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

Thomas Vogel and Ryan Rogers

Vol. 3 (2010), No. 4, 349–361
Abstract

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.

Keywords
variational, Hamilton's principle, solitons, embedded solitons
Mathematical Subject Classification 2000
Primary: 49S05
Secondary: 49M99, 34K35
Milestones
Received: 1 October 2009
Accepted: 12 November 2010
Published: 6 January 2011

Proposed: Zuhair Nashed
Communicated by Zuhair Nashed
Authors
Thomas Vogel
Department of Mathematics and Computer Science
Stetson University
421 N. Woodland Blvd., Unit 8332
DeLand, FL 32723
United States
Ryan Rogers
Department of Mathematics and Computer Science
Stetson University
421 N. Woodland Blvd., Unit 8332
DeLand, FL 32723
United States