# Definition Consider the [[Hamiltonian Vector Field]] and the map used to generate it: $ R: Z^* \rightarrow Z $ where $Rs matrix components are given by the [[Symplectic Matrix]] $\mathbb{J}$. We can define a [[bilinear form]] $ \begin{align} B(\alpha, \beta): Z^* \times Z^* &\rightarrow \mathbb{R}\\ B(\alpha, \beta) &\mapsto \braket{\alpha|R(\beta)} \end{align} $ where $\braket{\cdot|\cdot}$ is the canonical pairing between $Z^*$ and $Z$. The bilinear form $B$ or its [[Bilinear Form#Associated Linear Map|associated linear map]] $B^\flat$ are called the *Poisson Structure*. $B$ is called the Poisson Form.