Nonimaging Optics

Appendix C CONSERVATION OF SKEW AND LINEAR MOMENTUM

C.1 SKEW INVARIANT

The conservation of skew in rotational symmetric systems or the conservation of linear momentum in the linear symmetric ones comes easily in the Hamiltonian formulation. The Hamiltonian system is given by Eq. (6.16) for a system i₁, i₂, i₃ of orthogonal coordinates. For a rotationally symmetric system, let us assume that θ is the coordinate of the symmetry—that is, ∂n/∂θ = 0, where n is the refractive index distribution function. We will identify θ with i₂ of Eq. (6.16). Then we have

     (C.1)

where the Hamiltonian function is

     (C.2)

a1, a2, and a3 are, respectively, ...