# Definition Consider a system with [[Hamiltonian]] $H$, [[generalized coordinates]] $q^i$ and [[canonical momentum|momenta]] $p_j$. Define $z^I = (p^i, q_j), \, i, j=1\ldots n,\, I=1, \ldots, 2n$. Consider the operation: $ \pmb{\text{d}}H(z) = \left(\frac{\partial H}{\partial q^i}, \frac{\partial H}{\partial p_i}\right) \mapsto \left(\frac{\partial H}{\partial p^i}, -\frac{\partial H}{\partial q_i}\right) \equiv X_H(z) $ $X_H(z)$ is a [[vector field]] formed from the differential of $H$, and it is known as the *Hamiltonian vector field*. It can also be formed from the composition of the linear map $R$ $ R: Z^* \rightarrow Z $ with the differential of $H$. The matrix representation of $R$ is: $ [R] = \begin{pmatrix} \mathbb{0} & \mathbb{1} \\ -\mathbb{1} & \mathbb{0} \end{pmatrix} \equiv \mathbb{J} $ where $\mathbb{1}$ is the $n\times n$ identity matrix, $\mathbb{0}$ is the $n \times n$ zero [[matrix]], and $\mathbb{J}$ is the $2n \times 2n$ [[Symplectic Matrix|symplectic matrix]]. Thus, we end up with: $ X_H(z) = R \cdot \pmb{\text{d}}H(z) $ or, denoting the [[Components|components]] of $X_H$ by $X^I$, we have: $ X^I = R^{IJ} \frac{\partial H}{\partial z^J} \iff X_H = \mathbb{J} \nabla H $ here $\nabla H$ denotes the *naive gradient* of $H$, i.e., the row vector $\pmb{\text{d}} H$ regarded as a column [[vector]].