The applications of the partial Hamiltonian approach to mechanics and other areas

TL;DR

This paper demonstrates the versatility of the partial Hamiltonian approach in deriving solutions and integrals for various systems in mechanics and applied mathematics, including economic and biological models.

Contribution

It extends the partial Hamiltonian method to new areas such as mechanics with non-holonomic constraints and biological models, showing its broad applicability.

Findings

Constructed first integrals for diverse systems

Derived closed-form solutions for optimal growth models

Applied the approach to mechanical and biological oscillators

Abstract

The partial 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 partial 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 partial 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 mechanics and other areas as well. The partial Hamiltonian approach 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.