%!PS-Adobe-2.0 %%Creator: dvips(k) 5.98 Copyright 2009 Radical Eye Software %%Title: arXiv:1003.1840v1 [math.DG] 9 Mar 2010 %%CreationDate: Wed Jul 25 18:38:49 2018 %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -K1 -z ghc_arch.dvi -o %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 2018.07.25:1838 %%BeginProcSet: texc.pro 0 0 /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 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118 511 y(them)j(nonholonomic)f(v)m(elo)s(cities,)j(is)d(obtained.)59 b(W)-8 b(e)37 b(shall)e(quote)i(the)f(conditions)f(from)h([)3379 511 y SDict begin H.S end 3379 511 a Fi(3)3425 453 y SDict begin H.R end 3425 453 a 3425 511 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 3425 511 a Fi(])g(ex-)118 624 y(plicitly)25 b(in)h(Section)h(2.)41 b(These)27 b(conditions)f(are)i(describ)s(ed)d(as)j(generalized)f (Helmholtz)g(conditions)118 737 y(to)32 b(distinguish)27 b(them)k(from)f(the)h(Helmholtz)g(conditions)e(discussed)g(in)g(our)i (op)s(ening)e(paragraph,)118 850 y(whic)m(h)f(ma)m(y)h(b)s(e)f(called)g (the)h(classical)f(Helmholtz)h(conditions;)f(these)h(m)m(ust)g(of)g (course)g(comprise)f(a)118 963 y(sp)s(ecial)h(case)i(of)g(the)g (generalized)f(conditions.)118 1109 y(The)35 b(main)f(purp)s(ose)f(of)j (the)f(presen)m(t)g(pap)s(er)g(is)f(to)i(p)s(oin)m(t)e(out)h(that)h (the)g(generalized)e(Helmholtz)118 1222 y(conditions)h(as)i(stated)h (in)e([)1095 1222 y SDict begin H.S end 1095 1222 a Fi(3)1140 1163 y SDict begin H.R end 1140 1163 a 1140 1222 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 1140 1222 a Fi(])h(are)h(not)f(indep)s(enden)m(t:)d (in)i(fact)i(t)m(w)m(o)g(of)f(them)g(are)g(redundan)m(t,)g(in)118 1335 y(that)28 b(they)g(can)f(b)s(e)g(deriv)m(ed)g(from)g(the)g (remaining)f(ones.)40 b(This)25 b(w)m(e)j(sho)m(w)g(in)e(Section)h(2)h (b)s(elo)m(w.)39 b(W)-8 b(e)118 1448 y(use)33 b(the)g(same)g(formalism) e(as)j([)1232 1448 y SDict begin H.S end 1232 1448 a Fi(3)1277 1389 y SDict begin H.R end 1277 1389 a 1277 1448 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 1277 1448 a Fi(].)49 b(Since)32 b(the)h(v)m(ersion)f (of)h(the)h(generalized)e(conditions)g(obtained)118 1561 y(in)26 b([)246 1561 y SDict begin H.S end 246 1561 a Fi(2)292 1502 y SDict begin H.R end 292 1502 a 292 1561 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ2) cvn H.B /ANN pdfmark end 292 1561 a Fi(])h(is)g(equiv)-5 b(alen)m(t)26 b(to)i(that)g(in)e([)1291 1561 y SDict begin H.S end 1291 1561 a Fi(3)1337 1502 y SDict begin H.R end 1337 1502 a 1337 1561 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 1337 1561 a Fi(])h(the)h(same)g (redundancy)d(is)h(presen)m(t)i(there)f(as)h(w)m(ell.)38 b(By)28 b(taking)118 1674 y(adv)-5 b(an)m(tage)26 b(of)e(the)g(impro)m (v)m(emen)m(t)g(in)e(the)i(form)m(ulation)f(of)h(the)g(generalized)f (Helmholtz)g(conditions)118 1787 y(that)29 b(w)m(e)h(ac)m(hiev)m(e,)g (w)m(e)f(are)h(able)e(to)h(giv)m(e)h(a)f(shorter)f(and)g(more)h(elegan) m(t)h(pro)s(of)e(of)h(their)f(su\016ciency)118 1900 y(than)i(the)h(one) f(to)i(b)s(e)d(found)g(in)g([)1274 1900 y SDict begin H.S end 1274 1900 a Fi(3)1320 1841 y SDict begin H.R end 1320 1841 a 1320 1900 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 1320 1900 a Fi(].)118 2046 y(There)k(are)h(in)e(fact)i(sev)m(eral)g(in)m(teresting) e(questions)h(raised)f(b)m(y)i(the)f(t)m(w)m(o)i(pap)s(ers)d([)3032 2046 y SDict begin H.S end 3032 2046 a Fi(2)3078 1987 y SDict begin H.R end 3078 1987 a 3078 2046 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ2) cvn H.B /ANN pdfmark end 3078 2046 a Fi(,)3136 2046 y SDict begin H.S end 3136 2046 a Fi(3)3182 1987 y SDict begin H.R end 3182 1987 a 3182 2046 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 3182 2046 a Fi(],)i(only)f(one)118 2159 y(of)g(whic)m(h)e(will)f(b)s(e)i(dealt)g(with)g(here.)47 b(In)32 b(the)g(third)f(and)h(\014nal)f(section)i(of)f(our)h(pap)s(er)e (w)m(e)i(giv)m(e)g(an)118 2271 y(outline)28 b(of)i(these)g(additional)e (p)s(oin)m(ts)h(of)g(in)m(terest,)i(whic)m(h)d(will)f(receiv)m(e)k(a)f (fuller)d(airing)h(elsewhere.)118 2418 y(W)-8 b(e)32 b(emplo)m(y)e(the)g(Einstein)f(summation)g(con)m(v)m(en)m(tion)i (throughout.)118 2564 y(W)-8 b(e)22 b(end)d(this)g(in)m(tro)s(duction)g (with)g(a)h(brief)f(summary)g(of)i(the)f(results)f(ab)s(out)h(the)h (classical)e(Helmholtz)118 2677 y(conditions)29 b(that)i(w)m(e)g(shall) d(need.)118 2823 y(The)i(classical)f(Helmholtz)h(conditions)f(are)i (that)g(the)f Fh(f)2076 2837 y Fg(a)2148 2823 y Fi(should)e(satisfy) 1479 2961 y Fh(@)5 b(f)1577 2975 y Fg(a)p 1479 3001 140 4 v 1483 3085 a Fh(@)12 b Fi(\177)-52 b Fh(q)1580 3058 y Fg(b)1654 3022 y Fi(=)1763 2961 y Fh(@)5 b(f)1861 2976 y Fg(b)p 1760 3001 139 4 v 1760 3085 a Fh(@)12 b Fi(\177)-52 b Fh(q)1857 3058 y Fg(a)3489 3022 y SDict begin H.S end 3489 3022 a 3489 3022 a SDict begin 16.6 H.A end 