Angle eikonal from reciprocal action and other optical-mechanical analogues

In this work, and within the frame of the optical-mechanical analogy, the g eneral angle eikonal xi (Legendre transform of phase or eikonal psi(r, t)) is identified as the optical analogue of the time integral of co-Lagrangian function F and its variation in the surface separating two media of differ ent refractive indices is identified as the increment of its mechanical ana logue when an impulsive constraint is suddenly imposed to a mechanical syst em. An extremal principle, dual tin the sense of Young) to that of Fermat i s stated and together with an additional dispersion relation allows to reob tain canonical ray equations. The optical analogue of co-Lagrangian functio n is introduced and its Euler-Lagrange equation is found in k-space. It is also shown that xi verifies an equation dual to the eikonal one, allowing t o broaden the number of known analytical solutions of the eikonal equation. An example illustrates this duality between eikonal and angle eikonal.