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A simple harmonic oscillator is an oscillator that is neither driven nor damped. It consists of a mass m, which experiences a single force F, which pulls the mass in the direction of the point x = 0 and depends only on the position x of the mass and a constant k. Balance of forces (Newton's second law) for the system is.
Simple harmonic motion can serve as a mathematical model for a variety of motions, but is typified by the oscillation of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's law. The motion is sinusoidal in time and demonstrates a single resonant frequency. Other phenomena can be modeled by simple ...
If it is assumed that the angle is much less than 1 radian (often cited as less than 0.1 radians, about 6°), or , then substituting for sin θ into Eq. 1 using the small-angle approximation, , yields the equation for a harmonic oscillator, + =
This is the equation for a simple harmonic oscillator with angular frequency: ... Which is a simple harmonic motion. General case. As seen above, the effective mass ...
Here we show that a necessary condition for stable, exactly closed non-circular orbits is an inverse-square force or radial harmonic oscillator potential. In the following sections, we show that those two force laws produce stable, exactly closed orbits (a sufficient condition ) [it is unclear to the reader exactly what is the sufficient ...
Unlike the equations of motion for the simple harmonic oscillator, these modified equations do not take the form of Hamilton's equations, and therefore we do not expect Liouville's theorem to hold. Instead, as depicted in the animation in this section, a generic phase space volume will shrink as it evolves under these equations of motion.
Lagrangian mechanics describes a mechanical system as a pair (M, L) consisting of a configuration space M and a smooth function within that space called a Lagrangian. For many systems, L = T − V, where T and V are the kinetic and potential energy of the system, respectively. [3]
For the harmonic oscillator, the time evolution of an arbitrary Wigner distribution is simple. An initial W ( x , p ; t = 0) = F ( u ) evolves by the above evolution equation driven by the oscillator Hamiltonian given, by simply rigidly rotating in phase space , [ 1 ]