3489 3022 a 3489 3022 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.3) cvn /DEST pdfmark end 3489 3022 a Fi(\(3\))1209 3206 y Fh(@)5 b(f)1307 3220 y Fg(a)p 1209 3246 140 4 v 1213 3330 a Fh(@)23 b Fi(_)-43 b Fh(q)1310 3304 y Fg(b)1379 3267 y Fi(+)1483 3206 y Fh(@)5 b(f)1581 3221 y Fg(b)p 1480 3246 139 4 v 1480 3330 a Fh(@)22 b Fi(_)-42 b Fh(q)1577 3303 y Fg(a)1654 3267 y Fi(=)25 b(2)1822 3206 y Fh(d)p 1805 3246 81 4 v 1805 3330 a(dt)1911 3139 y Ff(\022)1991 3206 y Fh(@)5 b(f)2089 3221 y Fg(b)p 1987 3246 139 4 v 1987 3330 a Fh(@)12 b Fi(\177)-52 b Fh(q)2084 3303 y Fg(a)2136 3139 y Ff(\023)3489 3267 y SDict begin H.S end 3489 3267 a 3489 3267 a SDict begin 16.6 H.A end 3489 3267 a 3489 3267 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.4) cvn /DEST pdfmark end 3489 3267 a Fi(\(4\))1209 3457 y Fh(@)5 b(f)1307 3471 y Fg(a)p 1209 3498 140 4 v 1213 3581 a Fh(@)g(q)1310 3555 y Fg(b)1379 3519 y Fe(\000)1483 3457 y Fh(@)g(f)1581 3472 y Fg(b)p 1480 3498 139 4 v 1480 3581 a Fh(@)g(q)1577 3555 y Fg(a)1654 3519 y Fi(=)1760 3483 y Fd(1)p 1760 3498 36 4 v 1760 3550 a(2)1831 3457 y Fh(d)p 1815 3498 81 4 v 1815 3581 a(dt)1920 3391 y Ff(\022)1997 3457 y Fh(@)g(f)2095 3471 y Fg(a)p 1997 3498 140 4 v 2001 3581 a Fh(@)22 b Fi(_)-42 b Fh(q)2098 3555 y Fg(b)2167 3519 y Fe(\000)2271 3457 y Fh(@)5 b(f)2369 3472 y Fg(b)p 2268 3498 139 4 v 2268 3581 a Fh(@)22 b Fi(_)-42 b Fh(q)2365 3555 y Fg(a)2416 3391 y Ff(\023)2498 3519 y Fh(:)3489 3519 y SDict begin H.S end 3489 3519 a 3489 3519 a SDict begin 16.6 H.A end 3489 3519 a 3489 3519 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.5) cvn /DEST pdfmark end 3489 3519 a Fi(\(5\))118 3731 y(It)31 b(is)e(a)i(consequence)g(of)f(these)h(\(and)f(not)h(an)f(extra)h (condition,)f(as)g(stated)h(in)f([)2909 3731 y SDict begin H.S end 2909 3731 a Fi(3)2954 3672 y SDict begin H.R end 2954 3672 a 2954 3731 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 2954 3731 a Fi(]\))h(that)1689 3886 y Fh(@)1742 3853 y Fd(2)1782 3886 y Fh(f)1827 3900 y Fg(a)p 1647 3927 264 4 v 1647 4010 a Fh(@)12 b Fi(\177)-52 b Fh(q)1744 3984 y Fg(b)1778 4010 y Fh(@)12 b Fi(\177)-52 b Fh(q)1875 3984 y Fg(c)1945 3948 y Fi(=)25 b(0)118 4158 y(\(this)32 b(follo)m(ws)g(from)g(the)g(v) -5 b(anishing)31 b(of)h(the)h(co)s(e\016cien)m(t)h(of)2212 4106 y(.)-8 b(.)g(.)2218 4158 y Fh(q)2269 4114 y Fg(c)2337 4158 y Fi(in)31 b(condition)g(\()2881 4158 y SDict begin H.S end 2881 4158 a Fi(4)2927 4099 y SDict begin H.R end 2927 4099 a 2927 4158 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.4) cvn H.B /ANN pdfmark end 2927 4158 a Fi(\)\).)48 b(Th)m(us)31 b(w)m(e)i(ma)m(y)118 4271 y(write)g Fh(f)399 4285 y Fg(a)471 4271 y Fi(=)e Fh(g)616 4286 y Fg(ab)695 4271 y Fi(\177)-52 b Fh(q)732 4238 y Fg(b)789 4271 y Fi(+)22 b Fh(h)934 4285 y Fg(a)976 4271 y Fi(,)35 b(the)f(co)s(e\016cien)m(ts)h(b)s(eing)d(of)i(\014rst)f (order,)h(with)f Fh(g)2727 4286 y Fg(ab)2830 4271 y Fi(=)e Fh(g)2975 4286 y Fg(ba)3081 4271 y Fi(as)j(a)g(result)f(of)118 4384 y(condition)h(\()557 4384 y SDict begin H.S end 557 4384 a Fi(3)603 4325 y SDict begin H.R end 603 4325 a 603 4384 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.3) cvn H.B /ANN pdfmark end 603 4384 a Fi(\).)54 b(The)35 b(Helmholtz)f(conditions)g(can)h(b)s(e)g(re-expressed)f(in)g(terms)h (of)g Fh(g)3128 4399 y Fg(ab)3235 4384 y Fi(\(assumed)118 4497 y(to)c(b)s(e)f(symmetric\))g(and)g Fh(h)1060 4511 y Fg(a)1102 4497 y Fi(,)g(when)g(they)g(reduce)g(to)h(the)g(follo)m (wing)e(three)h(conditions:)1275 4640 y Fh(@)5 b(g)1371 4655 y Fg(ab)p 1275 4680 169 4 v 1293 4764 a Fh(@)23 b Fi(_)-43 b Fh(q)1390 4737 y Fg(c)1474 4701 y Fe(\000)1575 4640 y Fh(@)5 b(g)1671 4654 y Fg(ac)p 1575 4680 V 1593 4764 a Fh(@)22 b Fi(_)-42 b Fh(q)1690 4738 y Fg(b)1779 4701 y Fi(=)25 b(0)3489 4701 y SDict begin H.S end 3489 4701 a 3489 4701 a SDict begin 16.6 H.A end 3489 4701 a 3489 4701 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.6) cvn /DEST pdfmark end 3489 4701 a Fi(\(6\))1325 4890 y Fh(@)5 b(h)1430 4904 y Fg(a)p 1325 4931 148 4 v 1333 5014 a Fh(@)22 b Fi(_)-42 b Fh(q)1430 4988 y Fg(b)1503 4952 y Fi(+)1603 4890 y Fh(@)5 b(h)1708 4905 y Fg(b)p 1603 4931 141 4 v 1604 5014 a Fh(@)22 b Fi(_)-42 b Fh(q)1701 4988 y Fg(a)1779 4952 y Fi(=)25 b(2)1963 4866 y(\026)1946 4890 y Fh(d)p 1930 4931 81 4 v 1930 5014 a(dt)2020 4952 y Fi(\()p Fh(g)2098 4967 y Fg(ab)2171 4952 y Fi(\))3489 4952 y SDict begin H.S end 3489 4952 a 3489 4952 a SDict begin 16.6 H.A end 3489 4952 a 3489 4952 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.7) cvn /DEST pdfmark end 3489 4952 a Fi(\(7\))1121 5202 y(2)1181 5074 y Ff(\022)1258 5141 y Fh(@)5 b(h)1363 5155 y Fg(a)p 1258 5181 148 4 v 1266 5265 a Fh(@)g(q)1363 5238 y Fg(b)1436 5202 y Fe(\000)1537 5141 y Fh(@)g(h)1642 5156 y Fg(b)p 1537 5181 141 4 v 1538 5264 a Fh(@)g(q)1635 5238 y Fg(a)1687 5074 y Ff(\023)1779 