The Sine-Gordon equation in classical mechanic and field theory

The Sine-Gordon equation is a nonlinear hyperbolic partial differential equation. The equation as well as several solution techniques, were known in the nineteenth century in the course of study of various problems of differential geometry. The Sine-Gordon equation appears in a number of physical application including applications in relativistic field theory, Josephson junctions, or mechanical transmission lines. We will discuss the Sine-Gordon equation in classical and filed theory. We will consider pendulum example in classical theory, we will derive the motion equation of pendulum and the period of rotation. In the end we will review the Modified Method for Sine-Gordon Equations in the field theory. We will derive the energy equation and discuss when the phase shift can happen.