Omega-limit sets and bounded solutions

TL;DR

This paper investigates the properties of omega-limit sets of bounded solutions in Hamiltonian systems, establishing that these sets contain full-time solutions and that certain time-averages converge to stationary points, implying unboundedness in the absence of stationary points.

Contribution

It proves that omega-limit sets of bounded solutions contain full-time solutions and that time-averages converge to stationary points, providing new insights into the behavior of Hamiltonian systems.

Findings

Omega-limit sets contain full-time solutions.

Time-averages of solutions converge to stationary points.

Unbounded solutions occur if no stationary points exist.

Abstract

We prove among other things that the omega-limit set of a bounded solution of a Hamilton system \[\left\{\begin{aligned} & \mathbf{\dot{p}}=\frac{\partial H}{\partial \mathbf{q}} & \mathbf{\dot{q}}=-\frac{\partial H}{\partial \mathbf{p}} \\ \end{aligned} \right.\] 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 $t\to\infty$ for any bounded solution $(\mathbf {p,q})$ 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.