5202 y Fi(=)1917 5117 y(\026)1901 5141 y Fh(d)p 1885 5181 81 4 v 1885 5264 a(dt)1990 5074 y Ff(\022)2067 5141 y Fh(@)g(h)2172 5155 y Fg(a)p 2067 5181 148 4 v 2075 5265 a Fh(@)22 b Fi(_)-42 b Fh(q)2172 5238 y Fg(b)2244 5202 y Fe(\000)2345 5141 y Fh(@)5 b(h)2450 5156 y Fg(b)p 2345 5181 141 4 v 2346 5264 a Fh(@)22 b Fi(_)-42 b Fh(q)2443 5238 y Fg(a)2495 5074 y Ff(\023)2577 5202 y Fh(;)3489 5202 y SDict begin H.S end 3489 5202 a 3489 5202 a SDict begin 16.6 H.A end 3489 5202 a 3489 5202 a SDict begin [/View [/XYZ H.V]/Dest (equation.1.8) cvn /DEST pdfmark end 3489 5202 a Fi(\(8\))1839 5538 y(2)p eop end end %%Page: 3 3 TeXDict begin HPSdict begin 3 2 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 110 -8 a SDict begin H.S end 110 -8 a 110 -8 a SDict begin H.R end 110 -8 a 110 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.3) cvn /DEST pdfmark end 110 -8 a 118 399 a Fi(where)1557 458 y(\026)1541 482 y Fh(d)p 1524 523 81 4 v 1524 606 a(dt)1640 544 y Fi(=)1762 482 y Fh(@)p 1746 523 87 4 v 1746 606 a(@)5 b(t)1862 544 y Fi(+)37 b(_)-42 b Fh(q)1997 506 y Fg(c)2081 482 y Fh(@)p 2042 523 132 4 v 2042 606 a(@)5 b(q)2139 580 y Fg(c)2183 544 y Fh(:)118 737 y Fi(That)29 b(is)f(to)i(sa)m(y)-8 b(,)31 b(the)f Fh(f)919 751 y Fg(a)989 737 y Fi(are)f(the)h(Euler-Lagrange)f (expressions)f(of)h(some)h(\014rst-order)e(Lagrangian)118 850 y(if)d(and)h(only)f(if)g Fh(f)689 864 y Fg(a)755 850 y Fi(=)g Fh(g)894 865 y Fg(ab)973 850 y Fi(\177)-51 b Fh(q)1011 817 y Fg(b)1056 850 y Fi(+)12 b Fh(h)1191 864 y Fg(a)1259 850 y Fi(for)26 b(some)g(\014rst-order)g(functions)e Fh(g)2469 865 y Fg(ab)2567 850 y Fi(and)i Fh(h)2792 864 y Fg(a)2860 850 y Fi(suc)m(h)g(that)g Fh(g)3296 865 y Fg(ab)3394 850 y Fi(=)f Fh(g)3533 865 y Fg(ba)118 963 y Fi(and)c(\()321 963 y SDict begin H.S end 321 963 a Fi(6)367 904 y SDict begin H.R end 367 904 a 367 963 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.6) cvn H.B /ANN pdfmark end 367 963 a Fi(\){\()482 963 y SDict begin H.S end 482 963 a Fi(8)529 904 y SDict begin H.R end 529 904 a 529 963 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.8) cvn H.B /ANN pdfmark end 529 963 a Fi(\))h(hold.)36 b(This)21 b(reform)m(ulation)f(can)i(b)s(e)g(found)e(in)h(the)h(b)s(o)s (ok)f(b)m(y)h(San)m(tilli)d([)3004 963 y SDict begin H.S end 3004 963 a Fi(6)3049 904 y SDict begin H.R end 3049 904 a 3049 963 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Sant) cvn H.B /ANN pdfmark end 3049 963 a Fi(],)24 b(for)e(example.)118 1119 y SDict begin H.S end 118 1119 a 118 1119 a SDict begin 13.6 H.A end 118 1119 a 118 1119 a SDict begin [/View [/XYZ H.V]/Dest (section.2) cvn /DEST pdfmark end 118 1119 a 160 x Fj(2)135 b(The)44 b(generalized)j(Helmholtz)f(conditions) 118 1515 y Fi(W)-8 b(e)42 b(next)f(turn)e(to)i(the)g(analysis)e(of)i (the)g(generalized)f(Helmholtz)g(conditions.)70 b(F)-8 b(ollo)m(wing)40 b(the)118 1628 y(notation)31 b(of)f([)610 1628 y SDict begin H.S end 610 1628 a Fi(3)656 1569 y SDict begin H.R end 656 1569 a 656 1628 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 656 1628 a Fi(])h(w)m(e)f(set)1108 1858 y Fh(r)1149 1873 y Fg(ab)1247 1858 y Fi(=)1352 1796 y Fh(@)5 b(f)1450 1810 y Fg(a)p 1352 1837 140 4 v 1356 1920 a Fh(@)g(q)1453 1894 y Fg(b)1522 1858 y Fe(\000)1626 1796 y Fh(@)g(f)1724 1811 y Fg(b)p 1623 1837 139 4 v 1623 1920 a Fh(@)g(q)1720 1894 y Fg(a)1792 1858 y Fi(+)1893 1822 y Fd(1)p 1893 1837 36 4 v 1893 1889 a(2)1964 1796 y Fh(d)p 1948 1837 81 4 v 1948 1920 a(dt)2053 1729 y Ff(\022)2133 1796 y Fh(@)g(f)2231 1811 y Fg(b)p 2130 1837 139 4 v 2130 1920 a Fh(@)22 b Fi(_)-42 b Fh(q)2227 1894 y Fg(a)2299 1858 y Fe(\000)2400 1796 y Fh(@)5 b(f)2498 1810 y Fg(a)p 2400 1837 140 4 v 2404 1920 a Fh(@)22 b Fi(_)-42 b Fh(q)2501 1894 y Fg(b)2549 1729 y Ff(\023)1107 2109 y Fh(s)1150 2124 y Fg(ab)1247 2109 y Fi(=)1352 2073 y Fd(1)p 1352 2088 36 4 v 1352 2140 a(2)1413 1981 y Ff(\022)1490 2048 y Fh(@)5 b(f)1588 2062 y Fg(a)p 1490 2088 140 4 v 1494 2171 a Fh(@)22 b Fi(_)-42 b Fh(q)1591 2145 y Fg(b)1659 2109 y Fi(+)1764 2048 y Fh(@)5 b(f)1862 2063 y Fg(b)p 1760 2088 139 4 v 1760 2171 a Fh(@)22 b Fi(_)-42 b Fh(q)1857 2145 y Fg(a)1909 1981 y Ff(\023)1996 2109 y Fe(\000)2113 2048 y Fh(d)p 2097 2088 81 4 v 2097 2171 a(dt)2202 1981 y Ff(\022)2282 2048 y Fh(@)5 b(f)2380 2063 y Fg(b)p 2279 2088 139 4 v 2279 2171 a Fh(@)12 b Fi(\177)-52 b Fh(q)2376 2145 y Fg(a)2428 1981 y Ff(\023)2510 2109 y Fh(:)118 2339 y Fi(The)40 b(generalized)g(Helmholtz)g(conditions)f(as)i(giv)m (en)f(in)f([)2197 2339 y SDict begin H.S end 2197 2339 a Fi(3)2243 2280 y SDict begin H.R end 2243 2280 a 2243 2339 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 2243 2339 a Fi(])i(are)f(that)h Fh(r)2718 2354 y Fg(ab)2831 2339 y Fi(and)e Fh(s)3060 2354 y Fg(ab)3172 2339 y Fi(are)i(of)g(\014rst)118 2452 y(order,)30 b(and)g(that)h(in)e(addition)1346 2608 y Fh(@)5 