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In [[optics]] the '''Lagrange invariant''' is a measure of the light propagating through an optical system. It is defined by
:<math>H = n\overline{u}y - nu\overline{y}</math>,
where ''y'' and ''u'' are the [[marginal ray]] height and angle respectively, and ''ȳ'' and ''ū'' are the [[chief ray]] height and angle. ''n'' is the ambient [[refractive index]].  In order to reduce confusion with other quantities, the symbol [[Ж]] may be used in place of H.<ref name=Greivenkamp28>{{cite book |first=John E. |last=Greivenkamp |year=2004 |title=Field Guide to Geometrical Optics |publisher=SPIE |others=SPIE Field Guides vol. '''FG01''' |isbn=0-8194-5294-7 |page=28}}</ref> ''Ж''<sup>2</sup> is proportional to the throughput of the optical system (related to [[étendue]]).<ref name=Greivenkamp28/>  For a given optical system, the Lagrange invariant is a constant throughout all space, that is, it is invariant upon [[refraction]] and transfer.

The '''optical invariant''' is a generalization of the Lagrange invariant which is formed using the [[ray (optics)|ray]] heights and angles of any two rays.  For these rays, the optical invariant is a constant throughout all space.<ref>[http://www.newport.com/Optics-Fundamentals/604533/1033/content.aspx Optics Fundamentals], Newport Corporation, retrieved 9/8/2011</ref>

==References==
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[[Category:Geometrical optics]]