3.1 Symplecticity and Flows

Hamiltonian Systems have another fundamental property, namely

The flow of a Hamiltonian system is symplectic

We are going to define symplecticity and flow next.

Definition 3.1 (Flow of a differential equation) For a parameter \(t\), the flow \[ \phi_t \colon y_0 \mapsto y(t) \] is defined as the function that maps a state \(y_0\) to \(y(t)\) as the solution of the differential equation \[ \dot y = f(y),~y(0)=y_0 \] evaluated at \(t\).

Definition 3.2 (Symplecticity of a linear map) A linear map, \(A\colon\mathbb R^{2d} \to \mathbb R^{2d}\) written via its matrix \(A\in \mathbb R^{2d\times 2d}\) is symplectic, if

\[ A^TJA = J. \]

A geometrical interpretation (and also a valid definition) is that a symplectic linear map \(A\) preserves areas and there orientations. For \(d=1\), i.e. for \(2D\) vectors this means that \(A\) preserves that area that is spanned by two vectors (and its orientation). For \(d>1\) one considers the areas that are spanned by the vectors projected into the respective coordinate planes.

Examples:

  • (anisotropic) scaling with area preservation: \(A(\alpha) = \begin{bmatrix} \alpha & 0 \\ 0 & 1/\alpha \end{bmatrix}\)
  • Rotations: \(R(\theta )=\begin{bmatrix} \cos\theta & -\sin \theta \\ \sin\theta & \cos\theta\end{bmatrix}\)
  • all concatenations \(R(\theta_2)A(\alpha_2)R(\theta_1)A(\alpha_1)\) of such operations
Definition 3.3 (Symplecticity of a nonlinear map) A smooth nonlinear map \(g\colon U \subset \mathbb R^{2d} \to \mathbb R^{2d}\) is symplectic, if its Jacobian \(g'(p,q)\) is symplectic, i.e. \[ g'(p,q)^TJg'(p,q) = J. \] for all \((p,q)\in U\).

For a Hamiltonian system \[ \dot y = J^{-1}\nabla H(y) \] with \(y=(p,q)\) we have the following theorem.

Theorem 3.1 (Poincare 1899) If \(H\) is two times differentiable on a subset \(U\) of \(\mathbb R^{2d}\), then the associated flow \(\phi_t\) is symplectic (as long as it stays in \(U\)).

The converse direction goes as follows:

Theorem 3.2 Let \(f\colon U \to \mathbb R^{2d}\) be smooth. Then the system \(\dot y = f(y)\) is locally Hamiltonian if, and only if, its flow \(\phi_t\) is symplectic for all \(y\in U\) and for all \(t\) sufficiently small.

The flow of a Hamiltonian system is symplectic.

If a flow of system is symplectic for t sufficiently small, then the system is (locally) Hamiltonian.

These observations motivate the use of symplectic time discretization schemes. For these schemes, if applied to a Hamiltonian system, it holds that the (discrete) flow is symplectic, i.e. this structure is preserved. This means that the discrete system can be seen as a discrete representation of a (locally) Hamiltonian system.