b(f)1444 2622 y Fg(a)p 1346 2648 140 4 v 1350 2732 a Fh(@)12 b Fi(\177)-52 b Fh(q)1447 2705 y Fg(b)1520 2669 y Fi(=)1630 2608 y Fh(@)5 b(f)1728 2623 y Fg(b)p 1626 2648 139 4 v 1626 2732 a Fh(@)12 b Fi(\177)-52 b Fh(q)1723 2705 y Fg(a)3489 2669 y SDict begin H.S end 3489 2669 a 3489 2669 a SDict begin 16.6 H.A end 3489 2669 a 3489 2669 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.9) cvn /DEST pdfmark end 3489 2669 a Fi(\(9\))1318 2846 y Fh(@)5 b(s)1414 2861 y Fg(ab)p 1318 2886 168 4 v 1335 2970 a Fh(@)23 b Fi(_)-43 b Fh(q)1432 2943 y Fg(c)1520 2907 y Fi(=)1626 2846 y Fh(@)5 b(s)1722 2860 y Fg(ac)p 1626 2886 169 4 v 1645 2970 a Fh(@)22 b Fi(_)-42 b Fh(q)1742 2943 y Fg(b)3443 2907 y SDict begin H.S end 3443 2907 a 3443 2907 a SDict begin 16.6 H.A end 3443 2907 a 3443 2907 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.10) cvn /DEST pdfmark end 3443 2907 a Fi(\(10\))1319 3084 y Fh(@)5 b(r)1413 3099 y Fg(ab)p 1319 3124 167 4 v 1336 3207 a Fh(@)22 b Fi(_)-42 b Fh(q)1433 3181 y Fg(c)1520 3145 y Fi(=)1626 3084 y Fh(@)5 b(s)1722 3098 y Fg(ac)p 1626 3124 169 4 v 1645 3208 a Fh(@)g(q)1742 3181 y Fg(b)1825 3145 y Fe(\000)1926 3084 y Fh(@)g(s)2022 3099 y Fg(bc)p 1926 3124 161 4 v 1937 3207 a Fh(@)g(q)2034 3181 y Fg(a)3443 3145 y SDict begin H.S end 3443 3145 a 3443 3145 a SDict begin 16.6 H.A end 3443 3145 a 3443 3145 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.11) cvn /DEST pdfmark end 3443 3145 a Fi(\(11\))1450 3383 y(0)25 b(=)1626 3322 y Fh(@)5 b(r)1720 3337 y Fg(ab)p 1626 3362 167 4 v 1644 3445 a Fh(@)g(q)1741 3419 y Fg(c)1823 3383 y Fi(+)1924 3322 y Fh(@)g(r)2018 3337 y Fg(bc)p 1924 3362 160 4 v 1934 3445 a Fh(@)g(q)2031 3419 y Fg(a)2113 3383 y Fi(+)2214 3322 y Fh(@)g(r)2308 3336 y Fg(ca)p 2214 3362 167 4 v 2231 3445 a Fh(@)g(q)2328 3419 y Fg(b)2390 3383 y Fh(:)3443 3383 y SDict begin H.S end 3443 3383 a 3443 3383 a SDict begin 16.6 H.A end 3443 3383 a 3443 3383 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.12) cvn /DEST pdfmark end 3443 3383 a Fi(\(12\))118 3607 y(Our)19 b(main)h(concern)h(will)c(b)s (e)j(with)g(analyzing)f(conditions)g(\()2166 3607 y SDict begin H.S end 2166 3607 a Fi(10)2258 3548 y SDict begin H.R end 2258 3548 a 2258 3607 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 2258 3607 a Fi(\){\()2373 3607 y SDict begin H.S end 2373 3607 a Fi(12)2465 3548 y SDict begin H.R end 2465 3548 a 2465 3607 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 2465 3607 a Fi(\),)k(whic)m(h)c(corresp)s (ond)g(to)j(\(2.3e\),)118 3719 y(\(2.3f)7 b(\))29 b(and)d(\(2.3g\))j (of)e([)924 3719 y SDict begin H.S end 924 3719 a Fi(3)969 3661 y SDict begin H.R end 969 3661 a 969 3719 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 969 3719 a Fi(];)i(w)m(e)e(shall)e(sho)m(w)h(that)i (conditions)d(\()2268 3719 y SDict begin H.S end 2268 3719 a Fi(10)2359 3661 y SDict begin H.R end 2359 3661 a 2359 3719 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 2359 3719 a Fi(\))i(and)f(\()2629 3719 y SDict begin H.S end 2629 3719 a Fi(12)2721 3661 y SDict begin H.R end 2721 3661 a 2721 3719 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 2721 3719 a Fi(\))h(are)g(redundan)m(t,)f(b)s(eing)118 3832 y(consequences)31 b(of)g(the)f(remaining)f(conditions.)118 3978 y(Our)f(\014rst)h(aim)g(is)g(to)h(understand)e(exactly)i(what)f (it)h(means)f(for)g Fh(r)2428 3993 y Fg(ab)2530 3978 y Fi(and)g Fh(s)2749 3993 y Fg(ab)2849 3978 y Fi(to)i(b)s(e)e(of)g (\014rst)g(order,)118 4091 y(b)s(earing)j(in)h(mind)e(condition)h(\() 1229 4091 y SDict begin H.S end 1229 4091 a Fi(9)1275 4033 y SDict begin H.R end 1275 4033 a 1275 4091 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.9) cvn H.B /ANN pdfmark end 1275 4091 a Fi(\))i(ab)s(o)m(v)m(e.)51 b(F)-8 b(rom)34 b(the)g(v)-5 b(anishing)31 b(of)j(the)g(co)s(e\016cien)m(t)g (of)3256 4040 y(.)-8 b(.)g(.)3262 4091 y Fh(q)3313 4047 y Fg(c)3381 4091 y Fi(in)33 b Fh(s)3534 4106 y Fg(ab)118 4204 y Fi(w)m(e)e(ha)m(v)m(e)1676 4288 y Fh(@)1729 4255 y Fd(2)1769 4288 y Fh(f)1814 4302 y Fg(a)p 1634 4329 264 4 v 1634 4412 a Fh(@)12 b Fi(\177)-52 b Fh(q)1731 4386 y Fg(b)1765 4412 y Fh(@)12 b Fi(\177)-52 b Fh(q)1862 4386 y Fg(c)1933 4350 y Fi(=)24 b(0)p Fh(:)118 4559 y Fi(As)34 b(b)s(efore)g Fh(f)578 4573 y Fg(a)651 4559 y Fi(=)d Fh(g)796 4574 y Fg(ab)875 4559 y Fi(\177)-52 b Fh(q)912 4526 y Fg(b)969 4559 y Fi(+)22 b Fh(h)1114 4573 y Fg(a)1156 4559 y Fi(,)35 b(the)g(co)s(e\016cien)m(ts)g(b)s(eing) d(of)j(\014rst)e(order)g(and)h Fh(g)2853 4574 y Fg(ab)2959 4559 y Fi(symmetric.)51 b(The)118 4672 y(co)s(e\016cien)m(t)31 b(of)648 4620 y(.)