Editing Hamiltonian system (section)

== Symplectic structure ==
One important property of a Hamiltonian dynamical system is that it has a [[symplectic structure]].<ref name=ott/> Writing

: <math>\nabla_{\boldsymbol{r}} H(\boldsymbol{r}) = \begin{bmatrix}
\frac{\partial H(\boldsymbol{q},\boldsymbol{p})}{\partial \boldsymbol{q}} \\
\frac{\partial H(\boldsymbol{q},\boldsymbol{p})}{\partial \boldsymbol{p}} \\
\end{bmatrix}</math>

the evolution equation of the dynamical system can be written as

:<math>\frac{d\boldsymbol{r}}{dt} = M_N \nabla_{\boldsymbol{r}} H(\boldsymbol{r})</math>

where

:<math>M_N =
\begin{bmatrix}
0 & I_N \\
-I_N & 0 \\
\end{bmatrix}</math>
and ''I''<sub>''N''</sub> is the ''N''&times;''N'' [[identity matrix]].

One important consequence of this property is that an infinitesimal phase-space volume is preserved.<ref name=ott/> A corollary of this is [[Liouville's theorem (Hamiltonian)|Liouville's theorem]], which states that on a Hamiltonian system, the phase-space volume of a closed surface is preserved under time evolution.<ref name=ott/>

:<math>\begin{align}
\frac{d}{dt}\oint_{\partial V} d\boldsymbol{r} &= \oint_{\partial V}\frac{d\boldsymbol{r}}{dt}\cdot d\hat{\boldsymbol{n}}_{\partial V} \\
&= \oint_{\partial V} \left(M_N \nabla_{\boldsymbol{r}} H(\boldsymbol{r})\right) \cdot d\hat{\boldsymbol{n}}_{\partial V} \\
&= \int_{V}\nabla_{\boldsymbol{r}}\cdot \left(M_N \nabla_{\boldsymbol{r}} H(\boldsymbol{r})\right) \, dV \\
&= \int_{V}\sum_{i=1}^N\sum_{j=1}^N\left(\frac{\partial^2 H}{\partial q_i \partial p_j} - \frac{\partial^2 H}{\partial p_i \partial q_j}\right) \, dV \\
&= 0 
\end{align}</math>

where the third equality comes from the [[divergence theorem]].