ. In other words, two points x,y ²ⁿ are equivalent if x - y has integer coordinates. Other important examples of symplectic manifolds include RIEMANN SURFACES [III.79], complex PROJECTIVE SPACE [III.72], and cotangent BUNDLES [IV.6 § 5]. However, it is a wide open problem to determine, given a compact manifold, whether it can be assigned a system of coordinate charts that makes it symplectic.

We have seen that in (²ⁿ, ω₀), one can assign an ²ⁿ “area” ω₀ (v, v′) to any parallelogram in the space ²ⁿ. In a symplectic manifold M, one can similarly assign an area ω_p (v v′), but only to infinitesimal parallelograms based at a point p M. The axes of such a parallelogram are two infinitesimal vectors (or more precisely tangent vectors) υ and υ′ . There is a unique way to do this so that all the coordinate charts for M are symplectic diffeomorphisms. In the language of DIFFERENTIAL FORMS [III.16], the map ω is an antisymmetric nondegenerate 2-form, which can then be used to compute the “area” ∫_s ω of nnrinfintesima1 two-dimensional surfaces S in M. One can show that for any sufficiently small closed surface S, the integral ∫_sω vanishes, so ω is a closed form. Indeed, one can define a symplectic manifold more abstractly (without reference to charts) as a smooth manifold equipped with a closed, antisymmetric nondegenerate 2-form ω; a classical theorem of Darboux asserts that this abstract definition is equivalent to the more concrete definition using coordinate charts.

Finally, a special class of symplectic manifolds is given by Kähler manifolds. These are symplectic manifolds that are also complex manifolds, in such a way that the two structures are naturally compatible, a condition that generalizes the relationship (1). Observe that if one identifies points (q, p) in ²ⁿ with points p + iq in ⁿ, then the linear transformation J : ²ⁿ → ²ⁿ becomes the operation of multiplication by i:

J: (z₁ , . . . , z_n) (iz₁ , . . . , iz_n).

Thus the identity (1) relates the symplectic structure (as given by ω₀), the complex structure (as given by J), and the Riemannian structure (as given by the dot product “·“). A complex manifold is a manifold that at small distance scales looks like regions of ⁿ, with the transition functions required to be HOLQMORPWC [I.3 §5.6]. (A smooth map f : U ⊂ ⁿ → ⁿ is said to be holomorphic if each coordinate component of f (z_1, , . . . , z_n) is holomorphic in each variable z_k.) On a complex manifold we can multiply tangent vectors by i. This gives us at each point p M a linear map J_p such that v = -v for all tangent vectors v at p. A Kähler manifold is a complex manifold M with a symplectic structure ω (which computes signed areas of infinitesimal parallelograms) and a Riemannian metric g (which computes an inner product g(υ, υ′) of any two tangent vectors υ, υ′ at p); these two structures are linked together by the analogue of (1), namely

ω (υ, υ′) = g_p(υ′ ,Jυ).

Examples of Kähler manifolds include complex vector spaces ⁿ, Riemann surfaces, and complex projective spaces ⁿ.

An example of a compact symplectic manifold that is not Kähler can be obtained by taking the quotient of ⁴ by a symplectic action of a group that looks like ⁴ but with a group operation that differs from the usual one. The change in the group structure manifests itself as a topological property (an odd first Betti number) that prevents the quotient being Kähler.

Further Reading

Arnold, V. I. 1989. Mathematical Methods of Classical Mechanics, 2nd edn. Graduate Texts in Mathematics, volume 60. New York: Springer.

McDuff, D., and D. Salamon. 1998. Introduction to Symplectic Topology, 2nd edn. Oxford Mathematical Monographs. Oxford: Clarendon Press/Oxford University Press.