-8 b(.)g(.)654 4672 y Fh(q)705 4628 y Fg(c)770 4672 y Fi(in)29 b Fh(r)917 4687 y Fg(ab)1019 4672 y Fi(is)1068 4861 y Fd(1)p 1068 4876 36 4 v 1068 4928 a(2)1162 4835 y Fh(@)p 1123 4876 132 4 v 1123 4959 a(@)12 b Fi(\177)-52 b Fh(q)1220 4933 y Fg(c)1280 4768 y Ff(\022)1357 4835 y Fh(@)5 b(g)1453 4850 y Fg(bd)p 1357 4876 168 4 v 1371 4959 a Fh(@)22 b Fi(_)-42 b Fh(q)1468 4933 y Fg(a)1541 4896 y Fi(\177)-52 b Fh(q)1578 4859 y Fg(d)1639 4896 y Fi(+)1740 4835 y Fh(@)5 b(h)1845 4850 y Fg(b)p 1740 4876 141 4 v 1741 4959 a Fh(@)21 b Fi(_)-41 b Fh(q)1838 4933 y Fg(a)1910 4896 y Fe(\000)2011 4835 y Fh(@)5 b(g)2107 4850 y Fg(ad)p 2011 4876 175 4 v 2032 4959 a Fh(@)22 b Fi(_)-42 b Fh(q)2129 4933 y Fg(b)2202 4896 y Fi(\177)-52 b Fh(q)2239 4859 y Fg(d)2300 4896 y Fe(\000)2401 4835 y Fh(@)5 b(h)2506 4849 y Fg(a)p 2401 4876 148 4 v 2409 4959 a Fh(@)22 b Fi(_)-42 b Fh(q)2506 4933 y Fg(b)2558 4768 y Ff(\023)2640 4896 y Fh(;)118 5121 y Fi(whence)1613 5193 y Fh(@)5 b(g)1709 5208 y Fg(bc)p 1613 5233 162 4 v 1625 5316 a Fh(@)22 b Fi(_)-42 b Fh(q)1722 5290 y Fg(a)1810 5254 y Fi(=)1916 5193 y Fh(@)5 b(g)2012 5207 y Fg(ac)p 1916 5233 169 4 v 1935 5316 a Fh(@)22 b Fi(_)-42 b Fh(q)2032 5290 y Fg(b)2095 5254 y Fh(:)1839 5538 y Fi(3)p eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 448 -8 a SDict begin H.S end 448 -8 a 448 -8 a SDict begin H.R end 448 -8 a 448 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end 448 -8 a 118 399 a Fi(The)30 b(co)s(e\016cien)m(t)h(of)38 b(\177)-52 b Fh(q)877 366 y Fg(c)942 399 y Fi(in)29 b Fh(s)1091 414 y Fg(ab)1162 399 y Fi(,)i(namely)1446 579 y Fh(@)5 b(g)1542 593 y Fg(ac)p 1446 620 169 4 v 1464 703 a Fh(@)23 b Fi(_)-43 b Fh(q)1561 677 y Fg(b)1645 640 y Fi(+)1746 579 y Fh(@)5 b(g)1842 594 y Fg(bc)p 1746 620 162 4 v 1757 703 a Fh(@)22 b Fi(_)-42 b Fh(q)1854 677 y Fg(a)1937 640 y Fe(\000)20 b Fi(2)2083 579 y Fh(@)5 b(g)2179 594 y Fg(ab)p 2083 620 169 4 v 2102 703 a Fh(@)22 b Fi(_)-42 b Fh(q)2199 677 y Fg(c)2262 640 y Fh(;)118 883 y Fi(v)-5 b(anishes)29 b(as)i(a)g(consequence.)41 b(The)30 b(co)s(e\016cien)m(t)h(of)37 b(\177)-51 b Fh(q)1970 850 y Fg(c)2035 883 y Fi(in)29 b Fh(r)2182 898 y Fg(ab)2284 883 y Fi(is)1110 1076 y Fh(@)5 b(g)1206 1090 y Fg(ac)p 1110 1117 V 1129 1200 a Fh(@)g(q)1226 1174 y Fg(b)1310 1138 y Fe(\000)1410 1076 y Fh(@)g(g)1506 1091 y Fg(bc)p 1410 1117 162 4 v 1422 1200 a Fh(@)g(q)1519 1174 y Fg(a)1602 1138 y Fe(\000)1703 1102 y Fd(1)p 1703 1117 36 4 v 1703 1169 a(2)1763 1010 y Ff(\022)1879 1076 y Fh(@)1932 1043 y Fd(2)1971 1076 y Fh(h)2023 1090 y Fg(a)p 1840 1117 264 4 v 1840 1200 a Fh(@)22 b Fi(_)-42 b Fh(q)1937 1174 y Fg(b)1972 1200 y Fh(@)22 b Fi(_)-42 b Fh(q)2069 1174 y Fg(c)2134 1138 y Fe(\000)2280 1076 y Fh(@)2333 1043 y Fd(2)2373 1076 y Fh(h)2425 1091 y Fg(b)p 2235 1117 271 4 v 2235 1200 a Fh(@)22 b Fi(_)-42 b Fh(q)2332 1174 y Fg(a)2373 1200 y Fh(@)23 b Fi(_)-43 b Fh(q)2470 1174 y Fg(c)2515 1010 y Ff(\023)2597 1138 y Fh(;)118 1387 y Fi(an)45 b(expression)e(whic)m(h) h(for)h(later)g(con)m(v)m(enience)g(w)m(e)h(write)e(as)h Fh(\032)2447 1402 y Fg(abc)2549 1387 y Fi(;)53 b(w)m(e)45 b(m)m(ust)g(of)g(course)g(ha)m(v)m(e)118 1500 y Fh(\032)165 1515 y Fg(abc)293 1500 y Fi(=)25 b(0.)41 b(The)30 b(remaining)e(terms)i (in)g Fh(r)1512 1515 y Fg(ab)1614 1500 y Fi(and)f Fh(s)1833 1515 y Fg(ab)1935 1500 y Fi(are)i(all)e(of)i(\014rst)e(order,)i(and)e (w)m(e)i(ha)m(v)m(e)1099 1754 y Fh(r)1140 1769 y Fg(ab)1237 1754 y Fi(=)1343 1693 y Fh(@)5 b(h)1448 1707 y Fg(a)p 1343 1733 148 4 v 1351 1817 a Fh(@)g(q)1448 1790 y Fg(b)1521 1754 y Fe(\000)1622 1693 y Fh(@)g(h)1727 1708 y Fg(b)p 1622 1733 141 4 v 1623 1817 a Fh(@)g(q)1720 1790 y Fg(a)1792 1754 y Fe(\000)1893 1719 y Fd(1)p 1893 1734 36 4 v 1893 1786 a(2)1980 1669 y Fi(\026)1964 1693 y Fh(d)p 1948 1733 81 4 v 1948 1817 a(dt)2053 1626 y Ff(\022)2130 1693 y Fh(@)g(h)2235 1707 y Fg(a)p 2130 1733 148 4 v 2138 1817 a Fh(@)22 b Fi(_)-42 b Fh(q)2235 1790 y Fg(b)2308 1754 y Fe(\000)2408 1693 y Fh(@)5 b(h)2513 1708 y Fg(b)p 2408 1733 141 4 v 2409 1817 a Fh(@)22 b Fi(_)-42 b Fh(q)2506 1790 y Fg(a)2558 1626 y Ff(\023)1098 2011 y Fh(s)1141 2026 y Fg(ab)1237 2011 y Fi(=)1343 1975 y Fd(1)p 1343 1990 36 4 v 1343 2042 a(2)1404 1883 y Ff(\022)1481 1949 y Fh(@)5 b(h)1586 1963 y Fg(a)p 1481 1990 148 4 v 1489 2073 a Fh(@)22 b Fi(_)-42 b Fh(q)1586 2047 y Fg(b)1658 2011 y Fi(+)1759 1949 y Fh(@)5 b(h)1864 1964 y Fg(b)p 1759 1990 141 4 v 1760 2073 a Fh(@)22 b Fi(_)-42 b Fh(q)1857 2047 y Fg(a)1909 1883 y Ff(\023)1996 2011 y Fe(\000)2129 1925 y Fi(\026)2113 1949 y Fh(d)p 2097 1990 81 4 v 2097 2073 a(dt)2187 2011 y Fi(\()p Fh(g)2265 2026 y Fg(ab)2338 2011 y Fi(\);)118 2265 y(compare)31 b(with)e(\()724 2265 y SDict begin H.S end 724 2265 a Fi(7)770 2207 y SDict begin H.R end 770 2207 a 770 2265 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.7) cvn H.B /ANN pdfmark end 770 2265 a Fi(\))i(and)e(\()1047 2265 y SDict begin H.S end 1047 2265 a Fi(8)1093 2207 y SDict begin H.R end 1093 2207 a 1093 2265 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.8) cvn H.B /ANN pdfmark end 1093 2265 a Fi(\),)i(and)f(also)g(with)f (equations)h(\(2.16b\))i(and)e(\(2.17c\))j(of)e([)3079 