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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

250 Def'mition A

250 Def'mition A generalized mechanical system is one represented on all time intervals [t,t'] by Lagrangian submanifolds R (t,t~) c_ (P,c0) in general position. Generally R(t,t') does not represent the graph of a symplectomorphism (cf. (I. 1)). As an example of a generalized mechanical system one can consider the constrained mechanical systems introduced by Dirac [1950]. An infinitesimal description of dynamics is given by the limit t' -~ t applied to (P,¢0). The infinitesimal limit of the Lagrangian submanifold R(t,t') is a Lagrangian dR(~t'). submanifold 1~ t (= ) of the space of vectors 13 tangent to phase space trajectories of the system, with coordinates (q(t), p(t), dl(t), lb(t)), and endowed with the symplectic form do~ = o~ = ~dlbi(t) ^ dqi(t ) + Y~dPi(t) A ddli(t) dt i i = limt (~dPi(t')^ dqi(t') - ZdPi(t) ^ dqi(t) ) . t%t t-t If we take the canonical fibration defined by the 1-form -- Z(lbi(t)dqi(t) - dli(t)dPi(t)), ob = dO 1 and 1~ t is pararnetrized by (q(t), p(t)), then there is a function Ht(q(t), p(t)) such that I~ t is described by the equation Z(Oi(t)dqi(t) - dli(t)dPi(t)) = -dHt(q(t), p(t)) 1 and the generating function H t is the Hamiltonian of the system. Remark The construction described above is successfully used in geometrical optics, mainly by composition of the universal phase spaces to reduce the differential equations of rays to their corresponding difference equations. Consider an optical instrument consisting of a number of refracting surfaces of revolution (Fig. A.1). (x,y) z~ t 1 n 1 z, (~ 1,Yl ) t3 n 3 Fig. A.1 Structure of a general optical instrument. (xk,Y k) z~÷2 *(x',y')

251 Assume that the surfaces have the z-axis as a common axis of revolution. The canonical differential equations are 1 = vp ,fo = - Dx 1 ~, = ~-q, tl = - Dy ~)(z) where the dot means ~, d "~(z) is the refractive index, D(z) = ~-), and R(z) is the radius of curvature of the surface. An alternative description is to introduce a system of canonical difference equations Xi+l - xi-1 -- 8iPi+ Pi+1 - Pi-1 = - Dixi Yi+l - Yi-1 = 8iqi - qi+l - qi-1 = - DiYi ti Vi+l - '#i-1 5i = R-~. ' Di = R i with composed generating function k+! k 2 2 1 8i(p~+qi2)_ I ~ Di(x i +Yi)" i=1,3... (Xi+l-xi- 1)Pi + (Yi+l-Yi- 1)qi - ~ 7 i:2,4... Appendix 2 Diffraction at a Circular Obstacle. When a glancing ray encounters a smooth obstacle the resulting wavefront has two components: one due to reflection from the 'front' of the boundary of the obstacle and one due to diffraction around the 'back'. We analyse the geometry of such a situation in the case of a circular obstacle (the result is much the same for any generic smooth obstacle but the circle is convenient for explicit calculations) showing in particular that the union of the two wavefronts is C 2 but not C 3. t=0 ~Y Fig.A.2 Notation for diffraction at a circular obstacle. /

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