The Hamilton equations of motion are given in terms of [conjugate momentum](conjugate%20momentum) as $\dot{q}=\frac{\partial\mathcal{H}}{\partial p}$ ^46213f and $\dot{p}=-\frac{\partial \mathcal{H}}{\partial q}$ ^7b021b which are written as [[1st order ODE]]s. # Hamilton equations of motion in 3 spatial dimensions For a 3 dimensional system we just solve the equations for each vector component, which are written in terms of vector components in [Euclidean space](Euclidean%20space.md) as: $\dot{q}_i=\frac{\partial\mathcal{H}_i}{\partial p}$ and $\dot{p}_i=-\frac{\partial \mathcal{H}_i}{\partial q_i}$ # Hamilton equations of motion for fields Since in field theory we deal in [fields](Field.md), which are themselves functions of coordinates in space and time the [Hamiltonians](Hamiltonians.md) and [Lagrangians](Lagrangian.md) become [functionals](Functional.md) (functions of functions). Thus the equations of motion are rewritten in terms of [functional derivatives](Functional%20derivative.md) as $\dot{\phi}(x,t)=\frac{\delta \mathscr{H}}{\delta \pi (x,t)}$ ^ea9ad1 and $\dot{\pi}(\mathbf{x},t)=-\frac{\partial\mathscr{H}}{\partial \phi(\mathbf{x},t)}+\partial_i\bigg(\frac{\partial \mathscr{H}}{\partial (\partial_i\phi(\mathbf{x},t))}\bigg)$ ^44d4df where we'd rewrite these equations as follows in order to make explicit that we are considering the 4-dimensional coordinate space of [Spacetime](Spacetime.md). $\partial_0\phi(\mathbf{x})=\frac{\delta \mathscr{H}}{\delta \pi (\mathbf{x})}$ and $\partial_0\pi(\mathbf{x})=-\frac{\partial\mathscr{H}}{\partial \phi(\mathbf{x})}+\partial_i\bigg(\frac{\partial \mathscr{H}}{\partial (\partial_i\phi(\mathbf{x}))}\bigg)$ #Mechanics