%!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: 1.dvi %%CreationDate: Fri Dec 22 10:56:02 1995 %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: C:\EMTEX\DVIPS.EXE 1 %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1995.10.16:1429 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 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1836 y Fh(^)g Fj(dz)1085 1842 y Fi(n)p Fg(+1)1149 1836 y Fj(;)440 b Fk(\(2\))0 1983 y(i.e.,)12 b(to)i(eac)o(h)h Fj(z)e Fh(2)e Fj(R)326 1968 y Fg(2)p Fi(n)379 1983 y Fk(a)j(bilinear)f(an)o(tisymmetric)e (form)678 2094 y Fj(!)704 2100 y Fi(J)727 2094 y Fk(\()p Fj(\030)r(;)c(\021)q Fk(\))820 2100 y Fi(z)851 2094 y Fk(=)12 b Fj(\030)915 2077 y Fe(0)926 2094 y Fj(J)t(\021)627 b Fk(\(3\))0 2205 y(for)14 b(eac)o(h)h(pair)f(of)f(tangen)o(t)i(v)o (ectors)g Fj(\030)f Fk(=)f([)p Fj(\030)691 2211 y Fg(1)709 2205 y Fj(;)7 b Fh(\001)g(\001)g(\001)12 b Fj(;)7 b(\030)827 2211 y Fg(2)p Fi(n)866 2205 y Fk(])878 2190 y Fe(0)889 2205 y Fk(,)14 b Fj(\021)f Fk(=)g([)p Fj(\021)1027 2211 y Fg(1)1044 2205 y Fj(;)7 b Fh(\001)g(\001)g(\001)12 b Fj(;)7 b(\021)1165 2211 y Fg(2)p Fi(n)1203 2205 y Fk(])1215 2190 y Fe(0)1241 2205 y Fk(at)14 b Fj(z)r Fk(,)g Fj(J)k Fk(is)c(the)h(standard)0 2260 y(an)o(tisymmetric)c(matrix)370 2401 y Fj(J)k Fk(=)452 2330 y Fd(")523 2372 y Fk(0)68 b Fj(I)630 2378 y Fi(n)497 2433 y Fh(\000)p Fj(I)547 2439 y Fi(n)622 2433 y Fk(0)673 2330 y Fd(#)704 2401 y Fj(;)48 b(J)791 2384 y Fe(0)815 2401 y Fk(=)11 b Fh(\000)p Fj(J)16 b Fk(=)c Fj(J)1000 2384 y Fe(\000)p Fg(1)1045 2401 y Fj(;)48 b Fk(det)7 b Fj(J)16 b Fk(=)c(1)p Fj(:)0 2542 y Fk(The)i(fundamen)o(tal)e(2-form)g Fj(!)484 2548 y Fi(J)521 2542 y Fk(is)h(non-singular)g(and)h(closed,)g(i.e.,)e Fj(d!)1141 2548 y Fi(J)1176 2542 y Fk(=)f(0.)83 2605 y(Let)f Fj(w)q Fk(:)17 b Fj(R)245 2589 y Fg(2)p Fi(n)295 2605 y Fh(!)11 b Fj(R)380 2589 y Fg(2)p Fi(n)429 2605 y Fk(b)q(e)g(a)f(di\013eren)o(tial)g(mapping,)f Fj(z)k Fh(2)f Fj(R)1001 2589 y Fg(2)p Fi(n)1051 2605 y Fh(!)f Fj(w)q Fk(\()p Fj(z)r Fk(\))h Fh(2)f Fj(R)1271 2589 y Fg(2)p Fi(n)1310 2605 y Fk(,)g(the)g(corresp)q(onding)p eop %%Page: 8 3 8 2 bop 14 42 a Fk(8)0 142 y(Jacobi)14 b(matrix)e(is)h(denoted)i(b)o(y) 551 299 y Fj(@)r(w)p 551 317 56 2 v 556 355 a(@)r(z)622 327 y Fk(=)666 193 y Fd(2)666 267 y(6)666 291 y(6)666 316 y(6)666 341 y(6)666 368 y(4)730 213 y Fj(@)r(w)784 219 y Fg(1)p 730 232 73 2 v 735 270 a Fj(@)r(z)778 276 y Fg(1)859 241 y Fh(\001)7 b(\001)g(\001)971 213 y Fj(@)r(w)1025 219 y Fg(1)p 966 232 83 2 v 966 270 a Fj(@)r(z)1009 276 y Fg(2)p Fi(n)760 305 y Fk(.)760 322 y(.)760 339 y(.)1002 305 y(.)1002 322 y(.)1002 339 y(.)720 385 y Fj(@)r(w)774 391 y Fg(2)p Fi(n)p 720 404 94 2 v 735 442 a Fj(@)r(z)778 448 y Fg(1)859 414 y Fh(\001)g(\001)g(\001)961 385 y Fj(@)r(w)1015 391 y Fg(2)p Fi(n)p 961 404 V 966 442 a Fj(@)r(z)1009 448 y Fg(2)p Fi(n)1080 193 y Fd(3)1080 267 y(7)1080 291 y(7)1080 316 y(7)1080 341 y(7)1080 368 y(5)0 516 y Fk(mapping)13 b Fj(w)k Fk(induces,)f(for)f(eac)o(h)i Fj(z)f Fh(2)e Fj(R)650 501 y Fg(2)p Fi(n)689 516 y Fk(,)h(a)h(linear)f (mapping)e Fj(w)1072 522 y Fe(\003)1107 516 y Fk(of)i(the)h(tangen)o(t) f(space)i(at)f Fj(z)h Fk(in)o(to)0 571 y(the)d(tangen)o(t)h(space)f(at) g Fj(w)q Fk(\()p Fj(z)r Fk(\))g(b)o(y)538 673 y Fj(\030)g Fk(=)d([)p Fj(\030)643 679 y Fg(1)662 673 y Fj(;)c Fh(\001)g(\001)g (\001)12 b Fj(;)7 b(\030)780 679 y Fg(2)p Fi(n)818 673 y Fk(])830 656 y Fe(0)853 673 y Fh(!)k Fj(w)936 679 y Fe(\003)955 673 y Fj(\030)i Fk(=)1035 645 y Fj(@)r(w)p 1035 663 56 2 v 1040 701 a(@)r(z)1095 673 y(\030)0 769 y(w)i Fk(also)e(induces,)h(for)g(eac)o(h)g(2-form)e Fj(!)j Fk(on)f Fj(R)708 754 y Fg(2)p Fi(n)747 769 y Fk(,)f(a)h(2-form)d Fj(w)970 754 y Fe(\003)989 769 y Fj(!)16 b Fk(on)d Fj(R)1120 754 y Fg(2)p Fi(n)1173 769 y Fk(b)o(y)h(the)g(form)o(ula)538 869 y Fj(w)569 851 y Fe(\003)588 869 y Fj(!)q Fk(\()p Fj(\030)r(;)7 b(\021)q Fk(\))709 881 y Fi(z)739 869 y Fk(=)12 b Fj(!)q Fk(\()831 841 y Fj(@)r(w)p 831 859 V 836 897 a(@)r(z)892 869 y(\030)r(;)935 841 y(@)r(w)p 935 859 V 940 897 a(@)r(z)995 869 y(\021)q Fk(\))1033 876 y Fi(w)q Fg(\()p Fi(z)q Fg(\))1104 869 y Fj(:)0 963 y Fk(If)i Fj(!)q Fk(\()p Fj(\030)r(;)7 b(\021)q Fk(\))162 969 y Fi(z)192 963 y Fk(=)12 b Fj(\030)r(A)p Fk(\()p Fj(z)r Fk(\))p Fj(\021)q Fk(,)i Fj(A)419 948 y Fe(0)431 963 y Fk(\()p Fj(z)r Fk(\))e(=)g Fh(\000)p Fj(A)p Fk(\()p Fj(z)r Fk(\),)i(then)h Fj(w)808 948 y Fe(\003)827 963 y Fj(!)q Fk(\()p Fj(\030)r(;)7 b(\021)q Fk(\))k(=)h Fj(\030)r(B)r Fk(\()p Fj(z)r Fk(\))p Fj(\021)q Fk(,)j(where)589 1070 y Fj(B)r Fk(\()p Fj(z)r Fk(\))d(=)g(\()752 1042 y Fj(@)r(w)p 752 1061 V 757 1099 a(@)r(z)813 1070 y Fk(\))829 1023 y Fe(0)840 1070 y Fj(A)p Fk(\()p Fj(w)q Fk(\()p Fj(z)r Fk(\)\))992 1042 y Fj(@)r(w)p 992 1061 V 997 1099 a(@)r(z)1053 1070 y(:)0 1167 y Fk(A)g(di\013eomorphism)e(\(di\013eren)o(tiable,)i (one-one,)h(on)o(to)f(mapping\))e(of)h Fj(R)1126 1152 y Fg(2)p Fi(n)1178 1167 y Fk(is)h(called)g(a)g(canonical)f(trans-)0 