2265 y SDict begin H.S end 3079 2265 a Fi(3)3124 2207 y SDict begin H.R end 3124 2207 a 3124 2265 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 3124 2265 a Fi(].)118 2411 y(The)j(redundancy)e(of)i(condition)f (\()1350 2411 y SDict begin H.S end 1350 2411 a Fi(10)1442 2353 y SDict begin H.R end 1442 2353 a 1442 2411 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 1442 2411 a Fi(\))h(is)f(a)i(consequence)g(of)f(the)g(v) -5 b(anishing)32 b(of)i Fh(\032)3030 2426 y Fg(abc)3133 2411 y Fi(,)h(as)f(w)m(e)h(no)m(w)118 2524 y(sho)m(w.)41 b(W)-8 b(e)31 b(ha)m(v)m(e)835 2700 y Fh(@)5 b(s)931 2714 y Fg(ac)p 835 2740 169 4 v 854 2823 a Fh(@)22 b Fi(_)-42 b Fh(q)951 2797 y Fg(b)1039 2761 y Fi(=)1145 2725 y Fd(1)p 1145 2740 36 4 v 1145 2792 a(2)1205 2633 y Ff(\022)1320 2700 y Fh(@)1373 2667 y Fd(2)1413 2700 y Fh(h)1465 2714 y Fg(a)p 1282 2740 264 4 v 1282 2823 a Fh(@)22 b Fi(_)-42 b Fh(q)1379 2797 y Fg(b)1413 2823 y Fh(@)23 b Fi(_)-43 b Fh(q)1510 2797 y Fg(c)1576 2761 y Fi(+)1722 2700 y Fh(@)1775 2667 y Fd(2)1814 2700 y Fh(h)1866 2714 y Fg(c)p 1676 2740 271 4 v 1676 2823 a Fh(@)23 b Fi(_)-43 b Fh(q)1773 2797 y Fg(a)1815 2823 y Fh(@)22 b Fi(_)-42 b Fh(q)1912 2797 y Fg(b)1957 2633 y Ff(\023)2044 2761 y Fe(\000)2177 2676 y Fi(\026)2161 2700 y Fh(d)p 2145 2740 81 4 v 2145 2823 a(dt)2250 2633 y Ff(\022)2327 2700 y Fh(@)5 b(g)2423 2714 y Fg(ac)p 2327 2740 169 4 v 2345 2823 a Fh(@)23 b Fi(_)-43 b Fh(q)2442 2797 y Fg(b)2506 2633 y Ff(\023)2593 2761 y Fe(\000)2694 2700 y Fh(@)5 b(g)2790 2714 y Fg(ac)p 2694 2740 V 2712 2823 a Fh(@)g(q)2809 2797 y Fg(b)2872 2761 y Fh(;)118 3010 y Fi(using)29 b(the)i(comm)m(utation)g(relation)1504 3265 y SDict begin H.S end 1504 3265 a 1504 3265 a SDict begin 13.6 H.A end 1504 3265 a 1504 3265 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.13) cvn /DEST pdfmark end 1504 3265 a 1519 3136 a Ff(\024)1620 3203 y Fh(@)p 1577 3244 139 4 v 1577 3327 a(@)22 b Fi(_)-42 b Fh(q)1674 3301 y Fg(a)1726 3265 y Fh(;)1809 3179 y Fi(\026)1792 3203 y Fh(d)p 1776 3244 81 4 v 1776 3327 a(dt)1866 3136 y Ff(\025)1939 3265 y Fi(=)2088 3203 y Fh(@)p 2045 3244 139 4 v 2045 3327 a(@)5 b(q)2142 3301 y Fg(a)2194 3265 y Fh(:)1224 b Fi(\(13\))118 3530 y(It)36 b(follo)m(ws)f(that)i Fh(@)5 b(s)825 3544 y Fg(ac)897 3530 y Fh(=@)23 b Fi(_)-43 b Fh(q)1039 3497 y Fg(b)1098 3530 y Fe(\000)24 b Fh(@)5 b(s)1289 3545 y Fg(bc)1354 3530 y Fh(=@)22 b Fi(_)-42 b Fh(q)1496 3497 y Fg(a)1573 3530 y Fi(=)34 b Fe(\000)p Fh(\032)1796 3545 y Fg(abc)1933 3530 y Fi(=)g(0.)58 b(That)36 b(is)g(to)g(sa)m(y)-8 b(,)39 b(condition)c(\()3235 3530 y SDict begin H.S end 3235 3530 a Fi(10)3326 3471 y SDict begin H.R end 3326 3471 a 3326 3530 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 3326 3530 a Fi(\))h(holds)118 3643 y(automatically)41 b(as)h(a)f(consequence)h(of)f(the)h(\014rst-order)e(prop)s(ert)m(y)-8 b(.)73 b(F)-8 b(urthermore,)44 b Fh(\032)3198 3658 y Fg(abc)3343 3643 y Fi(=)f(0)f(is)118 3756 y(equiv)-5 b(alen)m(t)31 b(to)i(equation)f(\(2.17b\))i(of)e([)1482 3756 y SDict begin H.S end 1482 3756 a Fi(3)1527 3697 y SDict begin H.R end 1527 3697 a 1527 3756 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 1527 3756 a Fi(];)i(in)c(other)i(w)m(ords,)g(the)h(redundancy)d (of)i(\()3038 3756 y SDict begin H.S end 3038 3756 a Fi(10)3129 3697 y SDict begin H.R end 3129 3697 a 3129 3756 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 3129 3756 a Fi(\))g(is)f(actually)118 3868 y(implicit)c(in)j([)583 3868 y SDict begin H.S end 583 3868 a Fi(3)628 3810 y SDict begin H.R end 628 3810 a 628 3868 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 628 3868 a Fi(],)h(though)f(not)h(apparen) m(tly)f(recognized)g(there.)118 4015 y(Before)f(pro)s(ceeding)e(to)i (consider)f(condition)e(\()1748 4015 y SDict begin H.S end 1748 4015 a Fi(12)1840 3956 y SDict begin H.R end 1840 3956 a 1840 4015 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 1840 4015 a Fi(\))i(w)m(e)h(turn)e (aside)h(to)h(mak)m(e)g(some)f(remarks)g(ab)s(out)118 4127 y(the)33 b(classical)f(Helmholtz)h(conditions.)47 b(The)32 b(calculations)g(just)g(carried)g(out)i(are)f(essen)m(tially)f (the)118 4240 y(same)37 b(as)g(those)g(whic)m(h)e(lead)h(to)i(the)e(v)m (ersion)g(of)h(the)g(classical)f(Helmholtz)g(conditions)f(giv)m(en)h (in)118 4353 y(equations)42 b(\()572 4353 y SDict begin H.S end 572 4353 a Fi(6)617 4295 y SDict begin H.R end 617 4295 a 617 4353 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.6) cvn H.B /ANN pdfmark end 617 4353 a Fi(\){\()732 4353 y SDict begin H.S end 732 4353 a Fi(8)779 4295 y SDict begin H.R end 779 4295 a 779 4353 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.8) cvn H.B /ANN pdfmark end 779 4353 a Fi(\))g(at)h(the)f(end)f(of)h(the)g(in)m(tro)s(duction.)73 b(It)42 b(is)f(easy)i(to)f(see)h(that)f(in)e(that)j(case)118 4466 y Fh(r)159 4481 y Fg(ab)260 4466 y Fi(=)30 b Fh(s)404 4481 y Fg(ab)505 4466 y Fi(=)f(0)k(are)h(necessary)f(conditions.)47 b(This)31 b(observ)-5 b(ation,)34 b(together)g(with)e(the)h(part)g(of)g (the)118 4579 y(argumen)m(t)g(concerning)f(the)g(v)-5 b(anishing)30 b(of)j(the)f(co)s(e\016cien)m(ts)h(of)2380 4527 y(.)