%!PS-Adobe-2.0 %%Creator: dvips 5.83 Copyright 1998 Radical Eye Software %%Title: r13lk2.dvi %%CreationDate: Mon Nov 3 23:07:54 2003 %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\EMTEX\DVIPS32.EXE -O0.0mm,-8.8mm r13lk2 %DVIPSParameters: dpi=300, compressed %DVIPSSource: TeX output 2003.11.03:2307 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed 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1139 y Fl(g)894 1119 y Fk(0)887 1153 y Fm(i)n Fi(\()e Fm(a)i Fi(\))956 1139 y Fo(=)14 b(0)50 b Fl(:)703 b Fo(\(12\))0 1226 y(Multiplying)15 b(\(7\))i(b)o(y)g Fl(g)440 1235 y Fm(i)n Fi(\()-7 b Fm(a)i Fi(\))512 1226 y Fo(and)16 b(considering)f(\(8\),)725 1314 y Fl(G)757 1323 y Fi(\()-7 b Fm(a)i Fi(\))815 1314 y Fo(=)13 b Fl(g)886 1323 y Fm(i)n Fi(\()-7 b Fm(a)i Fi(\))942 1314 y Fo(\000)973 1321 y Fm(ij)1022 1314 y Fl(g)1041 1323 y Fm(j)p Fi(\()e Fm(a)i Fi(\))1151 1314 y Fl(:)636 b Fo(\(13\))0 1401 y(Multiplying)15 b(\(10\))i(b)o(y)g Fl(g)465 1410 y Fm(i)n Fi(\()-7 b Fm(a)i Fi(\))537 1401 y Fo(and)16 b(considering)f(\(7\),)i(\(8\),)g (\(12\),)732 1488 y Fl(G)771 1468 y Fk(0)764 1502 y Fi(\()-7 b Fm(a)i Fi(\))822 1488 y Fo(=)13 b Fl(g)893 1497 y Fm(i)n Fi(\()-7 b Fm(a)i Fi(\))949 1488 y Fo(\000)980 1468 y Fk(0)980 1501 y Fm(ij)1015 1488 y Fl(g)1034 1497 y Fm(j)p Fi(\()e Fm(a)i Fi(\))1145 1488 y Fl(:)642 b Fo(\(14\))0 1576 y(Multiplying)15 b(\(11\))i(b)o(y)g Fl(g)465 1585 y Fm(i)n Fi(\()-7 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1838 y Fl(g)618 1817 y Fk(0)611 1851 y Fm(i)n Fi(\()e Fm(a)i Fi(\))667 1838 y Fl(G)699 1847 y Fi(\()e Fm(a)i Fi(\))757 1838 y Fo(=)14 b Fl(g)829 1847 y Fm(i)n Fi(\()-7 b Fm(b)i Fi(\))881 1838 y Fo(\000)912 1817 y Fk(0)912 1850 y Fm(ij)947 1838 y Fl(g)966 1847 y Fm(j)p Fi(\()e Fm(a)i Fi(\))1037 1838 y Fo(+)11 b Fl(g)1106 1847 y Fm(i)n Fi(\()-7 b Fm(b)i Fi(\))1158 1838 y Fl(g)1184 1817 y Fk(0)1177 1851 y Fm(i)n Fi(\()e Fm(a)i Fi(\))1233 1838 y Fl(G)1265 1847 y Fi(\()e Fm(b)i Fi(\))1355 1838 y Fl(:)432 b Fo(\(16\))0 1936 y(Equations)16 b(\(8\),)h(\(9\),)g (\(12\))g(and)f(\(16\))h(yield)g(\()876 1924 y(\024)870 1936 y(Cerv)o(en)q(\023)-26 b(y)16 b(2001,)g(eqs.)h(4.14.9)f(and)g (4.14.10\).)645 2059 y Fl(g)671 2039 y Fk(0)664 2073 y Fm(i)n Fi(\()-7 b Fm(a)i Fi(\))734 2059 y Fo(=)786 2012 y Fh(X)787 2119 y Fm(b)p Fk(6)p Fi(=)p Fm(a)866 2059 y Fl(g)885 2068 y Fm(i)n Fi(\()e Fm(b)i Fi(\))943 2019 y Fl(g)962 2028 y Fm(j)p Fi(\()e Fm(b)i Fi(\))1019 2019 y Fo(\000)1050 2001 y Fk(0)1050 2033 y Fm(j)r(k)1093 2019 y Fl(g)1112 2028 y Fm(k)o Fi(\()e Fm(a)i Fi(\))p 943 2048 