# 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]] $\mathbb{J}$. The [[bilinear form]] associated with $R$ is the [[Poisson Structure]], but we can associate a [[bilinear form]] with $R^{-1}$ as: $ \begin{align} \Omega: Z \times Z &\rightarrow \mathbb{R}\\ \Omega(v, w) &\mapsto \braket{R^{-1}(v)|w} \end{align} $ where $\braket{\cdot|\cdot}$ is the canonical pairing between $Z^*$ and $Z$. The bilinear form $\Omega$ or its [[Bilinear Form#Associated Linear Map|associated linear map]] $\Omega^\flat$ are called the *Symplectic Structure*. $\Omega$ is called the [[Symplectic Form|symplectic form]]. # Associated Linear Maps We have $\Omega^\flat: Z \rightarrow Z^*$ defined by $\Omega^\flat(v)(w) = \Omega(v,w)$, or equivalently, $\Omega^\flat = R^{-1}$. A summary is given below: ![[Pasted image 20210714143206.png]]