# Definition (Vector Spaces) Given two [[Symplectic Vector Space|symplectic vector spaces]] $(Z, \Omega)$ and $(Y, \Xi)$, a [[smoothness|smooth]] [[map]] $f:Z \rightarrow Y$ is called a *symplectomorphism*, *canonical transformation*, *symplectic transformation*, or *Poisson transformation* if it preserves the [[Symplectic Form|symplectic forms]], i.e., $ \Xi(\pmb{D}f(z) \cdot z_1, \pmb{D}f(z)\cdot z_2)) = \Omega(z_1, z_2) $ where $\pmb{D}f(z)$ denotes the derivative of $f(z)$, and $\pmb{D}f(z)\cdot z_1$ means $\pmb{D}f(z)$ applied to $z_1$ as a [[linear map]]. In [[Pull-Back and Push-Forward|pull-back notation]], this reads: $ f^* \Xi = \Omega $ ## Motivation Consider [[Hamilton's Equations]]: $ \begin{align} \dot{q}^i &= \frac{\partial H}{\partial p_i}\\ \dot{p}_i &= -\frac{\partial H}{\partial q^i} \end{align} $ and a [[map]] from the [[phase space]] $Z$ to itself: $f:Z \rightarrow Z$, written as: $ (\tilde{p},\tilde{q}) = \phi(p, q) $ or equivalently $\tilde{z} = \phi(z)$, where $z(t) = (q(t). p(t))$ satisfies Hamilton's equations above, i.e., $ \begin{align} \dot{z} = X_H(z) = \Omega^\sharp \pmb{\text{d}}H(z) \end{align} $ where $X_H$ denotes the [[Hamiltonian Vector Field]] and $\Omega^\sharp: Z^* \rightarrow Z$ is the [[Symplectic Structure#Associated Linear Maps|associated linear map]] of the [[Symplectic Form]]. Its matrix has canonical form $\mathbb{J}$ (the [[Symplectic Matrix]]), and we denote its entries for now by $B^{JK}$. Using the chain rule: $ \dot{\tilde{z}}^I = \frac{\partial f^I}{\partial z^J} \dot{z}^J \equiv A^{I}_{\,J} \dot{z}^J $ But we have: $ \dot{z}^J = B^{JK} \frac{\partial H}{\partial z^K} $ and $ \frac{\partial H}{\partial \tilde{z}^K} = A^{L}_{\,\,K} \frac{\partial H}{\partial z^L} $ therefore we end up with: $ \dot{\tilde{z}}^I = A^{I}_{\,\,J} B^{JK} A^{L}_{\,\,K} \frac{\partial H}{\partial \tilde{z}^L} $ For these equations to be Hamiltonian, we require: $ A^{I}_{\,\,J} B^{JK} A^{L}_{\,\,K} = B^{IL} $ or equivalently $ A \mathbb{J} A^T = \mathbb{J} $ That is, these matrices belong to the [[Symplectic Group|symplectic group]].