A Solution Nonlinear Wilberforce Pendulum

This work presents solutions to the strongly coupled nonlinear Wilberforce pendulum model. We begin by presenting three specific cases of the equations of motion of the nonlinear Wilberforce Lagrangian. We find solutions governed by Jacobi elliptic functions, Mathieu functions, Weierstrass elliptic functions and obtaining beating behavior of plane waves when their frequencies are close and the phase is proportional to their amplitudes. Similarly, we find analytical solutions showing the power of these methods using the tanh, Riccati, Jacobi, and generalized Jacobi solitary wave solution techniques, without any restrictions.