-8 b(.)g(.)2386 4579 y Fh(q)42 b Fi(and)d(\177)-52 b Fh(q)s Fi(,)33 b(leads)f(to)h(the)f(follo)m(wing)1839 5538 y(4)p eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 110 -8 a SDict begin H.S end 110 -8 a 110 -8 a SDict begin H.R end 110 -8 a 110 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.5) cvn /DEST pdfmark end 110 -8 a 118 399 a Fi(conditions:)1308 555 y Fh(@)5 b(g)1404 569 y Fg(ac)p 1308 596 169 4 v 1327 679 a Fh(@)22 b Fi(_)-42 b Fh(q)1424 653 y Fg(b)1507 617 y Fi(+)1608 555 y Fh(@)5 b(g)1704 570 y Fg(bc)p 1608 596 162 4 v 1619 679 a Fh(@)23 b Fi(_)-43 b Fh(q)1716 653 y Fg(a)1805 617 y Fi(=)25 b(2)1956 555 y Fh(@)5 b(g)2052 570 y Fg(ab)p 1956 596 169 4 v 1975 679 a Fh(@)22 b Fi(_)-42 b Fh(q)2072 653 y Fg(c)3443 617 y SDict begin H.S end 3443 617 a 3443 617 a SDict begin 16.6 H.A end 3443 617 a 3443 617 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.14) cvn /DEST pdfmark end 3443 617 a Fi(\(14\))1351 806 y Fh(@)5 b(h)1456 820 y Fg(a)p 1351 846 148 4 v 1359 929 a Fh(@)22 b Fi(_)-42 b Fh(q)1456 903 y Fg(b)1529 867 y Fi(+)1630 806 y Fh(@)5 b(h)1735 821 y Fg(b)p 1630 846 141 4 v 1631 929 a Fh(@)21 b Fi(_)-41 b Fh(q)1728 903 y Fg(a)1805 867 y Fi(=)25 b(2)1989 782 y(\026)1973 806 y Fh(d)p 1956 846 81 4 v 1956 929 a(dt)2046 867 y Fi(\()p Fh(g)2124 882 y Fg(ab)2197 867 y Fi(\))3443 867 y SDict begin H.S end 3443 867 a 3443 867 a SDict begin 16.6 H.A end 3443 867 a 3443 867 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.15) cvn /DEST pdfmark end 3443 867 a Fi(\(15\))1301 1043 y Fh(@)5 b(g)1397 1058 y Fg(ab)p 1301 1084 169 4 v 1320 1167 a Fh(@)22 b Fi(_)-42 b Fh(q)1417 1141 y Fg(c)1500 1105 y Fe(\000)1601 1043 y Fh(@)5 b(g)1697 1057 y Fg(ac)p 1601 1084 V 1619 1167 a Fh(@)23 b Fi(_)-43 b Fh(q)1716 1141 y Fg(b)1805 1105 y Fi(=)25 b(0)3443 1105 y SDict begin H.S end 3443 1105 a 3443 1105 a SDict begin 16.6 H.A end 3443 1105 a 3443 1105 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.16) cvn /DEST pdfmark end 3443 1105 a Fi(\(16\))1143 1294 y Fh(@)1196 1261 y Fd(2)1236 1294 y Fh(h)1288 1308 y Fg(a)p 1105 1335 264 4 v 1105 1418 a Fh(@)d Fi(_)-42 b Fh(q)1202 1392 y Fg(b)1236 1418 y Fh(@)22 b Fi(_)-42 b Fh(q)1333 1392 y Fg(c)1398 1356 y Fe(\000)1545 1294 y Fh(@)1598 1261 y Fd(2)1637 1294 y Fh(h)1689 1309 y Fg(b)p 1499 1335 271 4 v 1499 1418 a Fh(@)22 b Fi(_)-42 b Fh(q)1596 1392 y Fg(a)1638 1418 y Fh(@)22 b Fi(_)-42 b Fh(q)1735 1392 y Fg(c)1805 1356 y Fi(=)25 b(2)1961 1227 y Ff(\022)2038 1294 y Fh(@)5 b(g)2134 1308 y Fg(ac)p 2038 1335 169 4 v 2057 1418 a Fh(@)g(q)2154 1392 y Fg(b)2237 1356 y Fe(\000)2338 1294 y Fh(@)g(g)2434 1309 y Fg(bc)p 2338 1335 162 4 v 2350 1418 a Fh(@)g(q)2447 1392 y Fg(a)2510 1227 y Ff(\023)3443 1356 y SDict begin H.S end 3443 1356 a 3443 1356 a SDict begin 16.6 H.A end 3443 1356 a 3443 1356 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.17) cvn /DEST pdfmark end 3443 1356 a Fi(\(17\))1147 1612 y(2)1207 1484 y Ff(\022)1284 1551 y Fh(@)g(h)1389 1565 y Fg(a)p 1284 1591 148 4 v 1292 1675 a Fh(@)g(q)1389 1648 y Fg(b)1462 1612 y Fe(\000)1563 1551 y Fh(@)g(h)1668 1566 y Fg(b)p 1563 1591 141 4 v 1564 1675 a Fh(@)g(q)1661 1648 y Fg(a)1713 1484 y Ff(\023)1805 1612 y Fi(=)1943 1527 y(\026)1927 1551 y Fh(d)p 1911 1591 81 4 v 1911 1675 a(dt)2016 1484 y Ff(\022)2093 1551 y Fh(@)g(h)2198 1565 y Fg(a)p 2093 1591 148 4 v 2101 1675 a Fh(@)22 b Fi(_)-42 b Fh(q)2198 1648 y Fg(b)2270 1612 y Fe(\000)2371 1551 y Fh(@)5 b(h)2476 1566 y Fg(b)p 2371 1591 141 4 v 2372 1675 a Fh(@)22 b Fi(_)-42 b Fh(q)2469 1648 y Fg(a)2521 1484 y Ff(\023)2603 1612 y Fh(:)3443 1612 y SDict begin H.S end 3443 1612 a 3443 1612 a SDict begin 16.6 H.A end 3443 1612 a 3443 1612 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.18) cvn /DEST pdfmark end 3443 1612 a Fi(\(18\))118 1860 y(These)39 b(are)h(the)f(conditions)f (quoted)h(in)f(Remark)h(3)g(of)h(Section)f(1)g(of)g([)2680 1860 y SDict begin H.S end 2680 1860 a Fi(3)2726 1802 y SDict begin H.R end 2726 1802 a 2726 1860 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 2726 1860 a Fi(].)67 b(Ho)m(w)m(ev)m(er,)44 b(it)39 b(is)f(no)m(w)118 1973 y(eviden)m(t)31 b(that)i(t)m(w)m(o)g(of)e(them)h (are)g(redundan)m(t.)43 b(Clearly)30 b(condition)g(\()2528 1973 y SDict begin H.S end 2528 1973 a Fi(14)2619 1915 y SDict begin H.R end 2619 1915 a 2619 1973 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.14) cvn H.B /ANN pdfmark end 2619 1973 a Fi(\))i(\(whic)m(h)f(is)g(the)g(v)-5 b(anishing)118 2086 y(of)44 b(the)g(co)s(e\016cien)m(t)g(of)50 b(\177)-51 b Fh(q)1003 2053 y Fg(c)1081 2086 y Fi(in)42 b