233 2 v 955 2093 a Fl(G)987 2102 y Fi(\()e Fm(a)i Fi(\))1042 2093 y Fj(\000)11 b Fl(G)1124 2102 y Fi(\()-7 b Fm(b)i Fi(\))1231 2059 y Fl(:)556 b Fo(\(17\))0 2201 y(W)l(e)15 b(transform)e(the)i(Christo\013el)e(matrix)h(\(1\),)i(its)e (\014rst{order)f(deriv)m(ativ)o(es)h(\(2\),)h(\(3\),)h(and)e(second{)0 2260 y(order)h(deriv)m(ativ)o(es)h(\(4\),)i(\(5\),)f(\(6\))g(in)o(to)f (the)h(eigen)o(v)o(ectors,)729 2348 y(\000)753 2357 y Fi(\()-7 b Fm(ab)i Fi(\))828 2348 y Fo(=)14 b Fl(g)900 2357 y Fm(i)n Fi(\()-7 b Fm(a)i Fi(\))956 2348 y Fo(\000)987 2355 y Fm(ij)1022 2348 y Fl(g)1041 2357 y Fm(j)p Fi(\()e Fm(b)i Fi(\))1147 2348 y Fl(;)640 b Fo(\(18\))729 2435 y(\000)760 2414 y Fk(0)753 2449 y Fi(\()-7 b Fm(ab)i Fi(\))828 2435 y Fo(=)14 b Fl(g)900 2444 y Fm(i)n Fi(\()-7 b Fm(a)i Fi(\))956 2435 y Fo(\000)987 2414 y Fk(0)987 2447 y Fm(ij)1022 2435 y Fl(g)1041 2444 y Fm(j)p Fi(\()e Fm(b)i Fi(\))1147 2435 y Fl(;)640 b Fo(\(19\))729 2521 y(\000)760 2500 y Fk(0)p Fm(?)753 2534 y Fi(\()-7 b Fm(ab)i Fi(\))828 2521 y Fo(=)14 b Fl(g)900 2530 y Fm(i)n Fi(\()-7 b Fm(a)i Fi(\))956 2521 y Fo(\000)987 2500 y Fk(0)p Fm(?)987 2533 y(ij)1022 2521 y Fl(g)1041 2530 y Fm(j)p Fi(\()e Fm(b)i Fi(\))1147 2521 y Fl(:)640 b Fo(\(20\))0 2599 y(Equation)16 b(\(13\))h(ma)o(y)f(then)h(b)q(e)g(expressed)e(as)797 2686 y Fl(G)829 2695 y Fi(\()-7 b Fm(a)i Fi(\))887 2686 y Fo(=)14 b(\000)964 2695 y Fi(\()-7 b Fm(aa)i Fi(\))1079 2686 y Fl(;)708 b Fo(\(21\))908 2806 y(121)p eop %%Page: 122 4 122 3 bop 0 -54 a Fo(equation)16 b(\(14\))i(as)797 12 y Fl(G)836 -9 y Fk(0)829 25 y Fi(\()-7 b Fm(a)i Fi(\))887 12 y Fo(=)14 b(\000)971 -9 y Fk(0)964 25 y Fi(\()-7 b Fm(aa)i Fi(\))1079 12 y Fl(;)708 b Fo(\(22\))0 97 y(and)16 b(equation)g(\(15\))h(with)g(\(17\))g(as)599 232 y Fl(G)638 212 y Fk(0)p Fm(?)631 246 y Fi(\()-7 b Fm(a)i Fi(\))689 232 y Fo(=)14 b(\000)773 212 y Fk(0)p Fm(?)766 246 y Fi(\()-7 b Fm(aa)i Fi(\))843 232 y Fo(+)11 b(2)926 185 y Fh(X)927 292 y Fm(b)p Fk(6)p Fi(=)p Fm(a)1031 188 y Fo(\000)1062 170 y Fk(0)1055 204 y Fi(\()-7 b Fm(ab)i Fi(\))1117 188 y Fo(\000)1148 170 y Fm(?)1141 204 y Fi(\()e Fm(ab)i Fi(\))p 1012 221 210 2 v 1012 266 a Fl(G)1044 275 y Fi(\()e Fm(a)i Fi(\))1099 266 y Fj(\000)11 b Fl(G)1181 275 y Fi(\()-7 b Fm(b)i Fi(\))1277 232 y Fl(:)510 b Fo(\(23\))0 382 y(Equations)14 b(\(2\){\(6\),)i(\(19\),)f(\(20\),)h(\(22\))f(and)f (\(23\))h(are)g(suitable)e(for)i(the)f(n)o(umerical)f(calculation)h(of) 0 442 y(the)i(\014rst)e(and)h(second)g(partial)f(phase{space)g(deriv)m (ativ)o(es)h(of)h(the)f(eigen)o(v)m(alues)g(of)g(the)h(Christo\013el)0 