1221 y(formation)g(if)i Fj(w)i Fk(preserv)o(es)h(the)f(standard)f (symplectic)f(structure,)j(i.e.,)c Fj(w)1199 1206 y Fe(\003)1218 1221 y Fj(!)1244 1227 y Fi(J)1279 1221 y Fk(=)g Fj(!)1349 1227 y Fi(J)1372 1221 y Fk(,)h(i.e.,)669 1327 y(\()690 1299 y Fj(@)r(w)p 690 1317 V 695 1355 a(@)r(z)750 1327 y Fk(\))766 1280 y Fe(0)778 1327 y Fj(J)t Fk(\()826 1299 y Fj(@)r(w)p 826 1317 V 831 1355 a(@)r(z)886 1327 y Fk(\))f(=)g Fj(J)620 b Fk(\(4\))0 1426 y(i.e.,)12 b(the)j(Jacobian)328 1410 y Fi(@)r(w)p 328 1417 45 2 v 332 1440 a(@)r(z)392 1426 y Fk(is)e(a)h(symplectic)f(matrix)f(for)i(eac)o(h)g Fj(z)r Fk(.)83 1483 y(F)m(or)g(ev)o(ery)h(pair)f(of)g(smo)q(oth)f (functions)i Fj(\036)p Fk(\()p Fj(z)r Fk(\),)f Fj( )q Fk(\()p Fj(z)r Fk(\))i(on)e Fj(R)1020 1468 y Fg(2)p Fi(n)1059 1483 y Fk(,)g(w)o(e)h(asso)q(ciate)g(a)f(smo)q(oth)g(function)0 1538 y Fj(\037)p Fk(\()p Fj(z)r Fk(\))e(=)g Fh(f)p Fj(\036;)7 b( )q Fh(g)p Fk(,)12 b(called)i(the)g(P)o(oisson)g(brac)o(k)o(et)h(b)o (y)656 1623 y Fh(f)p Fj(\036;)7 b( )q Fh(g)k Fk(=)g Fj(\036)849 1606 y Fe(0)849 1633 y Fi(z)868 1623 y Fj(J)895 1606 y Fe(\000)p Fg(1)940 1623 y Fj( )967 1629 y Fi(z)986 1623 y Fj(;)0 1708 y Fk(where)20 b Fj(\036)150 1693 y Fe(0)150 1718 y Fi(z)189 1708 y Fk(=)h([)264 1689 y Fi(@)r(\036)p 258 1698 52 2 v 258 1722 a(@)r(z)294 1726 y Fc(1)315 1708 y Fj(;)7 b Fh(\001)g(\001)g(\001)12 b Fj(;)427 1689 y Fi(@)r(\036)p 419 1698 56 2 v 419 1722 a(@)r(z)455 1726 y Fb(n)480 1708 y Fk(].)33 b(The)20 b(P)o(oisson)f(brac)o(k)o(ets) h(are)f(an)o(ti-symmetric)e(and)i(satisfy)g(Jacobi)0 1763 y(iden)o(tit)o(y)m(.)83 1820 y(Cho)q(ose)h(a)g(smo)q(oth)g (function)f Fj(H)s Fk(\()p Fj(z)r Fk(\))k(=)f Fj(H)s Fk(\()p Fj(z)837 1826 y Fg(1)856 1820 y Fj(;)7 b Fh(\001)g(\001)g(\001) 12 b Fj(;)7 b(z)975 1826 y Fg(2)p Fi(n)1014 1820 y Fk(\))22 b(=)p Fj(H)s Fk(\()p Fj(p)1159 1826 y Fg(1)1178 1820 y Fj(;)7 b Fh(\001)g(\001)g(\001)12 b Fj(;)7 b(p)1299 1826 y Fi(n)1321 1820 y Fj(;)g(q)1359 1826 y Fg(1)1376 1820 y Fj(;)g Fh(\001)g(\001)g(\001)12 b Fj(;)7 b(q)1495 1826 y Fi(n)1517 1820 y Fk(\).)37 b(The)0 1875 y(equations)14 b(\(1\),)f(or)h(written)h(alternativ)o(ely)e(as)709 1948 y Fj(dz)p 709 1967 43 2 v 712 2005 a(dt)769 1976 y Fk(=)e Fj(J)839 1959 y Fe(\000)p Fg(1)884 1976 y Fj(H)919 1982 y Fi(z)938 1976 y Fj(;)651 b Fk(\(5\))0 2070 y(is)11 b(called)g(the)h(canonical)e(system)h(of)g(equations)g(with)g (Hamiltonia)o(n)e Fj(H)s Fk(\()p Fj(z)r Fk(\).)17 b(According)12 b(to)f(the)g(general)0 2125 y(theory)j(of)g(ODE,)f(for)h(eac)o(h)g (Hamiltonian)d(system)i(\(5\),)h(there)h(corresp)q(onds)h(a)d (one-parameter)h(group)0 2180 y(of)f(di\013eomorphisms)f Fj(g)377 2165 y Fi(t)405 2180 y Fk(at)i(least)g(lo)q(cally)f(in)g Fj(t)h Fk(and)g Fj(z)r Fk(,)f(of)g Fj(R)968 2165 y Fg(2)p Fi(n)1021 2180 y Fk(suc)o(h)i(that)526 2265 y Fj(g)547 2248 y Fg(0)577 2265 y Fk(=)d(iden)o(tit)o(y)o Fj(;)48 b(g)841 2248 y Fi(t)854 2252 y Fc(1)870 2248 y Fg(+)p Fi(t)908 2252 y Fc(2)938 2265 y Fk(=)12 b Fj(g)1003 2248 y Fi(t)1016 2252 y Fc(1)1043 2265 y Fh(\001)d Fj(g)1085 2248 y Fi(t)1098 2252 y Fc(2)1116 2265 y Fj(;)0 2350 y Fk(suc)o(h)15 b(that,)e(if)g Fj(z)r Fk(\(0\))h(is)g(tak)o(en)g(as)g (the)g(initial)e(condition,)h(then)i(the)f(solution)f(of)g(\(5\))h(is)g (generated)h(b)o(y)704 2435 y Fj(z)r Fk(\()p Fj(t)p Fk(\))d(=)g Fj(g)849 2417 y Fi(t)864 2435 y Fj(z)r Fk(\(0\))p Fj(:)0 2520 y Fk(The)i(basic)g(prop)q(ert)o(y)h(of)e(Hamiltonian)e(system)j (\(1\))g(is)f(that)h Fj(g)1002 2504 y Fi(t)1031 2520 y Fk(are)g(canonical)g(transformations)718 2605 y Fj(g)739 2587 y Fi(t)p Fe(\003)771 2605 y Fj(!)797 2611 y Fi(J)831 2605 y Fk(=)e Fj(!)901 2611 y Fi(J)924 2605 y Fj(;)665 b Fk(\(6\))p eop %%Page: 9 4 9 3 bop 1619 42 a Fk(9)0 142 y(for)14 b(all)e Fj(t)p Fk(.)18 b(This)c(leads)g(to)g(the)g(follo)o(wing)d(class)k(of)e (phase-area)i(conserv)n(ation)f(la)o(ws)415 185 y Fd(Z)438 279 y Fi(g)455 271 y Fb(t)469 279 y Fi(\033)489 271 y Fc(2)514 241 y Fj(!)540 247 y Fi(J)575 241 y Fk(=)619 185 y Fd(Z)642 279 y Fi(\033)662 271 y Fc(2)687 241 y Fj(!)713 247 y Fi(J)736 241 y Fj(;)48 b Fk(ev)o(ery)15 b(2-c)o(hain)e Fj(\033)1075 226 y Fg(2)1105 241 y Fh(\032)f Fj(R)1181 226 y Fg(2)p Fi(n)1220 241 y Fj(;)320 303 y Fd(Z)343 397 y Fi(g)360 389 y Fb(t)374 397 y Fi(\033)394 389 y Fc(4)419 359 y Fj(!)445 365 y Fi(J)478 359 y Fh(^)c Fj(!)540 365 y Fi(J)575 359 y Fk(=)619 303 y Fd(Z)642 397 y Fi(\033)662 389 y Fc(4)687 359 y Fj(!)713 365 y Fi(J)746 359 y 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1116 a(2)559 1088 y Fj(\030)579 1070 y Fe(0)591 1088 y Fj(K)s Fk(\()p Fj(z)r Fk(\))p Fj(\021)q(;)48 b(K)802 1070 y Fe(0)814 1088 y Fk(\()p Fj(z)r Fk(\))12 b(=)g Fh(\000)p Fj(K)s Fk(\()p Fj(z)r Fk(\))p Fj(;)49 b Fk(det)7 b Fj(K)s Fk(\()p Fj(z)r Fk(\))13 b Fh(6)p Fk(=)e(0)p Fj(:)0 1184 y Fk(A)j(di\013eren)o(tiable)g(mapping)e Fj(w)q Fk(:)18 b Fj(R)558 1169 y Fg(2)p Fi(n)608 1184 y Fh(!)11 b Fj(R)693 1169 y Fg(2)p Fi(n)746 1184 y Fk(is)j(called)f (K-canonical)h(if)f