%!PS-Adobe-2.0 %%Creator: dvips 5.83 Copyright 1998 Radical Eye Software %%Title: r19lk1.dvi %%CreationDate: Sat Jun 20 10:04:19 2009 %%Pages: 12 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: C:\EMTEX\DVIPS32.EXE -O0.0mm,-8.8mm r19lk1 %DVIPSParameters: dpi=300, compressed %DVIPSSource: TeX output 2009.06.20:1004 %%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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974 y Fh(V)346 981 y Fg(;mj)423 974 y Fe(\000)g Fh(V)500 981 y Fg(;im)564 974 y Fh(H)609 950 y Fg(;m)605 988 y(;j)668 974 y Fe(\000)f Fh(V)744 981 y Fg(;im)808 974 y Fh(H)853 954 y Fg(;mn)928 974 y Fh(V)957 981 y Fg(;nj)1023 974 y Fe(\000)h Fh(V)1111 954 y Fa(\000)p Ff(1)1165 974 y Fj(\()p Fh(H)1225 981 y Fg(;i)1254 974 y Fh(V)1283 981 y Fg(;j)1325 974 y Fj(+)f Fh(V)1401 981 y Fg(;i)1430 974 y Fh(H)1471 981 y Fg(;j)1504 974 y Fj(\))h Fe(\000)g Fh(V)1620 954 y Fa(\000)p Ff(2)1673 974 y Fh(V)1702 981 y Fg(;i)1731 974 y Fh(V)1760 981 y Fg(;j)1793 934 y Fd(\003)1819 954 y Fg(@)r(x)1865 939 y Fc(j)p 1819 963 65 2 v 1820 991 a Fg(@)c Ff(~)-23 b Fg(y)1864 996 y Fb(~)-18 b Fc(c)229 1086 y Fj(+\()p Fh(V)316 1093 y Fg(;ij)375 1086 y Fj(+)10 b Fh(V)464 1066 y Fa(\000)p Ff(1)518 1086 y Fh(V)547 1093 y Fg(;i)575 1086 y Fh(V)604 1093 y Fg(;j)637 1086 y Fj(\))e(\()p Fh(H)728 1062 y Fg(;j)724 1101 y(;k)769 1066 y(@)r(x)815 1051 y Fc(k)p 769 1075 68 2 v 772 1103 a Fg(@)d Ff(~)-23 b Fg(y)816 1108 y Fb(~)-18 b Fc(c)853 1086 y Fj(+)11 b Fh(H)948 1066 y Fg(;j)r(k)1009 1064 y(@)r(y)1052 1069 y Fc(k)p 1009 1075 66 2 v 1011 1103 a Fg(@)5 b Ff(~)-23 b Fg(y)1055 1108 y Fb(~)-18 b Fc(c)1081 1086 y Fj(\))14 b(=)g Fe(\000)p Fh(H)1247 1093 y Fg(;ik)1303 1066 y(@)r(x)1349 1051 y Fc(k)p 1303 1075 68 2 v 1306 1103 a Fg(@)5 b Ff(~)-23 b Fg(y)1350 1108 y Fb(~)-18 b Fc(c)1388 1086 y Fe(\000)11 b Fh(H)1483 1062 y Fg(;k)1479 1100 y(;i)1526 1064 y(@)r(y)1569 1069 y Fc(k)p 1526 1075 66 2 v 1528 1103 a Fg(@)5 b Ff(~)-23 b Fg(y)1572 1108 y Fb(~)-18 b Fc(c)1647 1086 y Fh(:)140 b Fj(\(73\))0 1161 y(W)l(e)17 b(consider)e(the)i(homogeneit)o(y)e(of)i(Hamiltonian)e (function)i Fh(H)t Fj(,)0 1218 y Fd(\002)30 1258 y Fe(\000)8 b Fh(H)118 1265 y Fg(;ij)174 1258 y Fe(\000)h Fh(H)267 1234 y Fg(;m)263 1272 y(;i)317 1258 y Fh(V)346 1265 y Fg(;mj)423 1258 y Fe(\000)g Fh(V)500 1265 y Fg(;im)564 1258 y Fh(H)609 1234 y Fg(;m)605 1272 y(;j)668 1258 y Fe(\000)f Fh(V)744 1265 y Fg(;im)808 1258 y Fh(H)853 1237 y Fg(;mn)928 1258 y Fh(V)957 1265 y Fg(;nj)1023 1258 y Fe(\000)h Fh(V)1111 1237 y Fa(\000)p Ff(1)1165 1258 y Fj(\()p Fh(H)1225 1265 y Fg(;i)1254 1258 y Fh(V)1283 1265 y Fg(;j)1325 1258 y Fj(+)f Fh(V)1401 1265 y Fg(;i)1430 1258 y Fh(H)1471 1265 y Fg(;j)1504 1258 y Fj(\))h Fe(\000)g Fh(V)1620 1237 y Fa(\000)p Ff(2)1673 1258 y Fh(V)1702 1265 y Fg(;i)1731 1258 y Fh(V)1760 1265 y Fg(;j)1793 1218 y Fd(\003)1819 1238 y Fg(@)r(x)1865 1223 y Fc(j)p 1819 1246 65 2 v 1820 1275 a Fg(@)c Ff(~)-23 b Fg(y)1864 1280 y Fb(~)-18 b Fc(c)20 1370 y Fj(+\()p Fh(V)107 1377 