Application of Jacobi’s last multiplier for construction of Hamiltonians of certain biological systems

Abstract

Abstract:The relationship between Jacobi's last multiplier and the Lagrangian of a second-order ordinary differential equation is quite well known. In this article we demonstrate the significance of the last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonians of certain well known first-order systems of differential equations arising in biology.PACS (

Nonlinear dynamics of a position-dependent mass-driven Duffing-type oscillator

“…While position-dependent (effective) mass (PDM) quantum mechanical systems have repeatedly received attention in many areas of physics (such as, for instance, in the problems of compositionally graded crystals [1], quantum dots [2], nuclei [3], quantum liquids [4], metal clusters [5], etc, see also [6][7][8][9][10][11][12][13] for theoretical developments and references therein), interest in classical problems having a PDM is a relatively recent and rapidly developing subject [14][15][16][17][18][19][20][21][22][23]. The simplest case of a PDM classical oscillator has been approached by several authors [16][17][18][19][20].…”

On time-dependent Hamiltonian realizations of planar and nonplanar systems

“…If the system (15) has a Hamiltonian realization, then one expects that the two-form α (1) ∧ α (2) presented in (16) must be proportional to the closed twoform β (1) ∧ β (2) in (18). That is, there exists a real valued function M = M (x, y, t), called as the Jacobi's last multiplier, on the extended space M = Q × R such that…”

“…Here is an incomplete list [8,18,19,21,43,44,45,47,46,49] of the studies on JLM. It is well known that, the method of JLM may result with time-dependent Hamiltonian functions even for autonomous systems.…”

Reduction and Hamiltonian aspects of a model for virus-tumour interaction in oncolytic virotherapy

“…As we are dealing with a system of three first-order ODEs and have succeeded in finding one first integral it follows that we can obtain another first integral provided there exists a Jacobi Last Multiplier (JLM) for the system. This is a consequence of the fact that the given a system of n first-order ODEs if we can find n − 2 first integrals and a JLM then the system may be reduced to quadrature [7,8]. The defining equation for the JLM for a non-autonomous system of first-order ODEs given in general by…”