Fj(w)1203 1169 y Fe(\003)1222 1184 y Fj(!)1248 1190 y Fi(K)1291 1184 y Fk(=)f Fj(!)1361 1190 y Fi(K)1393 1184 y Fk(,)h(i.e.)583 1290 y(\()604 1262 y Fj(@)r(w)p 604 1280 56 2 v 609 1318 a(@)r(z)664 1290 y Fk(\))680 1243 y Fe(0)692 1290 y Fj(K)s Fk(\()p Fj(w)q Fk(\()p Fj(z)r Fk(\)\))851 1262 y Fj(@)r(w)p 851 1280 V 856 1318 a(@)r(z)924 1290 y Fk(=)e Fj(K)s Fk(\()p Fj(z)r Fk(\))p Fj(:)531 b Fk(\(8\))83 1384 y(The)14 b(P)o(oisson)g (brac)o(k)o(et)g(is)g(de\014ned)h(as)608 1470 y Fh(f)p Fj(\036;)7 b( )q Fh(g)722 1476 y Fi(K)764 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Fk(\()p Fj(z)r Fk(\))d(=)732 2565 y(1)p 732 2583 21 2 v 732 2621 a(2)758 2593 y Fj(z)779 2575 y Fe(0)791 2593 y Fj(S)r(z)r(;)90 b(S)968 2575 y Fe(0)992 2593 y Fk(=)12 b Fj(S;)507 b Fk(\(10\))p eop %%Page: 10 5 10 4 bop 14 42 a Fk(10)0 142 y(and)12 b Fj(K)j Fk(b)q(e)e(an)e(an)o (ti-symmetric)f(non-singular)h(constan)o(t)h(matrix,)e(then)j(the)f Fj(K)s Fk(-canonical)g(system)g(\(9\))0 197 y(b)q(ecomes)i(linear)595 234 y Fj(dz)p 595 252 43 2 v 598 290 a(dt)655 262 y Fk(=)d Fj(B)r(z)r(;)90 b(B)15 b Fk(=)c Fj(K)981 245 y Fe(\000)p Fg(1)1026 262 y Fj(S;)517 b Fk(\(11\))0 336 y(the)19 b(generating)g(one-parameter)f(group)g(is)g(a)h(group)f(of)g(linear)g (transformations)f(whic)o(h)h(coincides)0 391 y(with)c(their)g(o)o(wn)f (Jacobians)504 469 y Fj(z)r Fk(\()p Fj(t)p Fk(\))f(=)g Fj(G)p Fk(\()p Fj(t)p Fk(\))p Fj(z)r Fk(\(0\))p Fj(;)90 b(G)p Fk(\()p Fj(t)p Fk(\))11 b(=)h(exp)7 b Fj(tB)r(:)431 b Fk(\(12\))0 547 y(The)14 b(matrix)e Fj(B)17 b Fk(is)d (in\014nitesimally)c Fj(K)s Fk(-symplectic)680 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b(can)g(b)q(e)h(sho)o(wn)f(that)g Fj(d)p Fk(\()457 1960 y Fd(P)507 1992 y Fj(k)529 1998 y Fi(ij)558 1992 y Fk(\()p Fj(z)r Fk(\))p Fj(dz)652 1998 y Fi(i)679 1992 y Fh(^)12 b Fj(dz)760 1998 y Fi(j)777 1992 y Fk(\))21 b(=)f(0)e(and)h(det)h Fj(k)h Fh(6)p Fk(=)f(0,)f(so)g Fj(K)j Fk(actually)c(de\014nes)j(a)0 2046 y(symplectic)13 b(structure.)21 b(Then)14 b Fj(w)q Fk(\()p Fj(t)p Fk(\))g(satis\014es)h (the)f Fj(K)s Fk(-canonical)f(system)602 2111 y Fj(dw)p 602 2129 53 2 v 610 2167 a(dt)671 2139 y Fk(=)f Fj(K)753 2122 y Fe(\000)p Fg(1)808 2128 y Fd(e)798 2139 y Fj(H)833 2145 y Fi(w)871 2139 y Fk(=)g Fj(K)953 2122 y Fe(\000)p Fg(1)1003 2128 y Fd(e)997 2139 y Fj(S)s(w)525 b Fk(\(26\))0 2233 y(with)14 b(Hamiltonia)o(n)343 2222 y Fd(e)333 2233 y Fj(H)s Fk(\()p Fj(w)q Fk(\))e(=)g Fj(H)s Fk(\()p Fj(T)574 2218 y Fe(\000)p Fg(1)618 2233 y Fj(w)q Fk(\))g(=)726 2216 y Fg(1)p 726 2223 17 2 v 726 2247 a(2)747 2233 y Fj(w)778 2218 y Fe(0)795 2222 y Fd(e)790 2233 y Fj(S)r(w)q Fk(,)691 2303 y Fd(e)685 2314 y Fj(S)i Fk(=)e Fj(T)798 2297 y Fe(\000)p Fg(1)843 2288 y Fe(0)855 2314 y Fj(S)r(T)912 2297 y Fe(\000)p Fg(1)957 2314 y Fj(;)0 2389 y Fk(so)i(the)g (transition)315 2477 y Fd(e)307 2487 y Fj(G)340 2493 y Fi(\034)372 2487 y Fk(:)d Fj(w)q Fk(\()p Fj(t)p Fk(\))g(=)528 2416 y Fd(")573 2459 y Fj(p)p Fk(\()p Fj(t)p Fk(\))573 2519 y Fj(q)q Fk(\()p Fj(t)f Fk(+)680 2503 y Fi(\034)p 680 2510 19 2 v 681 2533 a Fg(2)704 2519 y Fk(\))741 2416 y Fd(#)776 2487 y Fh(!)h Fj(w)q Fk(\()p Fj(t)f Fk(+)f Fj(\034)c Fk(\))12 b(=)1037 2416 y Fd(")1081 2459 y Fj(p)p Fk(\()p Fj(t)e Fk(+)f Fj(\034)c Fk(\))1081 2519 y Fj(q)q Fk(\()p Fj(t)10 b Fk(+)f Fj(\034)14 b Fk(+)1262 2503 y Fi(\034)p 1262 2510 V 1263 2533 a Fg(2)1286 2519 y Fk(\))1322 2416 y Fd(#)0 2605 y Fk(is)g(linear)f(and)h Fj(K)s Fk(-symplectic,)f(i.e.,)592 2594 y Fd(e)584 2605 y Fj(G)617 2589 y Fe(0)617 2615 y Fi(\034)638 2605 y Fj(K)684 2594 y Fd(e)676 2605 y Fj(G)709 2611 y Fi(\034)741 2605 y Fk(=)f Fj(K)s Fk(.)p eop %%Page: 13 8 13 7 bop 1598 42 a Fk(13)83 142 y(It)22 b(can)h(b)q(e)g(pro)o(v)o(ed)f (that)h Fj(F)555 148 y Fi(\034)598 142 y Fk(is)f(also)g Fj(K)s Fk(-symplectic)f(as)i(is)f(exp)q(ected.)46 b(Ho)o(w)o(ev)o(er,) 24 b(the)f(en-)0 197 y(ergy)18 b(conserv)n(ation)f(prop)q(erties)i(are) f(someho)o(w)f(di\013eren)o(t,)i Fj(F)997 203 y Fi(\034)1034 197 y Fk(do)q(es)g(preserv)o(e)1307 186 y Fd(e)1297 197 y Fj(H)s Fk(\()p Fj(w)q Fk(\),)f(whic)o(h)g(is)f(the)0 252 y(true)k(energy)g(after)g(sync)o(hronizing)f Fj(p)626 237 y Fi(m)657 252 y Fk(,)i Fj(q)711 237 y Fi(m)p Fg(+)771 225 y Fc(1)p 771 230 15 2 v 771 247 a(2)812 252 y Fk(at)e(staggered)h (momen)o(ts)d(b)o(y)i Fj(T)1337 237 y Fe(\000)p Fg(1)1402 252 y Fk(to)g Fj(p)1480 237 y Fi(m)1512 252 y Fk(,)h Fj(q)1565 237 y Fi(m)1617 252 y Fk(at)0 306 y(the)e(same)e(momen)o(t.) 27 b Fj(F)399 312 y Fi(\034)438 306 y Fk(preserv)o(es,)20 b(instead,)f(a)f(mo)q(di\014ed)e(Hamiltonian)1258 296 y(^)1247 306 y Fj(H)s Fk(\()p Fj(w)q Fk(\))i(=)1422 290 y Fg(1)p 1422 297 17 2 v 1422 321 a(2)1444 306 y Fj(w)1475 291 y Fe(0)1493 296 y Fk(^)1486 306 y Fj(S)s(w)q Fk(,)1582 296 y(^)1575 306 y Fj(S)j Fk(=)0 406 y Fj(T)30 391 y Fe(\000)p Fg(1)75 383 y Fe(0)86 406 y Fj(J)t(T)143 391 y Fe(\000)p Fg(1)195 335 y Fd(")254 378 y Fj(I)81 b Fh(\000)390 361 y Fi(\034)p 390 368 19 2 v 391 