502 y(matrix.)0 636 y Fq(3.)25 b(Ham)o(iltonian)19 b(and)f(its)i (partial)g(phase{space)f(deriv)m(ativ)n(es)0 717 y Fo(F)l(or)14 b(arithmetic)g(a)o(v)o(eraging)f(of)h(the)h(Hamiltonians)f(of)g(b)q (oth)h(S{w)o(a)o(v)o(e)f(p)q(olarizations,)f(w)o(e)h(consider)0 776 y(homogeneous)j(Hamiltonians)h(of)h(arbitrary)f(degree)h Fl(N)5 b Fo(.)30 b(The)19 b(a)o(v)o(eraged)f(Hamiltonian)g(is)h(then)0 836 y(also)j(homogeneous)f(and)h(of)h(degree)f Fl(N)5 b Fo(.)41 b(According)21 b(to)i(Euler's)f(theorem)g(on)g(homogeneous)0 896 y(functions,)e(the)g(parameter)e(along)i(ra)o(ys)e(is)i(then)g (prop)q(ortional)e(to)i(the)g(tra)o(v)o(el)f(time.)32 b(W)l(e)20 b(th)o(us)0 956 y(c)o(ho)q(ose)13 b(the)h(v)m(alue)g(of)g (the)h(Hamiltonian)d(in)i(the)g(stationary)f(Hamilton{Jacobi)f (equation)i(so)g(that)0 1015 y(the)j(parameter)e(along)h(ra)o(ys)f(is)i (equal)f(to)h(the)g(tra)o(v)o(el)f(time,)738 1132 y Fl(H)t Fo(\()p Fl(x)830 1139 y Fm(m)869 1132 y Fl(;)8 b(\034)906 1139 y Fm(;n)945 1132 y Fo(\))14 b(=)1047 1098 y(1)p 1037 1120 46 2 v 1037 1166 a Fl(N)1138 1132 y(;)649 b Fo(\(24\))0 1244 y(where)22 b Fl(N)28 b Fo(is)23 b(the)g(degree)f(of)h (the)g(Hamiltonian)e Fl(H)t Fo(\()p Fl(x)1067 1251 y Fm(m)1107 1244 y Fl(;)8 b(p)1154 1251 y Fm(n)1181 1244 y Fo(\),)24 b(homogeneous)d(with)i(resp)q(ect)g(to)0 1304 y(slo)o(wness)15 b(v)o(ector)h Fl(p)373 1311 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Fk(0)1078 2623 y Fi(\()-7 b(1)i(\))1121 2610 y Fl(G)1160 2589 y Fm(?)1153 2623 y Fi(\()e(1)i(\))1195 2610 y Fo(\()p Fl(G)1246 2619 y Fi(\()e(1)i(\))1289 2610 y Fo(\))1314 2576 y Fg(N)p 1315 2582 30 2 v 1321 2601 a Ff(2)1350 2589 y Fk(\000)p Fi(2)1405 2610 y Fo(+)r Fl(G)1485 2589 y Fk(0)1478 2623 y Fi(\()e(2)i(\))1520 2610 y Fl(G)1559 2589 y Fm(?)1552 2623 y Fi(\()e(2)i(\))1595 2610 y Fo(\()p Fl(G)1646 2619 y Fi(\()e(2)i(\))1689 2610 y Fo(\))1714 2576 y Fg(N)p 1714 2582 V 1720 2601 a Ff(2)1749 2589 y Fk(\000)p Fi(2)1803 2554 y Fh(i)1876 2610 y Fl(:)1801 2686 y Fo(\(30\))908 2806 y(122)p eop %%Page: 123 5 123 4 bop 0 -54 a Fo(W)l(e)16 b(express)g(the)g(second)f(deriv)m(ativ)o (es)h(of)g(the)g(Hamiltonian)f(in)h(terms)f(of)i(the)f(deriv)m(ativ)o (es)g(of)g(the)0 6 y(Christo\013el)g(matrix)g(using)f(equations)h (\(22\))h(and)f(\(23\),)60 149 y Fl(H)105 128 y Fk(0)p Fm(?)153 149 y Fo(=)212 115 y(1)p 212 137 25 2 v 212 183 a(4)243 93 y Fh(h)266 149 y Fo(\000)297 128 y Fk(0)p Fm(?)290 162 y 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1146 955 V 1146 975 a(2)1183 980 y Fo(+)11 b(\()p Fl(G)1284 989 y Fi(\()-7 b(2)i(\))1326 