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A mathematical term introduced by J.L. Lagrange [[#References|[1]]] to denote a small displacement of an independent variable or of a functional. The method of variations is a method of studying an extremal problem in which the variations of the functional caused by small displacements of the argument are studied. This is one of the principal methods for studying extremal problems (hence the name [[Variational calculus|variational calculus]]).

Let a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v0961001.png" /> be given on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v0961002.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v0961003.png" /> be a parameter space. A variation of the argument <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v0961004.png" /> is an ordinary curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v0961005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v0961006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v0961007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v0961008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v0961009.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v09610010.png" /> which passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v09610011.png" /> in a certain neighbourhood defined by the restrictions that are in force. Let the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v09610012.png" /> correspond to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v09610013.png" />. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v09610014.png" /> runs through the set of all parameters, the variations run through a certain family of curves issuing from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v09610015.png" />. In finite-dimensional and infinite-dimensional analysis, beginning with Lagrange, it is usual to employ the directional variation where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v09610016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v09610017.png" />. In this case the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096100/v09610018.png" /> is referred to as the variation. However, other classes of variations are employed in geometry, in variational calculus and, in particular, in the theory of optimal control; these include polygonal variations, needle-shaped or spiked variations and variations connected with sliding regimes [[#References|[2]]], [[#References|[3]]]. The choice of the space of variations and the construction of the variations themselves are a very important element in obtaining necessary conditions for an extremum. See also [[Variation of a functional|Variation of a functional]]; [[Gâteaux derivative|Gâteaux derivative]]; [[Fréchet derivative|Fréchet derivative]]; [[Functional derivative|Functional derivative]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.L. Lagrange,   "Essai d'une nouvelle methode pour déterminer les maximas et les minimas des formules intégrales indéfinies" , ''Oevres'' , '''1''' , G. Olms , New York  (1973)  pp. 333–362</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.A. Bliss,   "Lectures on the calculus of variations" , Chicago Univ. Press  (1947)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.S. Pontryagin,   V.G. Boltayanskii,   R.V. Gamkrelidze,   E.F. Mishchenko,   "The mathematical theory of optimal processes" , Wiley  (1962)  (Translated from Russian)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.M. Gel'fand,   S.V. Fomin,   "Calculus of variations" , Prentice-Hall  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Cesari,   "Optimization - Theory and applications" , Springer  (1983)</TD></TR></table>