Some Generalized Trigonometric Sine Functions and Their Applications

In this paper, it is shown that D. Shelupsky’s generalized sine function, and various general sine functions developed by P. Drábek, R. Manásevich and M. Ôtani, P. Lindqvist, including the generalized Jacobi elliptic sine function of S. Takeuchi can be defined by systems of first order nonlinear ordinary differential equations with initial conditions. The structure of the system of differential equations is shown to be related to the Hamilton System in Lagrangian Mechanics. Numerical solutions of the ODE systems are solved to demonstrate the sine functions graphically. It is also demonstrated that the some of the generalized sine functions can be used to obtain analytic solutions to the equation of a nonlinear spring-mass system.