Two Lagrange-like optical invariants and some applications.

Geometric optics can be completely derived from Fermat's principle, as classical mechanics can be obtained by the application of the Hamilton principle. In Lagrangian optics, for optical systems with rotational symmetry, is known the invariant L₃, the Lagrange optical invariant. For systems built only with spherical lenses, we demonstrate there are two other optical invariants, L₁ and L₂, analogous to L₃. A proof based on Snell's law, the Weierstrass-Erdman jump condition, and the expression of the ray between two optical surfaces in the Hamiltonian formalism is reported. The presence of a conserved vector, L, allows us to write the equation of an emerging ray without any approximation.