392 a Fg(2)413 378 y Fj(B)219 438 y Fh(\000)256 422 y Fi(\034)p 256 429 V 257 452 a Fg(2)280 438 y Fj(A)d(I)447 335 y Fd(#)471 344 y Fe(\000)p Fg(1)523 335 y Fd(")552 378 y Fk(0)47 b Fh(\000)p Fj(B)547 438 y(A)64 b Fk(0)685 335 y Fd(#)710 406 y Fk(.)26 b(Note)18 b(that)e(in)h(practical)f(computation,)g(the)h (sync)o(hro-)0 502 y(nization)i(is)g(done)g(b)o(y)h(de\014ning)f Fj(q)562 487 y Fi(m)614 502 y Fk(=)672 485 y Fg(1)p 672 492 17 2 v 672 516 a(2)694 502 y Fk(\()p Fj(q)730 487 y Fi(m)p Fe(\000)790 475 y Fc(1)p 790 480 15 2 v 790 497 a(2)824 502 y Fk(+)13 b Fj(q)889 487 y Fi(m)p Fg(+)949 475 y Fc(1)p 949 480 V 949 497 a(2)971 502 y Fk(\),)20 b(then)g(all)e(the)i(\014rst)g(in)o(tegrals)f Fj(')p Fk(\()p Fj(p;)7 b(q)q Fk(\),)0 556 y(including)13 b(the)h(Hamiltonian)d Fj(H)s Fk(\()p Fj(p;)c(q)q Fk(\))13 b(are)i(conserv)o(ed)g(appro)o (ximately)d(as)451 679 y Fj(\036)p Fk(\()p Fj(p)513 662 y Fi(m)p Fg(+1)586 679 y Fj(;)7 b(q)625 662 y Fi(m)p Fg(+1)698 679 y Fk(\))12 b(=)g Fj(\036)p Fk(\()p Fj(p)832 662 y Fi(m)863 679 y Fj(;)7 b(q)902 662 y Fi(m)933 679 y Fk(\))42 b(mo)q(d)12 b Fj(O)q Fk(\()p Fj(\034)1156 662 y Fg(3)1175 679 y Fk(\))p Fj(:)83 812 y Fk(The)21 b(eigen)o(v)n(alue)f Fj(\025)h Fk(of)f Fj(F)506 818 y Fi(\034)547 812 y Fk(is)g(related)h(to)g(the)g(eigen)o(v)n(alue)f Fj(\026)h Fk(of)f(the)h(p)q(encil)g Fj(L)14 b Fh(\000)g Fj(\026M)1545 797 y Fe(\000)p Fg(1)1610 812 y Fk(b)o(y)0 867 y Fj(\025)24 852 y Fg(2)49 867 y Fk(+)6 b Fj(\025)p Fk(\()p Fj(\034)150 852 y Fg(2)169 867 y Fj(\026)g Fh(\000)g Fk(2\))g(+)g(1)12 b(=)f(0,)h(this)h(leads)f(to)g(the)h(Jordan)f(normal) f(form)f(of)i Fj(F)6 b Fk(,)11 b(consisting)h(again)g(of)f(three)0 922 y(p)q(ossible)j(t)o(yp)q(es)254 1044 y(T)o(yp)q(e)g(1)584 b(T)o(yp)q(e)14 b(2)101 1210 y Fj(F)128 1216 y Fi(\034)231 1139 y Fd(")276 1182 y Fk(1)41 b(0)276 1242 y(0)g(1)380 1139 y Fd(#)487 1114 y(2)487 1188 y(6)487 1214 y(4)535 1171 y Fk(1)9 b(+)612 1143 y Fj(!)639 1128 y Fg(2)658 1143 y Fj(\034)681 1128 y Fg(2)p 612 1162 88 2 v 645 1200 a Fk(2)714 1171 y(+)760 1143 y Fj(i!)q(\034)p 760 1162 65 2 v 782 1200 a Fk(2)829 1131 y Fd(p)p 871 1131 159 2 v 40 x Fk(4)g Fh(\000)g Fj(!)969 1160 y Fg(2)988 1171 y Fj(\034)1011 1160 y Fg(2)1308 1171 y Fk(0)772 1262 y(0)278 b(1)9 b(+)1148 1234 y Fj(!)1175 1219 y Fg(2)1194 1234 y Fj(\034)1217 1219 y Fg(2)p 1148 1252 88 2 v 1181 1290 a Fk(2)1250 1262 y Fh(\000)1296 1234 y Fj(i!)q(\034)p 1296 1252 65 2 v 1318 1290 a Fk(2)1365 1221 y Fd(p)p 1407 1221 159 2 v 41 x Fk(4)g Fh(\000)g Fj(!)1505 1250 y Fg(2)1524 1262 y Fj(\034)1547 1250 y Fg(2)1587 1114 y Fd(3)1587 1188 y(7)1587 1214 y(5)700 1380 y Fk(T)o(yp)q(e)14 b(3)101 1546 y Fj(F)128 1552 y Fi(\034)231 1450 y Fd(2)231 1523 y(6)231 1550 y(4)279 1507 y Fk(1)9 b(+)356 1479 y Fj(a)378 1464 y Fg(2)396 1479 y Fj(\034)419 1464 y Fg(2)p 356 1497 83 2 v 387 1535 a Fk(2)452 1507 y(+)499 1479 y Fj(a\034)p 499 1497 45 2 v 511 1535 a Fk(2)548 1466 y Fd(p)p 590 1466 154 2 v 41 x Fk(4)g(+)g Fj(a)683 1495 y Fg(2)702 1507 y Fj(\034)725 1495 y Fg(2)1007 1507 y Fk(0)501 1597 y(0)263 b(1)9 b(+)861 1569 y Fj(a)883 1554 y Fg(2)902 1569 y Fj(\034)925 1554 y Fg(2)p 861 1588 83 2 v 892 1626 a Fk(2)958 1597 y(+)1004 1569 y Fj(a\034)p 1004 1588 45 2 v 1016 1626 a Fk(2)1054 1557 y Fd(p)p 1095 1557 154 2 v 1095 1597 a Fk(4)g(+)h Fj(a)1189 1585 y Fg(2)1208 1597 y Fj(\034)1231 1585 y Fg(2)1270 1450 y Fd(3)1270 1523 y(7)1270 1550 y(5)83 1743 y Fk(T)o(yp)q(e)h(2:)17 b(When)12 b Fj(\034)k(<)442 1727 y Fg(2)p 439 1734 22 2 v 439 1758 a Fi(!)466 1743 y Fk(,)11 b(the)h(t)o(w)o(o)f(eigen)o(v)n (alues)g(are)h(unimo)q(dular,)e(complex-conjugate,)g(distinct.)0 1798 y(They)j(collide)e(at)h Fh(\000)p Fk(1)g(when)h Fj(\034)j Fk(=)540 1782 y Fg(2)p 537 1789 V 537 1813 a Fi(!)564 1798 y Fk(.)h(As)c Fj(\034)k(>)739 1782 y Fg(2)p 736 1789 V 736 1813 a Fi(!)775 1798 y Fk(they)c(b)q(ecome)f (distinct)h(and)f(real,)g(one)g(with)g(mo)q(dulus)0 1853 y Fj(>)k Fk(1)g(and)g(other)h(with)f(mo)q(dulus)f Fj(<)h Fk(1.)24 b(T)o(yp)q(e)17 b(3:)23 b(the)17 b(t)o(w)o(o)e(eigen)o(v)n (alues)i(are)f(real)g(and)h(distinct,)f(one)0 1908 y(with)c(mo)q(dulus) e Fj(>)i Fk(1)g(and)g(other)h(with)f(mo)q(dulus)e Fj(<)i Fk(1.)17 b(In)12 b(case)h Fj(L)g Fk(b)q(eing)f(non-negativ)o(e)f (de\014nite,)i(T)o(yp)q(e)0 1963 y(3)i(is)g(missing,)e(then)j(all)d (eigen)o(v)n(alues)i(of)g Fj(F)686 1969 y Fi(\034)721 1963 y Fk(are)h(unimo)q(dular)d(and)h(b)q(elonging)h(to)f(linear)h (elemen)o(tary)0 2017 y(divisors)f(when)g Fj(\034)i(<)372 2001 y Fg(2)p 343 2008 75 2 v 343 2032 a Fi(!)364 2036 y Fc(max)83 2083 y Fk(W)m(e)d(apply)g(the)i(ab)q(o)o(v)o(e)e(sc)o(heme) h(to)g(the)h(1-D)e(w)o(a)o(v)o(e)g(equation)389 2197 y Fj(@)413 2182 y Fg(2)432 2197 y Fj(u)p 389 2215 67 2 v 393 2253 a(@)r(t)432 2241 y Fg(2)472 2225 y Fk(=)f Fj(c)534 2208 y Fg(2)558 2197 y Fj(@)582 2182 y Fg(2)601 2197 y Fj(u)p 558 2215 V 558 2253 a(@)r(x)606 2241 y Fg(2)629 2225 y Fj(;)48 b Fk(0)12 b Fj(<)f(x)h(<)f Fk(1)p Fj(;)48 b(u)p Fk(\(0)p