y Fg(;ij)154 1370 y Fh(H)199 1346 y Fg(;j)195 1385 y(;k)243 1370 y Fj(+)11 b(2)p Fh(V)358 1349 y Fa(\000)p Ff(1)412 1370 y Fh(V)441 1377 y Fg(;i)469 1370 y Fh(H)510 1377 y Fg(;k)547 1370 y Fj(\))581 1350 y Fg(@)r(x)627 1335 y Fc(k)p 581 1358 68 2 v 584 1387 a Fg(@)5 b Ff(~)-23 b Fg(y)628 1392 y Fb(~)-18 b Fc(c)665 1370 y Fj(+)11 b(\()p Fh(V)763 1377 y Fg(;ij)811 1370 y Fh(H)856 1349 y Fg(;j)r(k)922 1370 y Fj(+)g Fh(V)1012 1349 y Fa(\000)p Ff(1)1065 1370 y Fh(V)1094 1377 y Fg(;i)1123 1370 y Fh(H)1168 1349 y Fg(;k)1205 1370 y Fj(\))1238 1348 y Fg(@)r(y)1281 1353 y Fc(k)p 1238 1358 66 2 v 1240 1387 a Fg(@)5 b Ff(~)-23 b Fg(y)1284 1392 y Fb(~)-18 b Fc(c)1323 1370 y Fj(=)14 b Fe(\000)p Fh(H)1456 1377 y Fg(;ik)1513 1350 y(@)r(x)1559 1335 y Fc(k)p 1513 1358 68 2 v 1516 1387 a Fg(@)5 b Ff(~)-23 b Fg(y)1560 1392 y Fb(~)-18 b Fc(c)1597 1370 y Fe(\000)11 b Fh(H)1692 1346 y Fg(;k)1688 1384 y(;i)1735 1348 y(@)r(y)1778 1353 y Fc(k)p 1735 1358 66 2 v 1737 1387 a Fg(@)5 b Ff(~)-23 b Fg(y)1781 1392 y Fb(~)-18 b Fc(c)1856 1370 y Fh(:)1801 1436 y Fj(\(74\))0 1505 y(W)l(e)17 b(collect)g(the)g(terms)e(con)o (taining)713 1485 y Fg(@)r(x)759 1470 y Fc(j)p 713 1493 65 2 v 714 1522 a Fg(@)5 b Ff(~)-23 b Fg(y)758 1527 y Fb(~)-18 b Fc(c)801 1505 y Fj(and)16 b(the)g(terms)g(con)o(taining)1372 1483 y Fg(@)r(y)1415 1488 y Fc(l)p 1370 1493 63 2 v 1370 1522 a Fg(@)5 b Ff(~)-23 b Fg(y)1414 1527 y Fb(~)-18 b Fc(c)1438 1505 y Fj(,)27 1575 y Fd(\002)59 1615 y Fe(\000)11 b Fh(H)154 1592 y Fg(;m)150 1629 y(;i)204 1615 y Fh(V)233 1622 y Fg(;mj)312 1615 y Fe(\000)g Fh(V)391 1622 y Fg(;im)455 1615 y Fh(H)500 1595 y Fg(;mn)575 1615 y Fh(V)604 1622 y Fg(;nj)672 1615 y Fe(\000)g Fh(V)762 1595 y Fa(\000)p Ff(1)816 1615 y Fh(H)857 1622 y Fg(;i)885 1615 y Fh(V)914 1622 y Fg(;j)958 1615 y Fj(+)g Fh(V)1048 1595 y Fa(\000)p Ff(1)1102 1615 y Fh(V)1131 1622 y Fg(;i)1159 1615 y Fh(H)1200 1622 y Fg(;j)1245 1615 y Fe(\000)f Fh(V)1335 1595 y Fa(\000)p Ff(2)1388 1615 y Fh(V)1417 1622 y Fg(;i)1446 1615 y Fh(V)1475 1622 y Fg(;j)1507 1575 y Fd(\003)1534 1596 y Fg(@)r(x)1580 1581 y Fc(j)p 1534 1604 65 2 v 1535 1633 a Fg(@)5 b Ff(~)-23 b Fg(y)1579 1638 y Fb(~)-18 b Fc(c)794 1706 y Fj(+)833 1665 y Fd(\002)853 1706 y Fh(V)882 1713 y Fg(;im)946 1706 y Fh(H)991 1685 y Fg(;mn)1077 1706 y Fj(+)11 b Fh(V)1167 1685 y Fa(\000)p Ff(1)1220 1706 y Fh(V)1249 1713 y Fg(;i)1278 1706 y Fh(H)1323 1685 y Fg(;n)1373 1706 y Fj(+)g Fh(H)1468 1682 y Fg(;n)1464 1720 y(;i)1507 1665 y Fd(\003)1534 1684 y Fg(@)r(y)1577 1689 y Fc(n)p 1534 1694 69 2 v 1537 1723 a Fg(@)5 b Ff(~)-23 b Fg(y)1581 1728 y Fb(~)-18 b Fc(c)1622 1706 y Fj(=)14 b(0)49 b Fh(:)38 b Fj(\(75\))0 1812 y(W)l(e)17 b(insert)f(\(34\))h(for)414 1789 y Fg(@)r(y)457 1794 y Fc(l)p 412 1800 63 2 v 412 1829 a Fg(@)5 b Ff(~)-23 b Fg(y)456 1834 y Fb(~)-18 b Fc(c)480 1812 y Fj(,)41 1876 y Fd(\002)73 1916 y Fe(\000)10 b Fh(H)167 1892 y Fg(;m)163 1930 y(;i)218 1916 y Fh(V)247 1923 y Fg(;mj)326 1916 y Fe(\000)h Fh(V)405 1923 y Fg(;im)468 1916 y Fh(H)513 