Lagrange invariant

In <a href="/facts/Optics/p0vq4S1T">optics</a> the Lagrange invariant is a measure of the light propagating through an optical system. It is defined by

H
 =
 n
 
 
 u
 ¯
 
 
 y
 −
 n
 u
 
 
 y
 ¯
 
 
 
 
 {\displaystyle H=n{\overline {u}}y-nu{\overline {y}}}
 
,
where y and u are the <a href="/facts/Marginal_ray/PgUEC7Sm">marginal ray</a> height and angle respectively, and ȳ and ū are the <a href="/facts/Chief_ray/PgUEC7Sm">chief ray</a> height and angle. n is the ambient <a href="/facts/Refractive_index/yjkzvgeE">refractive index</a>. In order to reduce confusion with other quantities, the symbol <a href="/facts/%25D0%2596/bthleFIE">Ж</a> may be used in place of H. Ж2 is proportional to the throughput of the optical system (related to <a href="/facts/%25C3%2589tendue/nApbAC9m">étendue</a>). For a given optical system, the Lagrange invariant is a constant throughout all space, that is, it is invariant upon <a href="/facts/Refraction/S8kvSKzM">refraction</a> and transfer.
The optical invariant is a generalization of the Lagrange invariant which is formed using the <a href="/facts/Ray_(optics)/PgUEC7Sm">ray</a> heights and angles of any two rays. For these rays, the optical invariant is a constant throughout all space.

Lagrange invariant open-in-new

Lagrange invariant