Characterization of partial Hamiltonian operators and related first integrals (Q1690364)

Modified Hamilton equations and certain properties of them are here discussed. A partial Hamiltonian system is defined as the set of Hamilton equations given by: \[ \begin{aligned} \dot q_i &= \frac{\partial H}{\partial p_i}, \\ \dot p_i &= -\frac{\partial H}{\partial q_i} + \Gamma_i,\end{aligned} \] \(i = 1, \ldots, n\), where \(\Gamma_i\) is a function of the canonical coordinates \(q_i\), \(p_i\) and time \(t\). For \(\Gamma_i = 0\), the system is a canonical Hamiltonian system. A first integral \(I\) is defined via a conservation law \(D_t I = 0\), for the total time derivative \(D_t\). The authors state a theorem on the existence of a function \(I\) in terms of a partial Hamiltonian operator associated with the above system (see the definition in the text). They derive an evolutionary form of this partial Hamiltonian operator and finally give a couple of applications. Among them one is the modified Emden equation, \(\dot p = -\tfrac{2}{t} p + 3 q^5\), and another one is the Ermakov system \[ \begin{aligned} \ddot r - r \dot \theta^2 &= \frac{F(\theta)}{r^3}, \\ r \ddot \theta + 2 \dot r \dot \theta &= \frac{G(\theta)}{r^3}, \end{aligned} \] with \(F\) and \(G\) arbitrary functions of \(\theta\). Finally, they discuss the inverse problem: given a first integral \(I\), how to construct the partial Hamiltonian operator out of it.