## Equivalence of the Etendue and Lagrange Invariants

Assuming an optical system with both ends in the same medium (n' = n) such as in air, an equivalence between the Lagrange invariant and the Etendue can be derived from the above relations.

If the entrance pupil is a square aperture, then A = 4x2, and if the field is a square solid angle, O = 4(^ax, then we obtain from (1.36a) and (1.34) the equivalence

If the entrance pupil and field of view are circular, then A = nx2 and O = 2n( 1 -cos (pmax). If the field angle is small then O = K(p(nax and we obtain similarly the equivalence

S. Carnot (1796-1832), in posthumous papers, noticed that (Bruhat [22]): "in a general thesis, the driving power is an invariable quantity in nature which strictly speaking is neither produced nor destroyed." This was brought to attention by Clapeyron. In 1845, J.R. Mayer gave the first formulation of the general law on energy conservation during a transfer process. These notions were given more accurate mathematical forms by Helmholtz (1847), Clausius (1850), who later introduced the term "entropy," and Lord Kelvin (1853).

Hence, one of the most fundamental theorems derived from Gaussian optics and applying to any optical system, may be stated as:

Aperture diameter D (m)

Fig. 1.19 Optical Etendue E = (n2/4) D2 q^ax of various telescopes equipped with multi-object spectroscopic facilities. Compared to the displayed curve, the gain in Etendue is greater than 6 with

Lamost

Aperture diameter D (m)

Fig. 1.19 Optical Etendue E = (n2/4) D2 q^ax of various telescopes equipped with multi-object spectroscopic facilities. Compared to the displayed curve, the gain in Etendue is greater than 6 with

### Lamost

^ The Etendue invariant E is related to the thermodynamics Carnot's law which expresses that, in a closed system, entropy does not decrease so the total energy transferred through a perfect optical system is conserved.

^ The Lagrange invariant H is a 1-dimensional representation of the optical Etendue.

Denoting D = 2x the input pupil diameter of a telescope, from (1.36c) and for a circular semi-field qmax, the Etendue is

For instance, this relation allows comparing the Etendue of various telescopes (Fig. 1.19).