The first integrals of generalized Hamiltonian systems

The generalized Hamiltonian systems of the form $\dot q^i=\frac{\partial H}{\partial p_i}, \dot p^i=-\frac{\partial H}{\partial q_i}+\Gamma^i(t,q^i,p_i)$ arise widely in different fields of the applied mathematics. The generalized Hamiltonian systems appear for a mechanical system with non-holonomic nonlinear constraints and non-potential generalized forces. In dynamic optimization problems of economic growth theory involving a non-zero discount factor the generalized Hamiltonian systems arise and are known as a current value Hamiltonian systems. It is shown that the partial Hamiltonian approach proposed earlier for the current value Hamiltonian systems arising in economic growth theory Naz et al \cite{naz} is applicable to the generalized Hamiltonian systems. The partial Hamiltonian approach is compared with the characteristic method and direct method. Moreover, a new class of generalized Hamiltonian systems known as a partial Hamiltonian system is introduced. The notion of partial Hamiltonian is developed for systems of first-order ordinary differential equations (ODEs) and is illustrated by several examples. Also, it is shown that every system of second-order ODEs can be expressed as a partial Hamiltonian system of first-order ODEs. In order to show effectiveness of partial Hamiltonian approach, it is utilized to construct first integrals and closed form solutions of optimal growth model with environmental asset, equations of motion for a mechanical system with non-potential forces,the force-free Duffing Van Der Pol Oscillator and Lotka-Volterra models.