## Download A Lemma on Systems of Knotted Curves by Alexander J. W. PDF

By Alexander J. W.

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C)). l-z. ,((z)q( /)p( c - . /. 5) with the ()ne set ()f E(ls. 3) for 1,11(; overa, ll ray-nl()ti()ii, tllrns into two 1D t)rot)l(;~11s, rift(;(1 t)y tll(; tta,miltonia, ns H:,, a,n(t Hy with the two sets of Eqs. 8) for tll(; "I)a,rtial" ray-i~()tions in tlm two sy~lm~(;t,ry planes of 1,11(;system. 2 Azially symmetric systems Tile henfi-sph(;ri(:a,1 lens i~l Fig. b) is a,n example of an optical system, which is symmetric under rotations around the ot)tical axis. 2]. 2]. It is convenient to transform the rectangular coordinates qx and qy to the polar coordinates p and 4) in the transverse plane, with q, - p c o s \$ , qy - psin 4).

F,g}} - A - k{:,,} - k{g,f}. 43) The cominutator of two Lie operators is therefore a Lie operator as well. 42). , the commutator bracket [kf, kg ]. 4 + {s, A ~2 and hence {f,g} - qp, we find: W1 Hamilton's equations for the light-ray Hamilton's equations can be cast in terms of Poisson brackets. 4 1) Linear Ray and Wave Optics in Phase Space 20 the ray-vector u(z) containing the z-dependent ray variables, whose known values ~t the entrance plane a,t zi are grouped into the vector ui. 2) t l';(Is.

Nt f()I'Iii dd- ~ U ( Z ) - AL . t()r ~l()tati()ll. Ft)r xl()t,a,titmal (:(m('ist;Iless, (rely tile t)()ssit)lc (tt;I)(nl(ttnl(:(; ()f k , (hi z is ('xl)li(:itly sll()wn. As a,ll'ea,tty rt;nm,rkt;(t, tll(; st)hltitnl ()f Eft. 3) is (,ll(', mlswer 1,() tll('~ flm(talllental t)I'()t)lt'~ni ()f gt;()metrit:al ()t)tit:s: 1,()(l(;tel'nline tile illt)ll(,-()ll(,t)ll(, rela,ti(ms f~n" a. giv(;~ ()t)ti(:al (lcvi(:c, wl~i(:l~ all()w ~s t() fin(1 ()~t tl~c lineal vc('t()rs u,, f()r tlmt the ray trm~sfer rela,ti(n~s ui --~ uo have s(m~(; (l(~sire(l 1)r()t)er(,ics.