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 and are the marginal ray height and angle respectively, and and are the chief ray height and angle. is the ambient refractive index. In order to reduce confusion with other quantities, the symbol may be used in place of .[1] 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]