View source for Energy integral

A quantity representing the sum of the kinetic and the potential energy of a mechanical system at a certain moment in time.

Suppose, for example, that in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035640/e0356401.png" /> with piecewise-smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035640/e0356402.png" /> for a hyperbolic partial differential 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/e/e035/e035640/e0356403.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

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/e/e035/e035640/e0356404.png" /></td> </tr></table>

one poses the mixed problem

<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/e/e035/e035640/e0356405.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</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/e/e035/e035640/e0356406.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</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/e/e035/e035640/e0356407.png" /></td> </tr></table>

A classical solution of (2)–(3) is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035640/e0356408.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035640/e0356409.png" /> satisfying (1) in the cylinder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035640/e03564010.png" />, the initial conditions (2) on the lower base, and the boundary condition (3) on the lateral surface of the cylinder.

Then the relation

<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/e/e035/e035640/e03564011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

holds, 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/e/e035/e035640/e03564012.png" /></td> </tr></table>

The energy integral is defined as the quantity

<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/e/e035/e035640/e03564013.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/e/e035/e035640/e03564014.png" /></td> </tr></table>

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035640/e03564015.png" /> equality (4) takes the form

<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/e/e035/e035640/e03564016.png" /></td> </tr></table>

The physical meaning of the energy integral consists in the fact that the total energy of an oscillating system in the absence of external perturbations does not change in time (the law of conservation of energy).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O.A. Ladyzhenskaya,   N.N. Ural'tseva,   "Equations aux dérivées partielles de type elliptique" , Dunod  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O.A. Ladyzhenskaya,   V.A. Solonnikov,   N.N. Ural'tseva,   "Linear and quasi-linear equations of parabolic type" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Bers,   F. John,   M. Schechter,   "Partial differential equations" , Interscience  (1966)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  F. John,   "Partial differential equations" , Springer  (1978)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.R. Garabedian,   "Partial differential equations" , Wiley  (1967)</TD></TR></table>