Omega-limit sets and bounded solutions

We prove among other things that the omega-limit set of a bounded solution of a Hamilton system $$\{\begin{array}{rlr}{\displaystyle }& {\displaystyle \dot{\mathbf{p}}=\frac{\mathrm{\partial }H}{\mathrm{\partial }\mathbf{q}}}& {\displaystyle \dot{\mathbf{q}}=-\frac{\mathrm{\partial }H}{\mathrm{\partial }\mathbf{p}}}\end{array}$$ is containing a full-time solution so there are the limits of $\frac 1t\int_0^t {\mathbf p}(s)ds$and$\frac 1t\int_0^t {\mathbf q}(s)ds$as$t\to\infty$foranyboundedsolution$(\mathbf {p,q})$ of the Hamilton system. These limits are stationary points of the Hamilton system so if a Hamilton system has no stationary point then every solution of this system is unbounded.