980 y Fo(\))1351 949 y Ff(1)p 1352 955 V 1352 975 a(2)1377 980 y Fo(]\()p Fl(G)1442 989 y Fi(\()e(2)i(\))1485 980 y Fo(\))1510 949 y Ff(1)p 1510 955 V 1510 975 a(2)1591 939 y Fl(:)196 b Fo(\(33\))0 1070 y(F)l(or)15 b Fl(N)c Fo(=)6 b Fj(\000)p Fo(1,)291 1166 y(\()p Fl(G)342 1175 y Fi(\()-7 b(1)i(\))385 1166 y Fo(\))410 1134 y Fg(N)p 410 1140 30 2 v 416 1160 a Ff(2)445 1148 y Fk(\000)p Fi(1)502 1166 y Fj(\000)s Fo(\()p Fl(G)595 1175 y Fi(\()e(2)i(\))637 1166 y Fo(\))662 1134 y Fg(N)p 662 1140 V 669 1160 a Ff(2)698 1148 y Fk(\000)p Fi(1)p 291 1192 461 2 v 416 1237 a Fl(G)448 1246 y Fi(\()e(1)i(\))502 1237 y Fj(\000)10 b Fl(G)583 1246 y Fi(\()-7 b(2)i(\))771 1203 y Fo(=)14 b Fj(\000)957 1166 y Fl(G)989 1175 y Fi(\()-7 b(1)i(\))1043 1166 y Fo(+)10 b(\()p Fl(G)1143 1175 y Fi(\()-7 b(1)i(\))1186 1166 y Fl(G)1218 1175 y Fi(\()e(2)i(\))1261 1166 y Fo(\))1286 1134 y Ff(1)p 1286 1140 17 2 v 1286 1160 a(2)1323 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Fm(N)277 2476 y Fi(3)314 2464 y Fo(,)h Fl(d)370 2446 y Fi(3)370 2477 y Fm(M)414 2464 y Fo(,)f Fl(d)469 2446 y Fi(3)469 2476 y(3)506 2464 y Fo(are)g(determined)f(b)o(y)g (de\014nition)g(\(51\))i(and)e(\(52\).)22 b(Ev)o(olution)15 b(equations)671 2550 y(_)664 2563 y Fl(h)693 2543 y Fm(i)693 2575 y Fi(3)729 2563 y Fo(=)f Fl(H)827 2539 y Fm(;i)823 2577 y(;j)856 2563 y Fl(H)901 2543 y Fm(;j)945 2563 y Fj(\000)d Fl(H)1040 2543 y Fm(;ij)1088 2563 y Fl(H)1129 2570 y Fm(;j)1212 2563 y Fl(;)575 b Fo(\(68\))808 2660 y(_)801 2673 y(^)800 2686 y Fl(h)829 2665 y Fi(3)829 2698 y Fm(i)865 2686 y Fo(=)14 b Fj(\000)p Fl(H)998 2693 y Fm(;i)1076 2686 y Fl(;)711 b Fo(\(69\))908 2806 y(126)p eop %%Page: 127 9 127 8 bop 0 -54 a Fo(follo)o(w)16 b(from)h(de\014nition)e(\(51\))j(and) e(\(52\))i(and)e(ra)o(y)h(tracing)f(system)h(\(39\))g(and)g(\(40\).)24 b(Co)q(e\016cien)o(ts)0 6 y Fl(d)26 -12 y Fm(a)26 18 y Fi(3)67 6 y Fo(and)15 b Fl(d)189 -12 y Fi(3)189 20 y Fm(b)228 6 y Fo(are)i(th)o(us)e(uniquely)h(determined,)619 108 y Fl(d)645 87 y Fm(a)645 120 y Fi(3)683 108 y Fo(=)736 95 y(^)736 108 y Fl(h)765 87 y Fm(a)765 120 y(i)788 108 y Fo(\()p Fl(H)852 84 y Fm(;i)848 122 y(;j)882 108 y Fl(H)927 87 y Fm(;j)971 108 y Fj(\000)11 b Fl(H)1066 87 y Fm(;ij)1114 108 y Fl(H)1155 115 y Fm(;j)1188 108 y Fo(\))50 b Fl(;)530 b Fo(\(70\))797 218 y Fl(d)823 198 y Fi(3)823 230 y Fm(b)859 218 y Fo(=)13 b Fl(H)952 225 y Fm(;i)981 218 y Fl(h)1010 198 y Fm(i)1010 230 y(b)1080 218 y Fl(;)707 b Fo(\(71\))784 309 y Fl(d)810 288 y Fi(3)810 321 y(3)846 309 y Fo(=)14 b Fl(H)940 316 y Fm(;i)968 309 y