Fj(;)7 b(t)p Fk(\))k(=)h Fj(u)p Fk(\()p Fj(l)q(;)7 b(t)p Fk(\))k(=)h(0)0 2354 y(with)i(\014nite)g (elemen)o(t)f(semi-discretization)392 2460 y Fj(d)414 2445 y Fg(2)432 2460 y Fj(u)456 2466 y Fi(k)p 392 2478 85 2 v 406 2516 a Fj(dt)443 2504 y Fg(2)493 2488 y Fk(=)544 2460 y Fj(c)562 2445 y Fg(2)p 542 2478 43 2 v 542 2516 a Fj(h)566 2504 y Fg(2)596 2488 y Fk([)o Fj(u)631 2494 y Fi(k)q Fe(\000)p Fg(1)703 2488 y Fh(\000)d Fk(2)p Fj(u)790 2494 y Fi(k)819 2488 y Fk(+)g Fj(u)885 2494 y Fi(k)q Fg(+1)947 2488 y Fk(])c Fj(;)48 b(k)13 b Fk(=)e(1)p Fj(;)c Fh(\001)g(\001)g(\001)12 b Fj(;)7 b(n)p Fk(;)387 2589 y Fj(u)411 2595 y Fg(0)441 2589 y Fk(=)12 b Fj(u)509 2595 y Fi(n)p Fg(+1)584 2589 y Fk(=)g(0)p Fj(;)48 b(h)11 b Fk(=)831 2561 y(1)p 793 2580 97 2 v 793 2618 a Fj(n)e Fk(+)h(1)895 2589 y Fj(:)p eop %%Page: 14 9 14 8 bop 14 42 a Fk(14)0 142 y(Let)14 b Fj(q)93 148 y Fi(k)125 142 y Fk(=)e Fj(u)193 148 y Fi(k)213 142 y Fk(,)h Fj(p)259 148 y Fi(k)291 142 y Fk(=)340 125 y Fi(@)r(u)380 129 y Fb(k)p 340 133 58 2 v 352 156 a Fi(@)r(t)402 142 y Fk(,)h(w)o(e)g(get)g(a)f(canonical)h(system)f(\(24\))h(with)391 378 y Fj(M)j Fk(=)11 b Fj(I)s(;)49 b(L)12 b Fk(=)664 350 y Fj(c)682 335 y Fg(2)p 661 369 43 2 v 661 407 a Fj(h)685 395 y Fg(2)716 208 y Fd(2)716 281 y(6)716 306 y(6)716 331 y(6)716 356 y(6)716 380 y(6)716 405 y(6)716 430 y(6)716 457 y(4)764 239 y Fk(2)74 b Fh(\000)p Fk(1)764 320 y Fh(\000)p Fk(1)58 b(2)960 291 y(.)976 303 y(.)993 316 y(.)863 371 y(.)880 384 y(.)896 396 y(.)960 371 y(.)976 384 y(.)993 396 y(.)1055 371 y(.)1071 384 y(.)1087 396 y(.)953 461 y Fh(\000)p Fk(1)74 b(2)41 b Fh(\000)p Fk(1)1048 521 y Fh(\000)p Fk(1)74 b(2)1216 208 y Fd(3)1216 281 y(7)1216 306 y(7)1216 331 y(7)1216 356 y(7)1216 380 y(7)1216 405 y(7)1216 430 y(7)1216 457 y(5)1251 378 y Fj(:)0 616 y Fk(The)21 b(t)o(yp)q(es)g(1)e(and)h(3)g(are)h(missing,)e Fj(!)645 622 y Fi(k)687 616 y Fk(=)747 600 y Fg(2)p Fi(c)p 747 607 32 2 v 753 630 a(h)797 616 y Fk(cos)891 600 y Fi(k)q(\031)p 871 607 80 2 v 871 630 a Fg(2)p Fi(n)p Fg(+1)977 616 y Fj(<)j(!)1057 622 y Fg(1)1098 616 y Fk(=)1157 600 y Fg(2)p Fi(c)p 1157 607 32 2 v 1163 630 a(h)1207 616 y Fk(cos)1311 600 y Fi(\031)p 1282 607 80 2 v 1282 630 a Fg(2)p Fi(n)p Fg(+1)1388 616 y Fj(<)1447 600 y Fg(2)p Fi(c)p 1447 607 32 2 v 1453 630 a(h)1483 616 y Fk(.)37 b(So)20 b(the)0 675 y(Couran)o(t)15 b(condition)f Fj(\034)19 b Fh(\024)435 658 y Fi(h)p 435 665 20 2 v 437 689 a(c)474 675 y Fk(ensures)e(stabilit)o(y)d(of)h(\(24\).)21 b(The)16 b(sc)o(heme)f(is)g(in)f(fact)i(equiv)n(alen)o(t)e(to)h(the)0 729 y(classical)c(5-p)q(oin)o(t)f(sc)o(heme)i(for)e(the)i(w)o(a)o(v)o (e)f(equation,)g(see)h([2].)k(There)c(is)f(an)g(in)o(teresting)h(study) g([3],)e(with)0 784 y(further)18 b(references)i(there,)e(on)f(computer) 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Fg(1)1128 1056 y Fj(p)1149 1062 y Fg(2)1168 1056 y Fj(q)1187 1062 y Fg(1)1205 1056 y Fj(q)1224 1062 y Fg(2)1242 1056 y Fk(\))p Fh(g)p Fj(;)345 1137 y Fk(1)p 344 1156 23 2 v 344 1194 a Fj(\034)372 1165 y Fk(\()r Fd(e)-25 b Fj(p)409 1171 y Fg(2)437 1165 y Fh(\000)9 b Fj(p)499 1171 y Fg(2)518 1165 y Fk(\))j(=)f Fh(\000)679 1137 y Fk(1)p 626 1156 126 2 v 628 1194 a Fd(e)-25 b Fj(q)645 1200 y Fg(2)673 1194 y Fh(\000)10 b Fj(q)734 1200 y Fg(2)757 1165 y Fh(f)p Fj(H)s Fk(\()r Fd(e)-25 b Fj(p)853 1171 y Fg(1)871 1165 y Fj(p)892 1171 y Fg(2)912 1165 y Fd(e)g Fj(q)929 1171 y Fg(1)950 1165 y Fd(e)g Fj(q)967 1171 y Fg(2)985 1165 y Fk(\))9 b Fh(\000)h Fj(H)s Fk(\()r Fd(e)-25 b Fj(p)1127 1171 y Fg(1)1145 1165 y Fj(p)1166 1171 y Fg(2)1187 1165 y Fd(e)g Fj(q)1204 1171 y Fg(1)1222 1165 y Fj(q)1241 1171 y Fg(2)1259 1165 y Fk(\))p Fh(g)p Fj(;)345 1246 y Fk(1)p 344 1265 23 2 v 344 1303 a Fj(\034)372 1274 y Fk(\()r Fd(e)g Fj(q)407 1280 y Fg(1)435 1274 y Fh(\000)9 b Fj(q)495 1280 y Fg(2)513 1274 y Fk(\))j(=)644 1246 y(1)p 590 1265 130 2 v 592 1303 a Fd(e)-25 b Fj(p)611 1309 y Fg(1)638 1303 y Fh(\000)10 b Fj(p)701 1309 y Fg(1)724 1274 y Fh(f)p Fj(H)s Fk(\()r Fd(e)-25 b Fj(p)820 1280 y Fg(1)839 1274 y Fj(p)860 1280 y Fg(2)880 1274 y Fd(e)g Fj(q)897 1280 y Fg(1)915 1274 y Fj(q)934 1280 y Fg(2)952 1274 y Fk(\))10 b Fh(\000)f Fj(H)s Fk(\()p Fj(p)1094 1280 y Fg(1)1113 1274 y Fj(p)1134 1280 y Fg(2)1154 1274 y Fd(e)-25 b Fj(q)1171 1280 y Fg(1)1190 1274 y Fj(q)1209 1280 y Fg(2)1227 1274 y Fk(\))p Fh(g)p Fj(;)304 b Fk(\(27\))345 1355 y(1)p 344 1374 23 2 v 344 1412 a Fj(\034)372 1384 y Fk(\()r Fd(e)-25 b Fj(q)407 1390 y Fg(2)435 1384 y Fh(\000)9 b Fj(q)495 1390 y Fg(2)513 1384 y Fk(\))j(=)644 1355 y(1)p 590 1374 130 2 v 592 1412 a Fd(e)-25 b Fj(p)611 1418 y Fg(2)638 1412 y Fh(\000)10 b Fj(p)701 1418 y Fg(2)724 1384 y Fh(f)p Fj(H)s Fk(\()r Fd(e)-25 b Fj(p)820 1390 y Fg(1)841 1384 y Fd(e)g Fj(p)860 1390 y Fg(2)880 1384 y Fd(e)g Fj(q)897 1390 y Fg(1)917 1384 y Fd(e)g Fj(q)934 1390 y Fg(2)952 1384 y Fk(\))10 b Fh(\000)f Fj(H)s Fk(\()r Fd(e)-25 b Fj(p)1094 1390 y Fg(1)1113 1384 y Fj(p)1134 1390 y Fg(2)1154 1384 y Fd(e)g