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

- ,

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.