In previous definitions, we've presented

some diffeomorphic spaces (i.e. \(T^*T^n\) and \((\CC^*)^n\)) which should be considered the same as symplectic manifolds.
some spaces which are diffeomorphic (i.e. \(S^2\) and itself) which can be equipped with clearly different symplectic structures (i.e. \(\text{vol}_g\) and \(\text{vol}_{2g}\)).

We now introduce an equivalence relation for symplectic structures on a manifold.

definition 0.0.1

Let \((X, \omega)\) and \((X', \omega')\) be symplectic manifolds, and \(\phi: X\to X' \) be a diffeomorphism. We call \(\phi\) a symplectomorphism if \[\phi^*\omega'=\omega.\] Given an embedding \(\psi: X\to X'\), we say that \(\psi\) is a symplectic embedding if \[\psi^*\omega'=\omega\] One property of the canonical symplectic structure on the cotangent bundle \(T^*Q\) is that the symplectomorphism type of \(T^*Q\) is only dependent on the diffeomorphism type of \(Q\) [dS01]. Let \(f: Q_1\to Q_2\) be a diffeomorphism. Define the lift of \(f\) to be a map \(f_\#: T^*Q_1\to T^* Q_2\) defined by \[f_\#(q,p)=(f(q), ((df_q)^*)^{-1}p\] The lift plays well with the tautological 1-form, in the sense that if \(\lambda_i\) is the tautological 1-form for \(T^*Q_i\), then \((f_\#)^*\lambda_2=\lambda_1\). A particularly interesting type of symplectomorphism is one which arises as an isotopy. A symplectic isotopy is a smooth 1-parameter family of maps \(\phi_t: X\times I\to X\) which for fixed values of \(t\) give a symplectomorphism, and start at the identity (in the sense that \(\phi_0=\operatorname{id}_X\)). These are examples of smooth isotopies and so they can equivalently be described as the flow of a time-dependent vector field.

definition 0.0.2

Given a time-dependent vector field \(V_t\), the flow associated to \(V\) is the function \(\phi_t: X\times \RR\to X\) which

for all \(t\), \(\phi_t:X\to X\) is a diffeomorphisms and;
is the identity at time 0 so that \(\phi_0=\operatorname{id}_M\) and;
generates the vector field \(V_t\) in the sense that \[(V_t)_p:=\frac{d}{ds}\phi_s(\phi^{-1}_t(p))|_{s=t}.\]

We say that \( V_t\) is the infinitesimal generator associated to \(\phi_t\). There is an equivalence between vector fields and flows.

theorem 0.0.3

On compact manifolds, every time-dependent vector field has a well defined flow. One can check that a smooth isotopy \(\phi_t\) is a symplectic isotopy by verifying that the symplectic form is preserved by the infinitesimal generator \(V_t\). This yields an easy to check criterion.

proposition 0.0.4

\(\phi_t\) is a symplectic isotopy if and only if its infinitesimal generator \(V_t\) satisfies \[d(\iota_{V_t}\omega)=0.\] We prove the forward direction, using that \(\phi^*_t\omega=\omega\) for all \(t\). By taking the derivative with respect to \(t\) of both sides, we obtain: \begin{align*} 0=&\frac{d}{dt}\phi^*\omega =\phi^*_t\mathcal L_{V_t}\omega \end{align*} Since \(\phi\) is a diffeomorphism, this is equivalent to the vanishing of \(\mathcal L_{V_t}\omega\), \begin{align*} 0=& \mathcal L_{V_t}\omega \end{align*} Applying Cartan's formula, and using that \(\omega\) is closed, \begin{align*} 0=&d(\iota_{V_t}\omega)+ \iota_{V_t}(d\omega)=d(\iota_{V_t}\omega). \end{align*} This means that \(\iota_{V_t}\omega\) is closed. One could ask for the stronger condition of exactness for the 1-form \(\iota_{V_t}\omega\). In this case, we can describe vector field \(V_t\) by a function on \(X\).

definition 0.0.5

Let \((X, \omega)\) be a symplectic manifold, and \(H: X\to \RR\) is a smooth function. The Hamiltonian vector field of \(H\) is the unique vector field \(V_H\) such that \(dH=\iota_{V_H}\omega\). The uniqueness of this vector field arises from the non-degeneracy of the symplectic form \(\omega\). This additionally means that to every exact symplectic isotopy we can associate a generating Hamiltonian function. The Hamiltonian isotopies give a large set of easy-to-describe symplectic isotopies, and the relation between Hamiltonian isotopies and all symplectic isotopies has a nice interpretation in terms of the topology of \(X\). If \(H^1(X, \RR)=0\), then every symplectic isotopy is Hamiltonian.

example 0.0.6

Suppose that we are working in \(\RR^{2n}=(q_1, \ldots , q_n,p_1, \ldots p_n) \), equipped with the standard symplectic form \(\omega= \sum_{i=1}^n dp_i \wedge dq_i\). We can compute the Hamiltonian vector field for \(H: \RR^{2n}\to \RR\), in local coordinates as \[V_H=\sum_{k=1}^n \left(a_k \partial_{q_k}+b_k \partial_{p_k}\right)\] where the functions \(a_k\) and \(b_k\) are given by the formulas \begin{align*} \frac{\partial p_k}{\partial H}=-b_k & & \frac{\partial q_k}{\partial H}=a_k. \end{align*} A notable feature of Hamiltonian flow \(V_H\) is that it preserves the level sets of \(H\), as \[V_H(H)=dH(V_H)=\omega(V_H, V_H)=0.\]

example 0.0.7

Consider \(S^2=\{(x_0, x_1, x_2)\;|\;x_0^2+x_1^2+x_2^2=1\}\) equipped with the symplectic form agreeing with the standard metric induced from \(\RR^3\). Take the Hamiltonian \begin{align*} H:S^2\to&\RR\\ (x_0,x_1,x_2)\mapsto& x_2 \end{align*} as drawn in figure 0.0.8. Since Hamiltonian flow preserves the level sets of \(H\), we know that the latitudinal slices are orbits under the action of the Hamiltonian flow. To show that the Hamiltonian flow uniformly rotates the sphere, consider the map \(\phi:S^2\setminus\{(0,0,1), (0,0,-1)\}\into S^1\times \RR\subset \RR^3\), where \(S^1\times \RR=\{(x_0, x_1, x_2)\;|\; x_0^2+x_1^2=1\}\), and the embedding is given by the latitudinal projection. This projection (the Gall-Peters map projection) is area-preserving, and so \(\phi\) is a symplectic embedding. In these new coordinates, \(\omega=d\theta\wedge dx_2\) and \(H=x_2\). In the Gall-Peters' coordinates, \(V_H=\partial_\theta\).

figure 0.0.8:The Hamiltonian flow of the standard height function rotates the sphere counterclockwise relative to the north pole

The Hamiltonian flow is sometimes called the symplectic gradient. In the setting where we have a compatible triple \((X, \omega,g, J)\), the Hamiltonian flow and gradient are related by the almost complex structure.

lemma 0.0.9

Let \((X, \omega, g, J)\) be a compatible triple. Let \(H:X\to \RR\) be a Hamiltonian. \[\grad H = J V_H\] This is a direct computation. On any test vector \(v\), \begin{align*} g(J V_H, v)=\omega(V_H, v)=dH(v)= g(\grad(H), v) \end{align*} Because \(g\) is nondegenerate, \(\grad(H)=JV_H\).

References

[dS01]

Ana Cannas da Silva. Lectures on symplectic geometry. Lecture Notes in Mathematics, 1764, 2001.