1895 y Fg(;ml)576 1916 y Fh(V)605 1923 y Fg(;lj)662 1916 y Fe(\000)g Fh(V)752 1895 y Fa(\000)p Ff(1)806 1916 y Fh(H)847 1923 y Fg(;i)875 1916 y Fh(V)904 1923 y Fg(;j)948 1916 y Fj(+)g Fh(V)1038 1895 y Fa(\000)p Ff(1)1092 1916 y Fh(V)1121 1923 y Fg(;i)1149 1916 y Fh(H)1190 1923 y Fg(;j)1235 1916 y Fe(\000)f Fh(V)1324 1895 y Fa(\000)p Ff(2)1378 1916 y Fh(V)1407 1923 y Fg(;i)1435 1916 y Fh(V)1464 1923 y Fg(;j)395 2008 y Fj(\()p Fh(V)443 2015 y Fg(;im)507 2008 y Fh(H)552 1987 y Fg(;mn)638 2008 y Fj(+)h Fh(V)728 1987 y Fa(\000)p Ff(1)781 2008 y Fh(V)810 2015 y Fg(;i)839 2008 y Fh(H)884 1987 y Fg(;n)934 2008 y Fj(+)g Fh(H)1029 1984 y Fg(;n)1025 2022 y(;i)1068 2008 y Fj(\)\()p Fh(V)1135 2015 y Fg(;nj)1205 2008 y Fj(+)f Fh(V)1295 1987 y Fa(\000)p Ff(1)1348 2008 y Fh(V)1377 2015 y Fg(;n)1416 2008 y Fh(V)1445 2015 y Fg(;j)1478 2008 y Fj(\))1497 1967 y Fd(\003)1524 1988 y Fg(@)r(x)1570 1973 y Fc(j)p 1524 1996 65 2 v 1525 2025 a Fg(@)5 b Ff(~)-23 b Fg(y)1569 2030 y Fb(~)-18 b Fc(c)1608 2008 y Fj(=)14 b(0)50 b Fh(:)h Fj(\(76\))0 2098 y(W)l(e)17 b(consider)e(the)i(homogeneit)o(y)e(of)i(Hamiltonian)e (function)i Fh(H)t Fj(,)29 2148 y Fd(\002)61 2188 y Fe(\000)11 b Fh(H)156 2164 y Fg(;m)152 2202 y(;i)206 2188 y Fh(V)235 2195 y Fg(;mj)314 2188 y Fe(\000)g Fh(V)393 2195 y Fg(;im)457 2188 y Fh(H)502 2168 y Fg(;mn)576 2188 y Fh(V)605 2195 y Fg(;nj)674 2188 y Fe(\000)g Fh(V)764 2168 y Fa(\000)p Ff(1)817 2188 y Fh(H)858 2195 y Fg(;i)887 2188 y Fh(V)916 2195 y Fg(;j)960 2188 y Fj(+)g Fh(V)1050 2168 y Fa(\000)p Ff(1)1104 2188 y Fh(V)1133 2195 y Fg(;i)1161 2188 y Fh(H)1202 2195 y Fg(;j)1246 2188 y Fe(\000)g Fh(V)1336 2168 y Fa(\000)p Ff(2)1390 2188 y Fh(V)1419 2195 y Fg(;i)1447 2188 y Fh(V)1476 2195 y Fg(;j)514 2272 y Fj(+)p Fh(V)582 2279 y Fg(;im)645 2272 y Fh(H)690 2251 y Fg(;mn)765 2272 y Fh(V)794 2279 y Fg(;nj)863 2272 y Fj(+)f Fh(V)953 2251 y Fa(\000)p Ff(1)1006 2272 y Fh(V)1035 2279 y Fg(;i)1064 2272 y Fh(H)1109 2251 y Fg(;m)1159 2272 y Fh(V)1188 2279 y Fg(;mj)1267 2272 y Fj(+)h Fh(H)1362 2248 y Fg(;m)1358 2286 y(;i)1412 2272 y Fh(V)1441 2279 y Fg(;mj)542 2364 y Fj(+)p Fh(V)621 2343 y Fa(\000)p Ff(1)674 2364 y Fh(V)703 2371 y Fg(;im)767 2364 y Fh(H)812 2343 y Fg(;m)862 2364 y Fh(V)891 2371 y Fg(;j)935 2364 y Fj(+)g Fh(V)1025 2343 y Fa(\000)p Ff(2)1079 2364 y Fh(V)1108 2371 y Fg(;i)1136 2364 y Fh(V)1165 2371 y Fg(;j)1209 2364 y Fj(+)g(2)p Fh(V)1324 2343 y Fa(\000)p Ff(1)1377 2364 y Fh(H)1418 2371 y Fg(;i)1447 2364 y Fh(V)1476 2371 y Fg(;j)1509 2323 y Fd(\003)1536 2344 y Fg(@)r(x)1582 2329 y Fc(j)p 1536 2352 V 1537 2381 a Fg(@)5 b Ff(~)-23 b Fg(y)1581 2386 y Fb(~)-18 b Fc(c)1620 2364 y Fj(=)14 b(0)50 b Fh(:)39 b Fj(\(77\))0 2461 y(W)l(e)17 b(no)o(w)f(execute)h(summation)e(in)h(\(77\))h(and)f(arriv)o(e)f(at)104 2519 y Fd(\002)125 2560 y Fh(V)165 2539 y Fa(\000)p Ff(1)218 2560 y Fh(H)259 2567 y Fg(;i)288 2560 y Fh(V)317 2567 y Fg(;j)361 2560 y Fj(+)c Fh(V)451 2539 y Fa(\000)p Ff(1)505 2560 y Fh(V)534 2567 y Fg(;i)562 2560 y Fh(H)603 2567 y