Fj(q)1171 1390 y Fg(1)1191 1384 y Fd(e)h Fj(q)1209 1390 y Fg(2)1227 1384 y Fk(\))p Fh(g)p Fj(:)83 1493 y Fk(By)18 b(addition)e(and)i(cancellation)f(w)o(e) h(ha)o(v)o(e)g(energy)h(conserv)n(ation)f(for)f(arbitrary)h(Hamiltonia) o(n)0 1548 y Fj(H)s Fk(\()r Fd(e)-25 b Fj(p)75 1554 y Fg(1)96 1548 y Fd(e)g Fj(p)115 1554 y Fg(2)135 1548 y Fd(e)g Fj(q)152 1554 y Fg(1)172 1548 y Fd(e)g Fj(q)189 1554 y Fg(2)207 1548 y Fk(\))12 b(=)g Fj(H)s Fk(\()p Fj(p)354 1554 y Fg(1)372 1548 y Fj(p)393 1554 y Fg(2)412 1548 y Fj(q)431 1554 y Fg(1)449 1548 y Fj(q)468 1554 y Fg(2)486 1548 y Fk(\).)83 1607 y(F)m(or)h(quadratic)h(Hamiltonian,)c Fj(H)15 b Fk(=)691 1590 y Fg(1)p 691 1597 17 2 v 691 1621 a(2)713 1607 y Fj(z)734 1592 y Fe(0)746 1607 y Fj(S)r(z)r Fk(,)f(w)o(e)g(get)302 1688 y(1)p 301 1707 23 2 v 301 1745 a Fj(\034)329 1716 y Fk(\()p Fj(z)366 1699 y Fi(m)p Fg(+1)449 1716 y Fh(\000)c Fj(z)512 1699 y Fi(m)543 1716 y Fk(\))i(=)g Fj(J)642 1699 y Fe(\000)p Fg(1)686 1716 y Fj(S)719 1688 y Fk(1)p 719 1707 21 2 v 719 1745 a(2)745 1716 y(\()p Fj(z)782 1699 y Fi(m)p Fg(+1)865 1716 y Fk(+)d Fj(z)927 1699 y Fi(m)959 1716 y Fk(\))g Fh(\000)1031 1688 y Fk(1)p 1031 1707 V 1031 1745 a(2)1057 1716 y Fj(J)t(R)p Fk(\()p Fj(z)1153 1699 y Fi(m)p Fg(+1)1236 1716 y Fh(\000)g Fj(z)1298 1699 y Fi(m)1330 1716 y Fk(\))p Fj(;)0 1814 y Fk(where)138 1986 y Fj(S)14 b Fk(=)221 1865 y Fd(2)221 1938 y(6)221 1963 y(6)221 1988 y(6)221 2015 y(4)269 1897 y Fj(s)288 1903 y Fg(11)365 1897 y Fj(s)384 1903 y Fg(12)462 1897 y Fj(s)481 1903 y Fg(13)558 1897 y Fj(s)577 1903 y Fg(14)365 1957 y Fj(s)384 1963 y Fg(22)462 1957 y Fj(s)481 1963 y Fg(23)558 1957 y Fj(s)577 1963 y Fg(24)365 2018 y Fh(\003)76 b Fj(s)481 2024 y Fg(33)558 2018 y Fj(s)577 2024 y Fg(34)558 2078 y Fj(s)577 2084 y Fg(44)633 1865 y Fd(3)633 1938 y(7)633 1963 y(7)633 1988 y(7)633 2015 y(5)672 1986 y Fk(=)12 b Fj(S)743 1969 y Fe(0)756 1986 y Fj(;)47 b(R)12 b Fk(=)903 1865 y Fd(2)903 1938 y(6)903 1963 y(6)903 1988 y(6)903 2015 y(4)951 1897 y Fk(0)41 b Fh(\000)p Fj(s)1064 1903 y Fg(12)1142 1897 y Fj(s)1161 1903 y Fg(13)1238 1897 y Fh(\000)p Fj(s)1289 1903 y Fg(14)1046 1957 y Fk(0)75 b Fj(s)1161 1963 y Fg(23)1254 1957 y Fj(s)1273 1963 y Fg(24)1046 2018 y Fh(\003)92 b Fk(0)58 b Fh(\000)p Fj(s)1289 2024 y Fg(34)1271 2078 y Fk(0)1346 1865 y Fd(3)1346 1938 y(7)1346 1963 y(7)1346 1988 y(7)1346 2015 y(5)1385 1986 y Fk(=)12 b Fh(\000)p Fj(R)1493 1969 y Fe(0)1504 1986 y Fj(;)0 2166 y Fk(and)213 2238 y Fj(z)234 2221 y Fi(m)p Fg(+1)319 2238 y Fk(=)g Fj(F)390 2244 y Fi(\034)411 2238 y Fj(z)432 2221 y Fi(m)463 2238 y Fj(;)48 b(F)550 2244 y Fi(\034)582 2238 y Fk(=)12 b(\()p Fj(I)h Fk(+)719 2210 y Fj(\034)p 719 2228 23 2 v 720 2266 a Fk(2)747 2238 y Fj(J)t(R)c Fh(\000)862 2210 y Fj(\034)p 862 2228 V 863 2266 a Fk(2)890 2238 y Fj(J)917 2221 y Fe(\000)p Fg(1)961 2238 y Fj(S)r Fk(\))1004 2221 y Fe(\000)p Fg(1)1050 2238 y Fk(\()p Fj(I)k Fk(+)1143 2210 y Fj(\034)p 1143 2228 V 1144 2266 a Fk(2)1171 2238 y Fj(J)t(R)c Fk(+)1285 2210 y Fj(\034)p 1285 2228 V 1286 2266 a Fk(2)1313 2238 y Fj(J)1340 2221 y Fe(\000)p Fg(1)1385 2238 y Fj(S)r Fk(\))p Fj(:)0 2331 y Fk(Let)85 2321 y(\026)74 2331 y Fj(K)15 b Fk(=)d Fj(J)h Fh(\000)251 2315 y Fi(\034)p 251 2322 19 2 v 252 2346 a Fg(2)274 2331 y Fj(R)p Fk(,)341 2321 y(\026)332 2331 y Fj(B)h Fk(=)431 2321 y(\026)420 2331 y Fj(K)458 2316 y Fe(\000)p Fg(1)503 2331 y Fj(S)r Fk(,)g(w)o(e)h(can)f(pro)o(v)o(e)g(that)g Fj(F)923 2337 y Fi(\034)957 2331 y Fk(is)g(the)g(Ca)o(yley)f(transform)537 2443 y Fj(F)564 2449 y Fi(\034)596 2443 y Fk(=)f Fj( )q Fk(\()p Fh(\000)721 2415 y Fj(\034)p 721 2434 23 2 v 722 2472 a Fk(2)759 2433 y(\026)749 2443 y Fj(B)s Fk(\))p Fj(;)48 b( )q Fk(\()p Fj(\025)p Fk(\))13 b(=)1004 2415 y(1)c Fh(\000)h Fj(\025)p 1004 2434 96 2 v 1004 2472 a Fk(1)f(+)h Fj(\025)1105 2443 y(;)0 2550 y Fk(so)k(w)o(e)h(ha)o(v)o(e) f(in)o(v)n(arian)o(t)e(\\symplectic)i(structures")835 2539 y(\026)824 2550 y Fj(K)859 2556 y Fg(1)890 2550 y Fk(=)946 2539 y(\026)935 2550 y Fj(K)s Fk(,)1010 2539 y(\026)999 2550 y Fj(K)1034 2556 y Fg(2)1053 2550 y Fj(;)1082 2539 y Fk(\026)1072 2550 y Fj(K)1107 2556 y Fg(3)1125 2550 y Fj(;)7 b Fh(\001)g(\001)g(\001)19 b Fk(and)14 b(in)o(v)n(arian)o(t)f(\\energies")7 2594 y(\026)0 2605 y Fj(S)25 2611 y Fg(1)44 2605 y Fj(;)7 b Fk(=)k Fj(S)r Fk(,)167 2594 y(\026)160 2605 y Fj(S)185 2611 y Fg(2)204 2605 y Fj(;)229 2594 y Fk(\026)223 2605 y Fj(S)248 2611 y Fg(3)266 2605 y Fj(;)c Fh(\001)g(\001)g(\001)19 b Fk(lik)o(e)13 b(\(17\),)g(\(18\).)p eop %%Page: 15 10 15 9 bop 1598 42 a Fk(15)209 142 y Fm(x)p Fl(4)26 b(Di\013erence)16 b(Sc)o(hemes)e(for)j(General)g(Canonical)g(System)o(s)83 237 y Fk(The)12 b(three)h(kinds)f(of)g(sc)o(hemes)g(for)g(linear)g (systems)g(in)f(the)i(previous)f(section)h(can)f(b)q(e)h(generalized)0 292 y(to)h(the)g(general)g(non-linear)f(case.)83 348 y(Sc)o(heme)g(I.)h(F)m(or)f(the)i(general)f(canonical)f(system)h (\(1\),)f(w)o(e)h(put)459 416 y(1)p 458 434 23 2 v 458 472 a Fj(\034)486 