Fl(H)1013 288 y Fm(;i)1092 309 y Fl(:)695 b Fo(\(72\))0 468 y Fq(6.)25 b(Basis)19 b(v)n(ectors)g(of)g(the)g(ra)n(y{cen)n(tred)h (co)r(ordinate)f(system)0 548 y Fo(There)13 b(are)h(man)o(y)f(p)q (ossibilities)f(ho)o(w)h(to)i(c)o(ho)q(ose)e(the)h(basis)f(v)o(ectors)h Fl(h)1334 530 y Fm(i)1334 560 y Fi(1)1356 548 y Fo(,)g Fl(h)1413 530 y Fm(i)1413 560 y Fi(2)1449 548 y Fo(\(or)1527 535 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Fo(\(78\))908 2806 y(127)p eop %%Page: 128 10 128 9 bop 0 -54 a Fq(6.2.)25 b(Notes)18 b(on)h(the)g(ev)n(olution)i (equations)f(for)e(the)h(basis)g(v)n(ectors)0 20 y Fo(The)14 b(ev)o(olution)f(di\013eren)o(tial)g(equations)h(for)g(the)g(basis)f(v) o(ectors)h(are)f(often)i(expressed)e(in)g(the)i(form)801 96 y(_)795 110 y Fl(h)824 89 y Fm(i)824 122 y(A)870 110 y Fo(=)e Fl(D)964 89 y Fm(i)963 122 y(j)984 110 y Fl(h)1013 86 y Fm(j)1013 124 y(A)1801 110 y Fo(\(79\))0 210 y(for)j(con)o(tra)o (v)m(arian)o(t)f(basis)g(v)o(ectors)h Fl(h)701 186 y Fm(j)701 223 y Fi(1)724 210 y Fo(,)g Fl(h)783 186 y Fm(j)783 223 y Fi(2)805 210 y Fo(,)g(or)g(in)h(the)f(form)801 291 y(_)794 304 y(^)794 317 y Fl(h)823 296 y Fm(A)823 329 y(i)869 317 y Fo(=)933 304 y(^)921 317 y Fl(D)963 293 y Fm(j)962 331 y(i)986 304 y Fo(^)985 317 y Fl(h)1014 296 y Fm(A)1014 329 y(j)1801 317 y Fo(\(80\))0 418 y(for)g(co)o(v)m (arian)o(t)f(basis)g(v)o(ectors)582 405 y(^)581 418 y Fl(h)610 400 y Fi(1)610 431 y Fm(j)632 418 y Fo(,)663 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Fl(H)1276 2691 y Fm(;j)1309 2684 y Fo(])g(+)1389 2671 y(^)1384 2684 y Fl(\030)1406 2691 y Fm(i)1422 2684 y Fl(H)1467 2664 y Fm(;j)1550 2684 y Fl(;)237 b Fo(\(91\))908 2806 y(128)p eop %%Page: 129 11 129 10 bop 0 -54 a Fo(where)149 -67 y(^)144 -54 y Fl(\030)166 -47 y Fm(i)199 -54 y Fo(is)16 b(an)g(arbitrary)f(v)o(ector)i(giv)o(en)f (along)g(the)h(ra)o(y)l(.)100 6 y(The)e(relations)f(b)q(et)o(w)o(een)h (co)q(e\016cien)o(ts)f Fl(d)869 -12 y Fm(A)869 19 y(B)904 6 y Fo(,)h Fl(D)975 -12 y Fm(i)974 19 y(j)1011 6 y Fo(and)1118 -7 y(^)1106 6 y Fl(D)1148 -18 y Fm(j)1147 20 y(i)1185 6 y Fo(follo)o(w)f(from)h(de\014nitions)f(\(65\),)h(\(79\))0 66 y(and)h(\(80\),)610 131 y Fl(d)636 110 y Fm(A)636 143 y(B)684 131 y Fo(=)737 118 y(^)737 131 y Fl(h)766 110 y Fm(A)766 143 y(i)798 131 y Fl(D)840 110 y Fm(i)839 143 y(j)860 131 y Fl(h)889 107 y Fm(j)889 145 y(B)937 131 y Fo(=)e Fj(\000)p Fl(h)1058 110 y Fm(i)1058 143 y(B)1103 118 y Fo(^)1091 131 y Fl(D)1133 107 y Fm(j)1132 145 y(i)1155 118 y Fo(^)1155 131 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