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 a, application to, application, conservative and/or force:
““Five o’clock tea” is a phrase our “rude forefathers,” even of the last generation, would scarcely have understood, so completely is it a thing of to-day; and yet, so rapid is the March of the Mind, it has already risen into a national institution, and rivals, in its universal application to all ranks and ages, and as a specific for “all the ills that flesh is heir to,” the glorious Magna Charta.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“If you would be a favourite of your king, address yourself to his weaknesses. An application to his reason will seldom prove very successful.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“Science is intimately integrated with the whole social structure and cultural tradition. They mutually support one other—only in certain types of society can science flourish, and conversely without a continuous and healthy development and application of science such a society cannot function properly.”
—Talcott Parsons (1902–1979)
“A Conservative government is an organised hypocrisy.”
—Benjamin Disraeli (1804–1881)
“March indefatigably on,
And for the last effect
Still keep thy Sword erect:
Besides the force it has to fright
The Spirits of the shady Night;
The same Arts that did gain
A Pow’r must it maintain.”
—Andrew Marvell (1621–1678)