View source for Free harmonic oscillation

A sinusoidal oscillation. If a mechanical or physical quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f0415701.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f0415702.png" /> denotes time, varies in accordance with the law

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f0415703.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

then it is said that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f0415704.png" /> performs a free harmonic oscillation. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f0415705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f0415706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f0415707.png" /> are real constants, called, respectively, the amplitude, frequency and phase of the free harmonic oscillation. The period is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f0415708.png" />. The following terminology is often used in physics and engineering. A free harmonic oscillation is called a harmonic oscillation, or a simple harmonic oscillation; functions of the form (1) are called [[Harmonics|harmonics]], the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f0415709.png" /> is called the instantaneous phase, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157010.png" /> is called the initial phase. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157011.png" /> is also called the circular or cyclic frequency, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157012.png" /> is called the frequency. A free harmonic oscillation (1) can be written as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157013.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157015.png" /> are connected by the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157016.png" /></td> </tr></table>

or by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157017.png" /></td> </tr></table>

Often the phase is taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157018.png" /> and not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157019.png" />.

Small oscillations of mechanical or physical systems with one degree of freedom near a stable non-degenerate equilibrium position are free harmonic oscillations, to a large degree of exactness. For example, small oscillations of a pendulum, oscillations of a load suspended by a string, oscillations of a tuning fork, the variation of the direction and strength of the current in an oscillating electrical circuit, the rolling of a ship, etc. A system performing free harmonic oscillation is called a linear harmonic oscillator, and its oscillation is described by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157020.png" /></td> </tr></table>

For a mathematical pendulum of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157021.png" /> and mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157023.png" />; for a load of mass <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157024.png" /> on a string with elasticity coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157026.png" />; for an oscillating electrical circuit of capacity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157027.png" /> and inductance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157029.png" />. The equilibrium position <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157031.png" /> in the phase plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157032.png" /> for a free harmonic oscillator is the centre, and the phase trajectories are circles.

The sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157033.png" /> of two free harmonic oscillations, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157034.png" /></td> </tr></table>

with commensurable frequencies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157036.png" /> is a free harmonic oscillation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157038.png" /> are incommensurable, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157039.png" /> is an [[Almost-periodic function|almost-periodic function]], and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157040.png" /></td> </tr></table>

The sum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157041.png" /> free harmonic oscillations with rationally-independent frequencies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157042.png" /> is also almost-periodic. For the sum of two free harmonic oscillations, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157043.png" /> is called the derangement. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157044.png" /> is small, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157045.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157047.png" /> have the same order of magnitude, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157048.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157049.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157050.png" /></td> </tr></table>

The  "amplitude"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157051.png" /> is a slowly-varying function of period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157052.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157053.png" /> varies from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157054.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157055.png" />. The oscillation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157056.png" /> is called a beat, and the  "amplitude"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157057.png" /> alternatingly increases and decreases. This case is important in the analysis of receiving devices.

Suppose one has a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157058.png" /> equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157059.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157061.png" /> are real symmetric positive-definite matrices with constant elements. By using an invertible transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157062.png" /> this system is transformed into the decomposed system

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157063.png" /></td> </tr></table>

The coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157064.png" /> are called normal. In normal coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041570/f04157065.png" /> is the vector sum of free harmonic oscillations along the coordinate axes.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Andronov,   A.A. Vitt,   A.E. Khaikin,   "Theory of oscillators" , Dover, reprint  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.S. Gorelik,   "Oscillations and waves" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.D. Landau,   E.M. Lifshits,   "Mechanics" , Pergamon  (1965)  (Translated from Russian)</TD></TR></table>