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

Abstract

The partial Hamiltonian systems of the formq i = ∂H ∂p i ,ṗ i = − ∂H ∂q i + Γ 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 [1] 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.

Exact solutions of a Black-Scholes model with time-dependent parameters by utilizing potential symmetries

“…One can also utilize the computer programmes in REDUCE by Wolf [37,38], GeM on Maple by Cheviakov [10,11] and SADE on Maple by Filho [32] for computation of conservation laws/first integrals. Recently, an elegant approach known as partial Hamiltonian approach [26,27,28,29] has been developed for derivation of first integrals of optimal control problems and dynamical systems.…”

“…We have established three new solutions given in (26), (28), (33) for the BST model (6) by utilizing the potential symmetries. These solutions are not reported in Naz and Johnpillai [23].…”

Closed‐form solutions of two‐sector Romer model of endogenous growth using partial Hamiltonian approach

“…Partial Hamiltonian approach is thus an effective tool, which can be applied to all mathematical growth models, which can be expressed into partial Hamiltonian systems. The partial Hamiltonian approach has been effectively utilized to construct the first integrals and exact solutions of various dynamical systems …”

On sufficiency issues, first integrals and exact solutions of Uzawa-Lucas model with unskilled labor

“…The partial Hamiltonian approach [20,21] and a partial Lagrangian approach [12,22] are elegant approaches to derive the exact solutions for the models arising in mathematical biology, physics, mechanics, economic growth theory and some other fields as well. The exact solutions for the basic Lucas-Uzawa model and its extension for the externalities case are studied in [23,24,4,25] by the partial Hamiltonian approach.…”

“…Let time t be the independent variable and (q, p) = (q 1 , ..., q n , p 1 , ..., p n ) the phase space coordinates. We provide an overview of partial Hamiltonian approach [20,21]. The following definitions and results are adapted from [20,21,22,6].…”