The Challenge of Absolute Instruments

This chapter illustrates the challenges associated with obtaining absolute instruments, i.e., devices that stigmatically image a region of 3D. The starting point is an analogy between mechanics and optics which relates the refractive index for an “optical” problem to a potential in a “mechanics” problem. There are deep mathematical connections between absolute instruments and superintegrable systems. We review some essentials of separable Hamiltonians and superintegrable systems, starting from Hamilton equations of motion. The main issue we encounter is that the relevant mechanical potential depends on velocity, so separation of variables (in order to obtain a separable Hamiltonian) is not straightforward. Moreover, if we wish to use liquid crystals to construct absolute instruments, the relevant Hamiltonian is actually quadratic in the momentum variables. Finally, we make an interesting connection between absolute instruments to Riemannian geometry (more precisely, Wiedersehen manifolds) through the so-called Wiedersehenmannigfaltigkeit.