Lagrange invariant

In optics the Lagrange invariant is a measure of the light propagating through an optical system. It is defined by
 * $$H = n\overline{u}y - nu\overline{y}$$,

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 $H$. $Ж^{2}$ is proportional to the throughput of the optical system (related to étendue). 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 heights and angles of any two rays. For these rays, the optical invariant is a constant throughout all space.