Fg(;j)647 2560 y Fj(+)g Fh(V)737 2539 y Fa(\000)p Ff(1)791 2560 y Fh(V)820 2567 y Fg(;i)848 2560 y Fh(H)893 2539 y Fg(;m)943 2560 y Fh(V)972 2567 y Fg(;mj)1052 2560 y Fj(+)g Fh(V)1142 2539 y Fa(\000)p Ff(1)1195 2560 y Fh(V)1224 2567 y Fg(;im)1288 2560 y Fh(H)1333 2539 y Fg(;m)1383 2560 y Fh(V)1412 2567 y Fg(;j)1445 2519 y Fd(\003)1472 2540 y Fg(@)r(x)1518 2525 y Fc(j)p 1472 2548 V 1473 2577 a Fg(@)5 b Ff(~)-23 b Fg(y)1517 2582 y Fb(~)-18 b Fc(c)1556 2560 y Fj(=)14 b(0)49 b Fh(:)104 b Fj(\(78\))0 2657 y(This)16 b(equation)g(holds)f(in)i(consequence)f(of)h(equation)f(\(56\).)908 2806 y(112)p eop %%Page: 113 11 113 10 bop 0 -54 a Fl(4.5.)25 b(Deriv)m(ativ)n(e)c(of)e(\(35\))0 20 y Fj(W)l(e)14 b(express)f(the)g(deriv)m(ativ)o(e)h(of)g(equation)f (\(35\))h(along)f(the)h(geo)q(desic)f(using)g(equations)g(\(63\),)i (\(64\),)0 80 y(\(65\),)i(\(67\))g(and)f(\(70\),)161 136 y Fd(\000)187 177 y Fe(\000)11 b Fh(V)266 184 y Fg(;)q Ff(~)-21 b Fg(a)o(m)337 177 y Fh(H)382 153 y Fg(;m)378 190 y(;j)443 177 y Fe(\000)11 b Fh(V)522 184 y Fg(;)q Ff(~)-21 b Fg(am)593 177 y Fh(H)638 156 y Fg(;mn)713 177 y Fh(V)742 184 y Fg(;nj)811 177 y Fe(\000)11 b Fh(V)901 156 y Fa(\000)p Ff(1)954 177 y Fh(V)983 184 y Fg(;)q Ff(~)-21 b Fg(a)1019 177 y Fh(H)1060 184 y Fg(;j)1104 177 y Fe(\000)11 b Fh(V)1194 156 y Fa(\000)p Ff(2)1248 177 y Fh(V)1277 184 y Fg(;)q Ff(~)-21 b Fg(a)1313 177 y Fh(V)1342 184 y Fg(;j)1375 136 y Fd(\001)1404 157 y Fg(@)r(x)1450 142 y Fc(j)p 1404 165 65 2 v 1405 194 a Fg(@)5 b Ff(~)-23 b Fg(y)1449 199 y Fb(~)-18 b Fc(c)600 272 y Fj(+)639 232 y Fd(\000)662 272 y Fh(V)691 279 y Fg(;)q Ff(~)d Fg(aj)757 272 y Fj(+)10 b Fh(V)847 252 y Fa(\000)p Ff(1)900 272 y Fh(V)929 279 y Fg(;)q Ff(~)-21 b Fg(a)965 272 y Fh(V)994 279 y Fg(;j)1027 232 y Fd(\001\000)1073 272 y Fh(H)1118 248 y Fg(;j)1114 287 y(;l)1158 253 y(@)r(x)1204 238 y Fc(l)p 1157 261 63 2 v 1157 290 a Fg(@)5 b Ff(~)-23 b Fg(y)1201 295 y Fb(~)-18 b Fc(c)1236 272 y Fj(+)11 b Fh(H)1331 252 y Fg(;j)r(l)1385 250 y(@)r(y)1428 255 y Fc(l)p 1383 261 V 1383 290 a Fg(@)5 b Ff(~)-23 b Fg(y)1427 295 y Fb(~)-18 b Fc(c)1451 232 y Fd(\001)1488 272 y Fj(=)14 b(0)49 b Fh(:)172 b Fj(\(79\))0 360 y(W)l(e)17 b(consider)e(the)i (homogeneit)o(y)e(of)i(Hamiltonian)e(function)i Fh(H)t Fj(,)162 414 y Fd(\000)188 455 y Fe(\000)11 b Fh(V)267 462 y Fg(;)q Ff(~)-21 b Fg(a)o(m)338 455 y Fh(H)383 431 y Fg(;m)379 468 y(;j)444 455 y Fe(\000)11 b Fh(V)523 462 y Fg(;)q Ff(~)-21 b Fg(am)595 455 y Fh(H)640 434 y Fg(;mn)714 455 y Fh(V)743 462 y Fg(;nj)812 455 y Fe(\000)11 b Fh(V)902 434 y Fa(\000)p Ff(1)955 455 y Fh(V)984 462 y Fg(;)q Ff(~)-21 b Fg(a)1020 455 y Fh(H)1061 462 y Fg(;j)1106 455 y Fe(\000)10 b Fh(V)1196 434 y Fa(\000)p Ff(2)1249 455 y Fh(V)1278 462 y Fg(;)q Ff(~)-21 b Fg(a)1314 455 y Fh(V)1343 462 y Fg(;j)1376 414 y Fd(\001)1405 435 y Fg(@)r(x)1451 420 y Fc(j)p 1405 443 65 2 v 1406 472 a Fg(@)5 b Ff(~)-23 b Fg(y)1450 477 y Fb(~)-18 b Fc(c)192 550 y Fj(+)231 510 y Fd(\000)254 550 y Fh(V)283 557 y Fg(;)q Ff(~)d