# Definition Given a finite-[[dimension|dimensional]] system with [[generalized coordinates]] $q^i$, [[canonical momenta]] $p_i$ and [[Hamiltonian]] $H$, Hamilton's equations take the form: $ \begin{align} \dot{q}^i &= \frac{\partial H}{\partial p_i}\\ \dot{p}_i &= -\frac{\partial H}{\partial q^i} \end{align} $ # Complex Notation Let $z = (p, q)$ or, equivalently $z = p + i q$. Then we have: $ \begin{align} \dot{z} &= \dot{q} + i \dot{p}\\ \frac{\partial H}{\partial \bar{z}} &\equiv \frac{1}{2} \left(\frac{\partial H}{\partial q} - i \frac{\partial H}{\partial p}\right)\\ \dot{z} &= -2 i \frac{\partial H}{\partial z} \end{align} $ # Symplectic Structure In terms of the [[Symplectic Form]] $\Omega$, its [[Symplectic Structure#Associated Linear Maps|associated linear maps]] $\Omega^\flat, \Omega^\sharp$, as well as the [[Hamiltonian Vector Field]] $X_H(z)$, we can write: $ \begin{align} \dot{z} &= X_H(z) = R \cdot \pmb{\text{d}}H(z)\\ &= (\Omega^\flat)^{-1} \pmb{\text{d}}H(z)\\ &= \Omega^\sharp \pmb{\text{d}}H(z) \end{align} $ or equivalently $ \Omega^\flat X_H(z) = \pmb{\text{d}}H(z) $ or in terms of $\Omega$ $ \Omega(X_H(z), v) = \pmb{\text{d}}H(z) \cdot v $ for all $z, v \in Z$.