Derivation the equation of motion of particles from nonstandard Lagrangian with Lennard-Jones Potential

Abstract

Derivation of the equation of motion of a particle from a nonstandard Lagrangian under the Leonard-Jones potential is presented. The objectives are to find the equation of motion of a particle under the Leonard-Jones potential by using a non-standard Lagrangian. And also discussing impact of the arbitrary time dependent function on dynamics of systems. There the arbitrary time dependent functions, 1, ,  and , were used. The results show that nonstandard Lagrangian,  , presented by El-Nabulsi substituted ​​into the standard Lagrangian equation that will got two additional terms, which are time dependent. However, if we had reduced the modified Euler-Lagrange equation by set the arbitrary time dependent function to 1, the standard Euler-Lagrange equation can be reversed. The dynamical system obtained from simulation found that the arbitrary time dependent function effect on the dynamics of the system is only the scaling of the range of interest as observed in the order rank in the vertical axis data.