Fg(a)o(j)337 550 y Fh(H)382 526 y Fg(;j)378 565 y(;l)427 550 y Fj(+)10 b(2)p Fh(V)541 530 y Fa(\000)p Ff(1)595 550 y Fh(V)624 557 y Fg(;)q Ff(~)-21 b Fg(a)660 550 y Fh(H)701 557 y Fg(;l)728 510 y Fd(\001)758 531 y Fg(@)r(x)804 516 y Fc(l)p 757 539 63 2 v 757 568 a Fg(@)5 b Ff(~)-23 b Fg(y)801 573 y Fb(~)-18 b Fc(c)837 550 y Fj(+)886 510 y Fd(\000)909 550 y Fh(V)938 557 y Fg(;)q Ff(~)d Fg(aj)993 550 y Fh(H)1038 530 y Fg(;j)r(l)1095 550 y Fj(+)11 b Fh(V)1185 530 y Fa(\000)p Ff(1)1238 550 y Fh(V)1267 557 y Fg(;)q Ff(~)-21 b Fg(a)1303 550 y Fh(H)1348 530 y Fg(;l)1376 510 y Fd(\001)1407 528 y Fg(@)r(y)1450 533 y Fc(l)p 1405 539 V 1405 568 a Fg(@)5 b Ff(~)-23 b Fg(y)1449 573 y Fb(~)-18 b Fc(c)1487 550 y Fj(=)13 b(0)50 b Fh(:)173 b Fj(\(80\))0 659 y(W)l(e)17 b(collect)g(the)g(terms) e(con)o(taining)713 639 y Fg(@)r(x)759 624 y Fc(j)p 713 647 65 2 v 714 676 a Fg(@)5 b Ff(~)-23 b Fg(y)758 681 y Fb(~)-18 b Fc(c)801 659 y Fj(and)16 b(the)g(terms)g(con)o(taining) 1372 637 y Fg(@)r(y)1415 642 y Fc(l)p 1370 647 63 2 v 1370 676 a Fg(@)5 b Ff(~)-23 b Fg(y)1414 681 y Fb(~)-18 b Fc(c)1438 659 y Fj(,)288 727 y Fd(\000)313 767 y Fe(\000)11 b Fh(V)392 774 y Fg(;)q Ff(~)-21 b Fg(am)464 767 y Fh(H)509 747 y Fg(;mn)583 767 y Fh(V)612 774 y Fg(;nj)681 767 y Fj(+)11 b Fh(V)771 747 y Fa(\000)p Ff(1)824 767 y Fh(V)853 774 y Fg(;)q Ff(~)-21 b Fg(a)889 767 y Fh(H)930 774 y Fg(;j)975 767 y Fe(\000)10 b Fh(V)1064 747 y Fa(\000)p Ff(2)1118 767 y Fh(V)1147 774 y Fg(;)q Ff(~)-21 b Fg(a)1183 767 y Fh(V)1212 774 y Fg(;j)1245 727 y Fd(\001)1274 748 y Fg(@)r(x)1320 732 y Fc(j)p 1274 756 65 2 v 1275 784 a Fg(@)5 b Ff(~)-23 b Fg(y)1319 789 y Fb(~)-18 b Fc(c)659 857 y Fj(+)698 817 y Fd(\000)721 857 y Fh(V)750 864 y Fg(;)q Ff(~)d Fg(a)o(m)821 857 y Fh(H)866 837 y Fg(;mn)952 857 y Fj(+)11 b Fh(V)1042 837 y Fa(\000)p Ff(1)1095 857 y Fh(V)1124 864 y Fg(;)q Ff(~)-21 b Fg(a)1160 857 y Fh(H)1205 837 y Fg(;n)1245 817 y Fd(\001)1274 835 y Fg(@)r(y)1317 840 y Fc(n)p 1274 846 69 2 v 1277 875 a Fg(@)5 b Ff(~)-23 b Fg(y)1321 880 y Fb(~)-18 b Fc(c)1361 857 y Fj(=)14 b(0)50 b Fh(:)298 b Fj(\(81\))0 961 y(W)l(e)17 b(insert)f(\(34\))h(for) 412 939 y Fg(@)r(y)455 944 y Fc(n)p 412 950 V 415 978 a Fg(@)5 b Ff(~)-23 b Fg(y)459 983 y Fb(~)-18 b Fc(c)486 961 y Fj(,)242 1021 y Fd(\002)266 1061 y Fe(\000)10 b Fh(V)344 1068 y Fg(;)q Ff(~)-21 b Fg(am)416 1061 y Fh(H)461 1041 y Fg(;mn)535 1061 y Fh(V)564 1068 y Fg(;nj)633 1061 y Fj(+)11 b Fh(V)723 1041 y Fa(\000)p Ff(1)776 1061 y Fh(V)805 1068 y Fg(;)q Ff(~)-21 b Fg(a)841 1061 y Fh(H)882 1068 y Fg(;j)927 1061 y Fe(\000)11 b Fh(V)1017 1041 y Fa(\000)p Ff(2)1070 1061 y Fh(V)1099 1068 y Fg(;)q Ff(~)-21 b Fg(a)1135 1061 y Fh(V)1164 1068 y Fg(;j)266 1150 y Fj(+)10 b(\()p Fh(V)363 1157 y Fg(;)q Ff(~)-21 b Fg(am)435 1150 y Fh(H)480 1129 y Fg(;mn)566 1150 y Fj(+)11 b Fh(V)656 1129 y Fa(\000)p Ff(1)709 1150 y Fh(V)738 1157 y Fg(;)q Ff(~)-21 b Fg(a)774 1150 y Fh(H)819 1129 y Fg(;n)859 1150 y Fj(\))8 b(\()p Fh(V)934 1157 y Fg(;nj)1003 1150 y Fj(+)j