Editing Poisson bracket (section)

== Hamilton's equations of motion ==
[[Hamilton's equations of motion]] have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that <math>f(p, q, t)</math> is a function on the solution's trajectory-manifold. Then from the multivariable [[chain rule]],
<math display="block">\frac{d}{dt} f(p, q, t) = \frac{\partial f}{\partial q} \frac{dq}{dt} + \frac {\partial f}{\partial p} \frac{dp}{dt} + \frac{\partial f}{\partial t}.</math>

Further, one may take <math>p = p(t)</math> and <math>q = q(t)</math> to be solutions to [[Hamilton's equations]]; that is,
<math display="block">\begin{align}
  \frac{d q}{d t} &=  \frac{\partial \mathcal H}{\partial p} = \{q, \mathcal H\}, \\ 
  \frac{d p}{d t} &= -\frac{\partial \mathcal H}{\partial q} = \{p, \mathcal H\}.
\end{align}</math>

Then 
<math display="block">\begin{align}
  \frac {d}{dt} f(p, q, t) &= \frac{\partial f}{\partial q} \frac{\partial \mathcal H}{\partial p} - \frac{\partial f}{\partial p} \frac{\partial \mathcal H}{\partial q} + \frac{\partial f}{\partial t} \\
                           &= \{f, \mathcal H\} + \frac{\partial f}{\partial t} ~.
\end{align}</math>

Thus, the time evolution of a function <math>f</math> on a [[symplectic manifold]] can be given as a [[flow (mathematics)|one-parameter family]] of [[symplectomorphism]]s (i.e., [[canonical transformations]], area-preserving diffeomorphisms), with the time <math>t</math> being the parameter:  Hamiltonian motion is a canonical transformation generated by the Hamiltonian. That is, Poisson brackets are preserved in it, so that ''any time <math>t</math>'' in the solution to Hamilton's equations,
<math display="block"> q(t) = \exp (-t \{ \mathcal H, \cdot \} ) q(0), \quad  p(t) = \exp (-t \{ \mathcal H, \cdot \}) p(0), </math>
can serve as the bracket coordinates. ''Poisson brackets are [[Canonical transformation|canonical invariants]]''.

Dropping the coordinates, 
<math display="block">\frac{d}{dt} f = \left(\frac{\partial}{\partial t} - \{\mathcal H, \cdot\}\right)f.</math>

The operator in the convective part of the derivative, <math>i\hat{L} = -\{\mathcal H, \cdot\}</math>, is sometimes referred to as the Liouvillian (see [[Liouville's theorem (Hamiltonian)]]).