Symmetries in Simple Harmonic Motion and Their Applications

The Simple Harmonic motion is ubiquitous in nature at every scale from macroscopic bodies to the very microscopic elementary particles like electrons and photons. In the usual macroscopic scales motion of particles obey Newtonian and Relativistic dynamics which is modeled very accurately by linear Mathematics developed over the last few centuries. But at the quantum scales we inherently have to deal with probabilities and the wave particle dual nature becomes apparent at microscopic scales. To describe the energy of photons in Quantum Field Theory we have to use the notion of frequency which is a Simple Harmonic Motion Notion which in turn requires Sines and Cosines. In this Paper I would like to formulate a Mathematical Model by generalizing some symmetry principles from Simple Harmonic motion and show how this Model reduces to linear Mathematics at high frequencies or longer time scales and how this model maybe useful in general for quantum physicists in particular for dealing with infinities.