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# Lagrange invariant

In optics the Lagrange invariant is a measure of the light propagating through an optical system. It is defined by

${\displaystyle H=n{\overline {u}}y-nu{\overline {y}}}$,

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.[1] Ж2 is proportional to the throughput of the optical system (related to étendue).[1] 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.[2]