Fh(V)1093 1129 y Fa(\000)p Ff(1)1147 1150 y Fh(V)1176 1157 y Fg(;n)1215 1150 y Fh(V)1244 1157 y Fg(;j)1277 1150 y Fj(\))1296 1110 y Fd(\003)1323 1130 y Fg(@)r(x)1369 1115 y Fc(j)p 1323 1138 65 2 v 1324 1167 a Fg(@)5 b Ff(~)-23 b Fg(y)1368 1172 y Fb(~)-18 b Fc(c)1407 1150 y Fj(=)14 b(0)49 b Fh(:)253 b Fj(\(82\))0 1238 y(W)l(e)17 b(consider)e(the)i (homogeneit)o(y)e(of)i(Hamiltonian)e(function)i Fh(H)t Fj(,)35 1285 y Fd(\000)61 1326 y Fe(\000)11 b Fh(V)140 1333 y Fg(;)q Ff(~)-21 b Fg(a)o(m)211 1326 y Fh(H)256 1305 y Fg(;mn)331 1326 y Fh(V)360 1333 y Fg(;nj)428 1326 y Fj(+)11 b Fh(V)518 1305 y Fa(\000)p Ff(1)572 1326 y Fh(V)601 1333 y Fg(;)q Ff(~)-21 b Fg(a)637 1326 y Fh(H)678 1333 y Fg(;j)722 1326 y Fe(\000)11 b Fh(V)812 1305 y Fa(\000)p Ff(2)865 1326 y Fh(V)894 1333 y Fg(;)q Ff(~)-21 b Fg(a)930 1326 y Fh(V)959 1333 y Fg(;j)61 1414 y Fj(+)11 b Fh(V)140 1421 y Fg(;)q Ff(~)-21 b Fg(a)o(m)211 1414 y Fh(H)256 1394 y Fg(;mn)331 1414 y Fh(V)360 1421 y Fg(;nj)428 1414 y Fj(+)11 b Fh(V)518 1394 y Fa(\000)p Ff(1)572 1414 y Fh(V)601 1421 y Fg(;)q Ff(~)-21 b Fg(a)637 1414 y Fh(H)682 1394 y Fg(;n)721 1414 y Fh(V)750 1421 y Fg(;nj)819 1414 y Fj(+)10 b Fh(V)909 1394 y Fa(\000)p Ff(1)962 1414 y Fh(V)991 1421 y Fg(;)q Ff(~)-21 b Fg(am)1062 1414 y Fh(H)1107 1394 y Fg(;m)1158 1414 y Fh(V)1187 1421 y Fg(;j)1230 1414 y Fj(+)11 b Fh(V)1320 1394 y Fa(\000)p Ff(2)1374 1414 y Fh(V)1403 1421 y Fg(;)q Ff(~)-21 b Fg(a)1439 1414 y Fh(V)1468 1421 y Fg(;j)1501 1374 y Fd(\001)1530 1395 y Fg(@)r(x)1576 1380 y Fc(j)p 1530 1403 V 1531 1432 a Fg(@)5 b Ff(~)-23 b Fg(y)1575 1437 y Fb(~)-18 b Fc(c)1614 1414 y Fj(=)14 b(0)49 b Fh(:)d Fj(\(83\))0 1509 y(W)l(e)17 b(no)o(w)f(execute)h(summation)e(in)h(\(83\))h(and)f(arriv)o(e)f(at)289 1565 y Fd(\000)312 1606 y Fh(V)352 1585 y Fa(\000)p Ff(1)405 1606 y Fh(V)434 1613 y Fg(;)q Ff(~)-21 b Fg(a)470 1606 y Fh(H)511 1613 y Fg(;j)556 1606 y Fj(+)10 b Fh(V)646 1585 y Fa(\000)p Ff(1)699 1606 y Fh(V)728 1613 y Fg(;)q Ff(~)-21 b Fg(a)764 1606 y Fh(H)809 1585 y Fg(;n)848 1606 y Fh(V)877 1613 y Fg(;nj)946 1606 y Fj(+)11 b Fh(V)1036 1585 y Fa(\000)p Ff(1)1089 1606 y Fh(V)1118 1613 y Fg(;)q Ff(~)-21 b Fg(am)1190 1606 y Fh(H)1235 1585 y Fg(;m)1285 1606 y Fh(V)1314 1613 y Fg(;j)1347 1565 y Fd(\001)1376 1586 y Fg(@)r(x)1422 1571 y Fc(j)p 1376 1594 V 1377 1623 a Fg(@)5 b Ff(~)-23 b Fg(y)1421 1628 y Fb(~)-18 b Fc(c)1460 1606 y Fj(=)14 b(0)49 b Fh(:)200 b Fj(\(84\))0 1700 y(This)16 b(equation)g(holds)f(in)i(consequence)f(of)h(equations)f(\(56\))h(and)f (\(57\).)0 1818 y Fl(4.6.)25 b(Deriv)m(ativ)n(e)c(of)e(\(36\))0 1892 y Fj(W)l(e)14 b(express)f(the)g(deriv)m(ativ)o(e)h(of)g(equation)f (\(36\))h(along)f(the)h(geo)q(desic)f(using)g(equations)g(\(63\),)i (\(65\),)0 1951 y(\(68\),)i(\(69\))g(and)f(\(70\),)140 2029 y Fg(@)r(x)186 2014 y Fc(i)p 138 2037 66 2 v 138 2065 a Fg(@)5 b Ff(~)-23 b Fg(y)182 2070 y Fb(~)-18 b Fc(a)209 2008 y Fd(\000)235 2048 y Fe(\000)11 b Fh(V)314 2055 y Fg(;)q Ff(~)-21 b Fg(a)o(m)385 2048 y Fh(H)430 2028 y