444 y Fk(\()p Fj(z)523 427 y Fi(m)p Fg(+1)606 444 y Fh(\000)c Fj(z)669 427 y Fi(m)700 444 y Fk(\))i(=)g Fj(J)799 427 y Fe(\000)p Fg(1)843 444 y Fj(H)878 450 y Fi(z)897 444 y Fk(\()918 416 y(1)p 918 434 21 2 v 918 472 a(2)944 444 y Fj(z)965 427 y Fi(m)p Fg(+1)1048 444 y Fk(+)1094 416 y(1)p 1094 434 V 1094 472 a(2)1120 444 y Fj(z)1141 427 y Fi(m)1173 444 y Fk(\))p Fj(:)379 b Fk(\(28\))0 536 y(The)14 b(transition)g Fj(z)296 521 y Fi(m)339 536 y Fh(!)d Fj(z)413 521 y Fi(m)p Fg(+1)501 536 y Fk(is)i(non-linear)g(in)h(general.)k(By)c(di\013eren)o(tiation,) 332 609 y Fj(@)r(z)377 594 y Fi(m)p Fg(+1)p 332 627 120 2 v 353 665 a Fj(@)r(z)398 653 y Fi(m)465 637 y Fh(\000)c Fj(I)15 b Fk(=)d Fj(\034)5 b(J)634 620 y Fe(\000)p Fg(1)678 637 y Fj(H)713 643 y Fi(z)q(z)749 603 y Fd(\000)773 609 y Fj(z)794 594 y Fi(m)p Fg(+1)877 609 y Fk(+)k Fj(z)939 594 y Fi(m)p 773 627 199 2 v 861 665 a Fk(2)976 603 y Fd(\001)1002 578 y(\024)1029 609 y Fk(1)p 1029 627 21 2 v 1029 665 a(2)1059 609 y Fj(@)r(z)1104 594 y Fi(m)p Fg(+1)p 1059 627 120 2 v 1080 665 a Fj(@)r(z)1125 653 y Fi(m)1193 637 y Fk(+)1239 609 y(1)p 1239 627 21 2 v 1239 665 a(2)1265 637 y Fj(I)1286 578 y Fd(\025)1315 637 y Fj(;)0 749 y Fk(here)15 b Fj(H)125 755 y Fi(z)q(z)161 715 y Fd(\000)185 733 y Fi(z)202 720 y Fb(m)p Fc(+1)265 733 y Fg(+)p Fi(z)307 720 y Fb(m)p 185 740 151 2 v 251 763 a Fg(2)340 715 y Fd(\001)373 749 y Fk(is)e(the)h(Hessian)h(matrix)d (of)h(the)h(function)g Fj(H)s Fk(\()p Fj(z)r Fk(\),)f(ev)n(aluated)h (at)f Fj(z)h Fk(=)1487 733 y Fi(z)1504 720 y Fb(m)p Fc(+1)1567 733 y Fg(+)p Fi(z)1609 720 y Fb(m)p 1487 740 V 1554 763 a Fg(2)1642 749 y Fk(,)5 794 y Fi(@)r(z)42 782 y Fb(m)p Fc(+1)p 5 801 100 2 v 23 825 a Fi(@)r(z)60 816 y Fb(m)123 810 y Fk(is)g(the)h(Jacobian)e(matrix)f Fj(F)572 816 y Fi(\034)593 810 y Fk(,)h(so)203 925 y Fj(F)230 931 y Fi(\034)262 925 y Fk(=)306 867 y Fd(\024)328 925 y Fj(I)g Fh(\000)405 897 y Fj(\034)p 405 916 23 2 v 406 954 a Fk(2)433 925 y Fj(J)460 908 y Fe(\000)p Fg(1)504 925 y Fj(H)539 931 y Fi(z)q(z)575 892 y Fd(\000)599 897 y Fj(z)620 882 y Fi(m)p Fg(+1)703 897 y Fk(+)d Fj(z)766 882 y Fi(m)p 599 916 199 2 v 688 954 a Fk(2)802 892 y Fd(\001)821 867 y(\025)843 874 y Fe(\000)p Fg(1)895 867 y Fd(\024)917 925 y Fj(I)i Fk(+)994 897 y Fj(\034)p 994 916 23 2 v 995 954 a Fk(2)1022 925 y Fj(J)1049 908 y Fe(\000)p Fg(1)1093 925 y Fj(H)1128 931 y Fi(z)q(z)1164 892 y Fd(\000)1188 897 y Fj(z)1209 882 y Fi(m)p Fg(+1)1292 897 y Fk(+)d Fj(z)1354 882 y Fi(m)p 1188 916 199 2 v 1277 954 a Fk(2)1391 892 y Fd(\001)1410 867 y(\025)1439 925 y Fj(:)0 1024 y Fk(When)k Fj(z)j Fk(remains)c(b)q(ounded)i(and)f (tak)o(e)g Fj(\034)18 b Fk(su\016cien)o(tly)13 b(small)e(w)o(e)i(can)h (k)o(eep)g(the)g(in\014nitesimally)c(sym-)0 1079 y(plectic)21 b(matrix)285 1063 y Fi(\034)p 285 1070 19 2 v 286 1093 a Fg(2)309 1079 y Fj(J)336 1064 y Fe(\000)p Fg(1)380 1079 y Fj(H)415 1085 y Fi(z)q(z)451 1045 y Fd(\000)475 1063 y Fi(z)492 1050 y Fb(m)p Fc(+1)555 1063 y Fg(+)p Fi(z)597 1050 y Fb(m)p 475 1070 151 2 v 542 1093 a Fg(2)630 1045 y Fd(\001)670 1079 y Fk(non-exceptional,)h(then)f Fj(F)1117 1085 y Fi(\034)1138 1079 y Fk(,)g(as)g(a)g(Ca)o(yley)f (transform,)h(is)0 1134 y(symplectic.)c(Th)o(us)c(all)e(the)h(conserv)n (ation)h(la)o(ws)f(for)g(phase)h(areas)f(remain)f(true.)19 b(Ho)o(w)o(ev)o(er,)12 b(unlik)o(e)g(the)0 1188 y(linear)g(case,)i(the) f(\014rst)h(in)o(tegrals)e Fj(\036)p Fk(\()p Fj(z)r Fk(\))h(including)f Fj(H)k Fk(itself)c(are)h(not)g(conserv)o(ed)h(exactly)m(.)k(Instead,)13 b(the)0 1243 y(appro)o(ximate)f(conserv)n(ation)545 1324 y Fj(')p Fk(\()p Fj(z)609 1307 y Fi(m)p Fg(+1)683 1324 y Fk(\))g(=)g Fj(')p Fk(\()p Fj(z)819 1307 y Fi(m)851 1324 y Fk(\))41 b(mo)q(d)13 b Fj(O)q Fk(\()p Fj(\034)1074 1307 y Fg(3)1092 1324 y Fk(\))0 1405 y(can)h(b)q(e)h(sho)o(wn.)83 1461 y(W)m(e)e(remark)g(that)h(the)h(analogous)d(a)o(v)o(eraged)i (implicit)d(Euler)k(sc)o(heme)387 1532 y(1)p 386 1550 23 2 v 386 1588 a Fj(\034)414 1560 y Fk(\()p Fj(z)451 1543 y Fi(m)p Fg(+1)534 1560 y Fh(\000)10 b Fj(z)597 1543 y Fi(m)628 1560 y Fk(\))i(=)g Fj(J)727 1543 y Fe(\000)p Fg(1)778 1501 y Fd(\024)805 1532 y Fk(1)p 805 1550 21 2 v 805 1588 a(2)831 1560 y Fj(H)866 1566 y Fi(z)885 1560 y Fk(\()p Fj(z)922 1543 y Fi(m)p Fg(+1)996 1560 y Fk(\))d(+)1067 1532 y(1)p 1067 1550 V 1067 1588 a(2)1093 1560 y Fj(H)1128 1566 y Fi(z)1147 1560 y Fk(\()p Fj(z)1184 1543 y Fi(m)1216 1560 y Fk(\))1232 1501 y Fd(\025)1261 1560 y Fj(;)307 b Fk(\(29\))0 1661 y(whic)o(h)11 b(reduces,)h(lik)o(e)e (\(28\),)h(to)f(the)i(same)e(symplectic)g(sc)o(heme)g(\(19\))h(with)f Fj(K)15 b Fk(=)d Fj(J)i Fk(for)d(linear)f(problems.)370 1749 y Fj(F)397 1755 y Fi(\034)429 1749 y Fk(=)473 1703 y Fd(h)493 1749 y Fj(I)i Fh(\000)570 1721 y Fj(\034)p 570 1740 23 2 v 571 1778 a Fk(2)598 1749 y Fj(J)625 1732 y Fe(\000)p Fg(1)669 1749 y Fj(H)704 1755 y Fi(z)q(z)740 1749 y Fk(\()p Fj(z)777 1732 y Fi(m)p Fg(+1)851 1749 y Fk(\))867 1703 y Fd(i)6 b(h)913 1749 y Fj(I)13 b Fk(+)990 