Identification of localized structure in a nonlinear damped harmonic oscillator using Hamilton’s principle

In the mid-seventeenth century Isaac Newton formalized the language necessary to describe the evolution of physical systems. Newton argued that the evolution of the state of a process can be described entirely in terms of the forces involved with the process. About a century and a half later, William Hamilton was able to establish the whole of Newtonian mechanics without ever using the concept of force. Rather, Hamilton argued that a physical system will evolve in such a way as to extremize the integral of the difference between the kinetic and potential energies. This paradigmatic reformulation allows for a type of reverse engineering of physical systems. This paper will use the Hamiltonian formulation of a nonlinear damped harmonic oscillator with third and fifth order nonlinearities to establish the existence of localized solutions of the governing model. These localized solutions are commonly known in mathematical physics as solitons. The data obtained from the variational method will be used to numerically integrate the equation of motion, and find the exact solution numerically.