In general, Hamiltonian Thermostats, as proposed in the model by Jellinek and Berry in the paper \cite{PhysRevA.38.3069}, are derived from the Hamiltonian $H=H(p,q)$ of a mechanical system using some elementary constructions. This thesis is dedicated to studying the properties of the Jellinek–Berry (JB) thermostat applied to an ideal gas.

An ideal gas is a mechanical system with zero potential energy. In an idealized gas, atoms or molecules do not interact, so they only possess kinetic energy. If, as is usually assumed, the kinetic energy is just the euclidean squared-norm of momenta, then the Hamiltonian is [ H(p,q)=\frac{1}{2}|p|^2 ]

where $q\in M,$ $p\in T^*_q(M)$ and $M$ is a flat manifold,. This Hamiltonian is completely integrable because $p$ is constant along the solution (conservation of momentum). Two major findings are:

\begin{itemize}

\item If $G(s,p_s)$  is the internal energy of the JB thermostat, then the thermostatted Hamiltonian $F_{\epsilon}(q,p,s,p_s)=H_{\epsilon}(q,\alpha(s)p)+G(s,p_s)$ for some scalar function $\alpha(s),$ is completely integrable when $H$ is the total energy of an ideal gas.

\item If $H_{\epsilon}$ is a small, real-analytic perturbation of the ideal-gas Hamiltonian $H=H_0$ sufficient conditions are determined that imply the existence of positive-measure sets of invariant tori for $F_{\epsilon}.$

\end{itemize}