Fg(;mn)505 2048 y Fh(V)534 2064 y Fg(;n)570 2055 y Ff(~)571 2064 y Fg(b)601 2048 y Fe(\000)11 b Fh(V)691 2028 y Fa(\000)p Ff(2)745 2048 y Fh(V)774 2055 y Fg(;)q Ff(~)-21 b Fg(a)810 2048 y Fh(V)839 2064 y Fg(;)850 2055 y Ff(~)851 2064 y Fg(b)870 2008 y Fd(\001)143 2139 y Fj(+)193 2098 y Fd(\000)215 2139 y Fh(H)260 2118 y Fg(;i)256 2151 y(;m)313 2119 y(@)r(x)359 2104 y Fc(m)p 313 2127 80 2 v 319 2156 a Fg(@)5 b Ff(~)-23 b Fg(y)363 2161 y Fb(~)-18 b Fc(a)409 2139 y Fj(+)10 b Fh(H)503 2118 y Fg(;im)574 2116 y(@)r(y)617 2121 y Fc(m)p 574 2127 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Fb(~)-18 b Fc(a)645 2642 y Fd(\000)668 2682 y Fh(V)697 2697 y Fg(;)q Ff(~)d Fg(a)730 2688 y Ff(~)731 2697 y Fg(b)762 2682 y Fj(+)11 b Fh(V)852 2661 y Fa(\000)p Ff(1)905 2682 y Fh(V)934 2689 y Fg(;)q Ff(~)-21 b Fg(a)970 2682 y Fh(V)999 2697 y Fg(;)1010 2688 y Ff(~)1011 2697 y Fg(b)1031 2642 y Fd(\001)1068 2682 y Fj(=)13 b Fh(H)1165 2661 y Fg(;im)1236 2660 y(@)r(y)1279 2665 y Fc(m)p 1236 2671 V 1242 2706 a Fg(@)5 b Ff(~)-23 b Fg(x)1287 2689 y Fb(~)1288 2696 y Fc(b)1368 2682 y Fh(:)419 b Fj(\(87\))908 2806 y(113)p eop %%Page: 114 12 114 11 bop 0 -52 a Fj(W)l(e)17 b(m)o(ultiply)e(equation)h(\(87\))i (from)d(the)i(righ)o(t{hand)e(side)g(b)o(y)i(matrix)1388 -72 y Fg(@)r(x)1434 -87 y Fc(k)p 1388 -63 68 2 v 1391 -35 a Fg(@)5 b Ff(~)-23 b Fg(y)1433 -30 y Fb(~)1434 -23 y Fc(b)1461 -52 y Fj(,)329 63 y Fe(\000)387 44 y Fg(@)r(x)433 29 y Fc(i)p 385 52 66 2 v 385 81 a Fg(@)5 b Ff(~)-23 b Fg(y)429 86 y Fb(~)-18 b Fc(a)456 23 y Fd(\000)479 63 y Fh(V)508 70 y Fg(;)q Ff(~)d Fg(a)o(m)590 63 y Fj(+)11 b Fh(V)680 43 y Fa(\000)p Ff(1)734 63 y Fh(V)763 70 y Fg(;)q Ff(~)-21 b Fg(a)799 63 y Fh(V)828 70 y Fg(;m)877 23 y Fd(\001)900 63 y Fh(H)945 43 y Fg(;mn)1020 23 y Fd(\000)1043 63 y Fh(V)1072 79 y Fg(;n)1108 70 y Ff(~)1109 79 y Fg(b)1139 63 y Fj(+)11 b Fh(V)1229 43 y Fa(\000)p Ff(1)1283 63 y Fh(V)1312 70 y Fg(;n)1351 63 y Fh(V)1380 79 y Fg(;)1391 70 y Ff(~)1392 79 y Fg(b)1412 23 y Fd(\001)1441 44 y Fg(@)r(x)1487 29 y Fc(k)p 1441 52 68 2 v 1444 81 a Fg(@)5 b Ff(~)-23 b Fg(y)1486 86 y Fb(~)1487 93 y Fc(b)329 164 y Fj(+)11 b Fh(H)424 143 y Fg(;im)494 142 y(@)r(y)537 147 y Fc(m)p 494 153 77 2 v 500 181 a Fg(@)5 b Ff(~)-23 b Fg(y)544 186 y Fb(~)-18 b Fc(a)577 124 y Fd(\000)599 164 y Fh(V)628 180 y Fg(;)q Ff(~)d Fg(a)661 170 y Ff(~)662 180 y Fg(b)693 164 y Fj(+)11 b Fh(V)783 143 y Fa(\000)p Ff(1)836 164 y Fh(V)865 171 y Fg(;)q Ff(~)-21 b Fg(a)901 164 y Fh(V)930 180 y Fg(;)941 170 y Ff(~)942 180 y Fg(b)962 124 y Fd(\001)991 144 y Fg(@)r(x)1037 129 y Fc(k)p 991 153 68 2 v 994 181 a Fg(@)5 b 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y Fc(k)p 968 703 68 2 v 971 731 a Fg(@)5 b Ff(~)-23 b Fg(y)1015 736 y Fb(~)-18 b Fc(c)1055 714 y Fj(=)14 b Fh(\016)1132 694 y Fg(k)1130 727 y(i)1206 714 y Fh(:)581 b Fj(\(91\))0 810 y(of)17 b(the)g(symplectic)f(prop)q(ert)o(y)f(of)i (matrix)f(\(29\).)0 930 y Fl(5.)25 b(Conclusions)0 1005 y Fj(Deriv)o(ed)19 b(relations)g(\(34\){\(36\))h(b)q(et)o(w)o(een)g (the)g(propagator)e(matrix)h(\(29\))h(of)g(geo)q(desic)f(deviation)0 1065 y(and)24 b(the)g(second{order)f(spatial)g(deriv)m(ativ)o(es)h(of)h (c)o(haracteristic)e(function)h(are)g(applicable)f(to)0 1124 y(Finsler)10 b(geometry)l(,)i(to)g(Riemann)e(geometry)l(,)i(and)e (to)i(their)f(v)m(arious)g(applications)f(lik)o(e)h(the)h(general)0 1184 y(relativit)o(y)17 b(or)f(the)g(high{frequency)g(appro)o (ximations)e(of)j(w)o(a)o(v)o(e)e(propagation.)0 1304 y Fl(Ac)n(kno)n(wledgemen)n(ts)0 1378 y Fj(The)21 b(researc)o(h)e(has)h (b)q(een)h(supp)q(orted)e(b)o(y)i(the)g(Gran)o(t)f(Agency)h(of)g(the)g (Czec)o(h)g(Republic)e(under)0 1438 y(con)o(tract)g(205/07/0032,)f(b)o (y)g(the)h(Ministry)f(of)h(Education)f(of)h(the)g(Czec)o(h)g(Republic)e (within)i(re-)0 1498 y(searc)o(h)12 b(pro)s(ject)i(MSM0021620860,)e (and)g(b)o(y)i(the)f(mem)o(b)q(ers)f(of)i(the)g(consortium)d(\\Seismic) i(W)l(a)o(v)o(es)0 1558 y(in)j(Complex)g(3{D)h(Structures")e(\(see)i (\\h)o(ttp:/)-8 b(/sw3d.cz"\).)0 1677 y Fl(References)6 1739 y Fj(\024)0 1752 y(Cerv)o(en)q(\023)-26 b(y,)16 b(V.)h(\(1972\):)22 b(Seismic)15 b(ra)o(ys)h(and)g(ra)o(y)f(in)o (tensities)h(in)g(inhomogeneous)e(anisotropic)h(me-)100 1812 y(dia.)21 b Fk(Ge)m(ophys.)f(J.)e(R.)h(astr.)f(So)m(c.)p Fj(,)g Fl(29)p Fj(,)d(1{13.)6 1859 y(\024)0 1871 y(Cerv)o(en)q(\023)-26 b(y,)16 b(V.,)h(Klime)m(\024)-22 b(s,)15 b(L.)h(&)g(P)m(\024)-22 b(sen)o(\024)e(c)-6 b(\023)-19 b(\020k,)16 b(I.)h(\(1984\):)22 b(P)o(araxial)15 b(ra)o(y)h(appro)o(ximations)e(in)i(the)h(com-)100 1931 y(putation)e(of)i(seismic)e(w)o(a)o(v)o(e\014elds)g(in)h (inhomogeneous)e(media.)21 b Fk(Ge)m(ophys.)e(J.)f(R.)h(astr.)f(So)m (c.)p Fj(,)100 1991 y Fl(79)p Fj(,)d(89{104.)0 2051 y(Finsler,)k(P)l(.) i(\(1918\):)441 2038 y Fk(\177)432 2051 y(Ub)m(er)h(Kurven)g(und)g (Fl\177)-25 b(achen)22 b(in)g(A)o(l)s(lgemeinen)f(R\177)-25 b(aumen.)36 b Fj(PhD)20 b(Thesis,)100 2111 y(G\177)-25 b(ottingen,)16 b(Reedition:)21 b(Birkh\177)-25 b(auser,)15 b(Basel,)h(1951.)0 2170 y(Hamilton,)e(W.R.)g(\(1837\):)21 b(Third)14 b(supplemen)o(t)e(to)j(an)f(essa)o(y)g(on)g(the)h(theory)g (of)g(systems)e(of)i(ra)o(ys.)100 2230 y Fk(T)l(r)m(ans.)k(R)m(oy.)g (Irish)h(A)m(c)m(ad.)p Fj(,)e Fl(17)p Fj(,)e(1{144.)0 2290 y(Klime)m(\024)-22 b(s,)12 b(L.)i(\(1994\):)21 b(T)l (ransformations)10 b(for)j(dynamic)g(ra)o(y)g(tracing)g(in)g (anisotropic)f(media.)20 b Fk(Wave)100 2350 y(Motion)p Fj(,)c Fl(20)p Fj(,)g(261{272.)0 2409 y(Klime)m(\024)-22 b(s,)26 b(L.)h(\(2002\):)43 b(Second{order)25 b(and)h(higher{order)f(p) q(erturbations)g(of)i(tra)o(v)o(el)g(time)f(in)100 2469 y(isotropic)15 b(and)h(anisotropic)f(media.)21 b Fk(Stud.)d(ge)m (ophys.)i(ge)m(o)m(d.)p Fj(,)f Fl(46)p Fj(,)c(213{248.)0 2529 y(Rund,)48 b(H.)25 b(\(1959\):)38 b Fk(The)26 b(Di\013er)m(ential) f(Ge)m(ometry)g(of)h(Finsler)f(Sp)m(ac)m(es.)47 b Fj(Springer,)25 b(Berlin{)100 2589 y(G\177)-25 b(ottingen{Heilderb)q(erg.)908 2806 y(114)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF