New role of null lagrangians in derivation of equations of motion for dynamical systems

Abstract

The space of null Lagrangians is the least investigated territory in dynamics as these Lagrangians are identically sent to zero by their Euler-Lagrange operator, and thereby they are having no effects on equations of motion. A procedure that significantly generalizes the previous work, which appeared in Physica Scripta 97, 125213 (2022), is developed and used to construct null Lagrangians and then the corresponding non-standard Lagrangians, which represent a range of interesting dynamical systems. By using the generalized procedure, derivation of equations of motion for a harmonic oscillator as well as for the Bateman and Duffing oscillators is presented. The obtained results demonstrate a new role played by the null Lagrangians and their corresponding non-standard Lagrangians in describing linear and nonlinear, and dissipative and non-dissipative dynamical systems.

Non-Standard and Null Lagrangians for Nonlinear Dynamical Systems and Their Role in Population Dynamics

“…However, there is another family of Lagrangians called null Lagrangians (NLs), which make the E-L equation identically zero and are given as the total derivative of a scalar function [7]; the latter is called here a gauge function. The NLs were extensively studied in mathematics (e.g., [7,[39][40][41][42][43]) and recently in physics [44][45][46][47], but to the best of our knowledge, NSLs have not yet been introduced to biology and specifically, to its population dynamics. Therefore, in the following, we present the first applications of NLs to biology and its population dynamics.…”

“…Recent studies of null Lagrangians demonstrated that there is a different condition that is obeyed by NLs and that this condition plays the same role for NLs as the E-L equation plays for standard and non-standard Lagrangians [46,47]. The condition can be written as…”

“…and it shows that the substitution of any NL into Equation (62) results in an equation of motion. However, the resulting equations of motion may be limited because their coefficients are required to obey relationships that are different for different equations of motion [46,47]. The previous work also demonstrated that an inverse of any null Lagrangian generates a non-standard Lagrangian, whose substitution into the E-L equation gives a new equation of motion.…”

“…The previous work also demonstrated that an inverse of any null Lagrangian generates a non-standard Lagrangian, whose substitution into the E-L equation gives a new equation of motion. However, the reverse is not always true, which means that not all NSLs have their corresponding NLs because of the so-called null condition that must be satisfied [46,47].…”

“…Another important characteristic of NLs is the fact that they can always be expressed as the total derivative of a scalar function [7], which has been called a gauge function [44][45][46][47]. Thus, we may write…”