1721 y Fj(\034)p 990 1740 V 991 1778 a Fk(2)1018 1749 y Fj(J)1045 1732 y Fe(\000)p Fg(1)1090 1749 y Fj(H)1125 1755 y Fi(z)q(z)1160 1749 y Fk(\()p Fj(z)1197 1732 y Fi(m)1229 1749 y Fk(\))1245 1703 y Fd(i)1272 1749 y Fj(;)0 1837 y Fk(whic)o(h)h(is)g(not)f(symplectic)h(in)f(general.)83 1893 y(Sc)o(heme)19 b(I)q(I.)h(F)m(or)f(the)h(canonical)f(system)h (with)f(general)h(separable)g(Hamiltonian)d Fj(H)s Fk(\()p Fj(p;)7 b(q)q Fk(\))20 b(=)0 1948 y Fj(U)5 b Fk(\()p Fj(p)p Fk(\))k(+)h Fj(V)f Fk(\()p Fj(q)q Fk(\),)14 b(w)o(e)g(ha)o(v)o (e)510 1996 y(1)p 509 2015 V 509 2053 a Fj(\034)536 2024 y Fk(\()p Fj(p)573 2007 y Fi(m)p Fg(+1)656 2024 y Fh(\000)c Fj(p)719 2007 y Fi(m)750 2024 y Fk(\))i(=)g Fh(\000)p Fj(V)878 2030 y Fi(q)896 2024 y Fk(\()p Fj(q)932 2007 y Fi(m)p Fg(+1)p Fi(=)p Fg(2)1040 2024 y Fk(\))p Fj(;)510 2097 y Fk(1)p 509 2116 V 509 2154 a Fj(\034)536 2125 y Fk(\()p Fj(q)572 2108 y Fi(m)p Fg(+1+1)p Fi(=)p Fg(2)731 2125 y Fh(\000)d Fj(q)792 2108 y Fi(m)p Fg(+1)p Fi(=)p Fg(2)900 2125 y Fk(\))i(=)h Fj(U)999 2131 y Fi(p)1019 2125 y Fk(\()p Fj(p)1056 2108 y Fi(m)p Fg(+1)1129 2125 y Fk(\))p Fj(:)1580 2072 y Fk(\(30\))0 2247 y(The)i(transition)275 2176 y Fd(")320 2219 y Fj(p)341 2204 y Fi(m)320 2279 y Fj(q)340 2264 y Fi(m)p Fg(+1)p Fi(=)p Fg(2)468 2176 y Fd(#)503 2247 y Fh(!)556 2176 y Fd(")601 2219 y Fj(p)622 2204 y Fi(m)p Fg(+1)601 2279 y Fj(q)621 2264 y Fi(m)p Fg(+1+1)p Fi(=)p Fg(2)791 2176 y Fd(#)829 2247 y Fk(has)g(Jacobian)496 2410 y Fj(F)523 2416 y Fi(\034)555 2410 y Fk(=)599 2339 y Fd(")683 2382 y Fj(I)84 b Fk(0)644 2442 y Fh(\000)p Fj(\034)5 b(M)46 b(I)827 2339 y Fd(#)851 2348 y Fe(\000)p Fg(1)903 2339 y Fd(")948 2382 y Fj(I)f Fh(\000)p Fj(\034)5 b(L)948 2442 y Fk(0)73 b Fj(I)1115 2339 y Fd(#)1146 2410 y Fj(;)0 2524 y Fk(whic)o(h)14 b(can)g(b)q(e)g(sho)o(wn)g(to)g Fj(K)s Fk(-symplectic)f(as)h(for)g(\(24\),)f(\(26\),)g(but)h(with)494 2605 y Fj(M)j Fk(=)11 b Fj(U)622 2611 y Fi(pp)659 2605 y Fk(\()p Fj(p)696 2587 y Fi(m)p Fg(+1)770 2605 y Fk(\))p Fj(;)48 b(L)12 b Fk(=)f Fj(V)953 2611 y Fi(q)q(q)988 2605 y Fk(\()p Fj(q)1024 2587 y Fi(m)p Fg(+1)p Fi(=)p Fg(2)1132 2605 y Fk(\))p Fj(:)p eop %%Page: 16 11 16 10 bop 14 42 a Fk(16)0 142 y(This)20 b(leads)g(to)g(a)g(class)h(of)e (mo)q(di\014ed)g(conserv)n(ation)h(la)o(ws)g(of)g(phase)g(areas,)i(but) f(with)e(Liouville's)0 197 y(theorem)14 b(unc)o(hanged.)83 252 y(The)g(\014rst)h(in)o(tegrals)e Fj(\036)p Fk(\()p Fj(p;)7 b(q)q Fk(\),)13 b(including)g Fj(H)s Fk(\()p Fj(p;)7 b(q)q Fk(\),)13 b(are)h(appro)o(ximately)d(conserv)o(ed)16 b(as)133 340 y Fj(\036)p Fk(\()p Fj(p)195 323 y Fi(m)p Fg(+1)269 340 y Fj(;)292 312 y Fk(1)p 292 331 21 2 v 292 369 a(2)318 340 y(\()p Fj(q)354 323 y Fi(m)p Fg(+1+1)p Fi(=)p Fg(2)512 340 y Fk(+)10 b Fj(q)574 323 y Fi(m)p Fg(+1)p Fi(=)p Fg(2)681 340 y Fk(\)\))i(=)g Fj(\036)p Fk(\()p Fj(p)831 323 y Fi(m)862 340 y Fj(;)885 312 y Fk(1)p 885 331 V 885 369 a(2)911 340 y(\()p Fj(q)947 323 y Fi(m)p Fg(+1)p Fi(=)p Fg(2)1064 340 y Fk(+)d Fj(q)1125 323 y Fi(m)p Fe(\000)p Fg(1)p Fi(=)p Fg(2)1233 340 y Fk(\))p Fj(;)48 b Fk(mo)q(d)12 b Fj(O)q Fk(\()p Fj(\034)1474 323 y Fg(3)1493 340 y Fk(\))p Fj(:)83 421 y Fk(Sc)o(heme)k(I)q(I)q(I.)h (This)f(has)h(already)f(b)q(een)i(constructed)h(for)d(the)h(nonlinear)f (case)i(in)e(the)h(previous)0 476 y(section,)12 b(the)h(Hamiltonia)o(n) c Fj(H)s Fk(\()p Fj(z)r Fk(\))j(is)f(alw)o(a)o(ys)g(conserv)o(ed)i (exactly)m(.)k(Ho)o(w)o(ev)o(er,)12 b(the)g(\014rst)h(in)o(tegrals)e Fj(\036)p Fk(\()p Fj(z)r Fk(\),)0 530 y(other)k(than)e(the)i (Hamiltonian,)10 b(are)k(appro)o(ximately)e(conserv)o(ed)j(to)f(a)g(lo) o(w)o(er)f(order)i(as)538 604 y Fj(\036)p Fk(\()p Fj(z)600 587 y Fi(m)p Fg(+1)674 604 y Fk(\))d(=)f Fj(\036)p Fk(\()p Fj(z)807 587 y Fi(m)839 604 y Fk(\))p Fj(;)48 b Fk(mo)q(d)13 b Fj(O)q Fk(\()p Fj(\034)1081 587 y Fg(2)1099 604 y Fk(\))0 678 y(due)j(to)g(some)e(kind)i(of)f(asymmetry)e(in)i(the)h(algorithm.) 21 b(Moreo)o(v)o(er,)16 b(except)h(in)e(the)i(linear)e(systems,)0 733 y(symplectic)e(prop)q(erties)j(for)d(the)i(Jacobian)e(of)h (transition)f(could)h(not)f(b)q(e)i(established)g(in)e(general.)83 788 y(The)h(details)g(of)f(the)h(results)h(and)f(some)f(other)i(dev)o (elopmen)o(ts)e(will)f(b)q(e)j(published)f(elsewhere.)704 919 y Fl(References)19 1005 y Fa(1)21 b(Arnold)14 b(V)f(I.)f (Mathematical)j(Metho)q(ds)g(of)d(Classical)k(Mec)o(hanics.)e(New)e(Y)m (ork:)17 b(Springer.)e(1978)19 1055 y(2)21 b(Ric)o(h)o(tm)o(y)o(er)13 b(R)f(D,)g(Morton)h(K)f(W.)g(Di\013erence)i(Metho)q(ds)f(for)e (Initial-V)m(alue)k(Problems,)f(2nd)e(edition.)i(New)59 1105 y(Y)m(ork:)j(In)o(terscience.)d(1967)19 1155 y(3)21 b(Buneman)14 b(O.)f(Adv)n(an)o(tages)h(of)f(Hamiltonian)i(F)m(orm)o (ulations)h(in)d(Computer)h(Sim)o(ulations.)i(In)d(T)m(ab)q(or,)f(ed.) 59 1206 y(Mathematical)k(Metho)q(ds)g(in)f(Hydro)q(dynamics)h(and)f(In) o(tegrabilit)o(y)i(of)c(Dynamical)k(Systems.)e(Amer)e(Inst)59 1256 y(Ph)o(ys.)g(USA.)g(1981)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF