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 3.2 (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
For a Hamiltonian system \[ \dot y = J^{-1}\nabla H(y) \] with \(y=(p,q)\) we have the following theorem.
The converse direction goes as follows:
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.