Lagrange invariant

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

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

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

See also

• Etendue
• Smith-Helmholtz invariant
• Abbe sine condition