Application To A Conservative Force
The problem of the simple harmonic oscillator occurs frequently in physics, because a mass at equilibrium under the influence of any conservative force, in the limit of small motions, behaves as a simple harmonic oscillator.
A conservative force is one that has a potential energy function. The potential energy function of a harmonic oscillator is:
Given an arbitrary potential energy function, one can do a Taylor expansion in terms of around an energy minimum to model the behavior of small perturbations from equilibrium.
Because is a minimum, the first derivative evaluated at must be zero, so the linear term drops out:
The constant term V(x0) is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:
Thus, given an arbitrary potential energy function with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.
Read more about this topic: Harmonic Oscillator
Famous quotes containing the words application to, application, conservative and/or force:
“The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“Great abilites are not requisite for an Historian; for in historical composition, all the greatest powers of the human mind are quiescent. He has facts ready to his hand; so there is no exercise of invention. Imagination is not required in any degree; only about as much as is used in the lowest kinds of poetry. Some penetration, accuracy, and colouring, will fit a man for the task, if he can give the application which is necessary.”
—Samuel Johnson (1709–1784)
“I never dared be radical when young
For fear it would make me conservative when old.”
—Robert Frost (1874–1963)
“If you were to destroy the belief in immortality in mankind, not only love but every living force on which the continuation of all life in the world depended, would dry up at once.”
—Feodor Dostoyevsky (1821–1881)