displaced from its equilibrium position, a restoring force that obeys Hooke's law tends to restore the system to equilibrium. Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, it accelerates and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, at , the mass has momentum because of the acceleration that the restoring force has imparted. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then slows it down until its velocity reaches zero, whereupon it is accelerated back to the equilibrium position again. As long as the system has no energy loss, the mass continues to oscillate. Thus simple harmonic motion is a type of periodic motion. If energy is lost in the system, then the mass exhibits damped oscillation. Note if the real space and phase space plot are not co-linear, the phase space motion becomes elliptical. The area enclosed depends on the amplitude and the maximum momentum. Dynamics In Newtonian mechanics, for one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear ordinary differential equation with constant coefficients, can be obtained by means of Newton's 2nd law and Hooke's law for a mass on a spring. where is the inertial mass of the oscillating body, is its displacement from the equilibrium (or mean) position, and is a constant (the spring constant for a mass on a spring). Therefore, Solving the differential equation above produces a solution that is a sinusoidal function: where The meaning of the constants and can be easily found: setting on the equation above we see that , so that is the initial position of the particle, ; taking the derivative of that equation and evaluating at zero we get that , so that is the initial speed of the particle divided by the angular frequency, . Thus we can write: This equation can also be written in the form: where or equivalently In the solution, and are two constants determined by the initial conditions (specifically, the initial position at time is , while the initial velocity is ), and the origin is set to be the equilibrium position. Each of these constants carries a physical meaning of the motion: is the amplitude (maximum displacement