Fh(s)1243 2101 y Fg(ab)1314 2086 y Fi(\))i(follo)m(ws)f(from)g (condition)f(\()2380 2086 y SDict begin H.S end 2380 2086 a Fi(16)2471 2028 y SDict begin H.R end 2471 2028 a 2471 2086 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.16) cvn H.B /ANN pdfmark end 2471 2086 a Fi(\))i(and)f(the)h(symmetry)f (of)h Fh(g)3508 2101 y Fg(ab)3580 2086 y Fi(.)118 2199 y(Condition)e(\()591 2199 y SDict begin H.S end 591 2199 a Fi(17)683 2140 y SDict begin H.R end 683 2140 a 683 2199 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.17) cvn H.B /ANN pdfmark end 683 2199 a Fi(\))j(is)e(the)i(condition)e Fh(\032)1499 2214 y Fg(abc)1650 2199 y Fi(=)49 b(0.)83 b(The)44 b(second)h(part)f(of)h(the)g(argumen)m(t)g(ab)s(o)m(v)m(e,)118 2312 y(that)32 b(leading)e(to)i(the)g(relation)e Fh(\032)1281 2327 y Fg(abc)1411 2312 y Fi(=)c Fh(@)5 b(s)1604 2327 y Fg(bc)1669 2312 y Fh(=@)23 b Fi(_)-43 b Fh(q)1811 2279 y Fg(a)1874 2312 y Fe(\000)21 b Fh(@)5 b(s)2062 2326 y Fg(ac)2134 2312 y Fh(=@)23 b Fi(_)-43 b Fh(q)2276 2279 y Fg(b)2311 2312 y Fi(,)32 b(sho)m(ws)f(that)h(in)e(the)h(classical)g (case)118 2425 y(condition)f(\()553 2425 y SDict begin H.S end 553 2425 a Fi(17)644 2366 y SDict begin H.R end 644 2366 a 644 2425 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.17) cvn H.B /ANN pdfmark end 644 2425 a Fi(\))h(follo)m(ws)f (from)h(the)g(other)g(conditions.)41 b(When)30 b(these)i(t)m(w)m(o)g (redundan)m(t)d(conditions)118 2538 y(are)35 b(remo)m(v)m(ed)g(w)m(e)g (obtain)f(the)g(classical)g(Helmholtz)g(conditions)e(in)h(the)i(form)f (giv)m(en)g(at)h(the)g(end)118 2651 y(of)c(the)f(in)m(tro)s(duction.) 118 2797 y(These)j(results)g(in)f(the)h(classical)g(case)h(are)g (actually)f(v)m(ery)h(w)m(ell)f(kno)m(wn,)h(though)f(not)h(apparen)m (tly)118 2910 y(to)k(the)f(authors)g(of)g([)870 2910 y SDict begin H.S end 870 2910 a Fi(3)915 2851 y SDict begin H.R end 915 2851 a 915 2910 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 915 2910 a Fi(],)i(and)e(ha)m(v)m(e)h(b)s(een)e(kno)m(wn)h(for)f(a)i(long)e (time:)h(they)g(are)h(to)f(b)s(e)g(found,)g(for)118 3023 y(example,)d(in)d(San)m(tilli's)g(b)s(o)s(ok)h(of)h(1978)i([)1559 3023 y SDict begin H.S end 1559 3023 a Fi(6)1605 2964 y SDict begin H.R end 1605 2964 a 1605 3023 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Sant) cvn H.B /ANN pdfmark end 1605 3023 a Fi(])e(\(whic)m(h)f(is)g(in)g(fact)i(referred)e(to)i (in)e([)2933 3023 y SDict begin H.S end 2933 3023 a Fi(3)2979 2964 y SDict begin H.R end 2979 2964 a 2979 3023 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 2979 3023 a Fi(]\).)49 b(F)-8 b(or)34 b(the)f(sak)m(e)118 3135 y(of)k(clarit)m(y)f(w)m(e)g(should)f(p)s(oin)m (t)g(out)i(a)f(di\013erence)g(b)s(et)m(w)m(een)h(the)g(t)m(w)m(o)g (cases:)h(in)d(the)h(classical)g(case)118 3248 y(condition)30 b(\()553 3248 y SDict begin H.S end 553 3248 a Fi(17)645 3190 y SDict begin H.R end 645 3190 a 645 3248 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.17) cvn H.B /ANN pdfmark end 645 3248 a Fi(\))i(is)e(completely)i(redundan)m(t;)f(in)f (the)i(generalized)f(case)i(it)e(is)g(not)h(redundan)m(t,)f(but)118 3361 y(o)s(ccurs)e(t)m(wice)g(in)f(the)h(form)m(ulation)f(of)h(the)h (conditions)d(in)h([)2201 3361 y SDict begin H.S end 2201 3361 a Fi(3)2247 3303 y SDict begin H.R end 2247 3303 a 2247 3361 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 2247 3361 a Fi(],)i(once)f(in)f(the)h (requiremen)m(t)g(that)g Fh(r)3533 3376 y Fg(ab)118 3474 y Fi(should)f(b)s(e)i(of)h(\014rst)e(order)h(and)g(once)h(as)g(the)f (condition)f Fh(@)5 b(s)2206 3489 y Fg(ab)2278 3474 y Fh(=@)23 b Fi(_)-43 b Fh(q)2420 3441 y Fg(c)2481 3474 y Fi(=)25 b Fh(@)5 b(s)2673 3489 y Fg(bc)2737 3474 y Fh(=@)23 b Fi(_)-43 b Fh(q)2879 3441 y Fg(a)2922 3474 y Fi(.)118 3620 y(W)-8 b(e)29 b(no)m(w)g(return)e(to)i(the)f (generalized)g(conditions,)f(and)h(pro)m(v)m(e)h(that)g(condition)d(\() 2969 3620 y SDict begin H.S end 2969 3620 a Fi(12)3061 3562 y SDict begin H.R end 3061 3562 a 3061 3620 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 3061 3620 a Fi(\))i(follo)m(ws)f(from)118 3733 y(condition)i(\()552 3733 y SDict begin H.S end 552 3733 a Fi(11)643 3675 y SDict begin H.R end 643 3675 a 643 3733 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.11) cvn H.B /ANN pdfmark end 643 3733 a Fi(\).)41 b(It)31 b(will)d(b)s(e)h(con)m(v)m(enien)m(t)j(to)f(write)e(condition)g(\()2365 3733 y SDict begin H.S end 2365 3733 a Fi(12)2457 3675 y SDict begin H.R end 2457 3675 a 2457 3733 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 2457 3733 a Fi(\))h(as)1599 3888 y Ff(X)1596 4085 y Fg(a;b;c)1759 3913 y Fh(@)5 b(r)1853 3928 y Fg(ab)p 1759 3953 167 4 v 1776 4036 a Fh(@)g(q)1873 4010 y Fg(c)1960 3974 y Fi(=)25 b(0)p Fh(;)118 4269 y Fi(where)378 4201 y Ff(P)474 4296 y Fg(a;b;c)644 4269 y Fi(stands)i(for)h(the)g(cyclic)f (sum)g(o)m(v)m(er)j Fh(a)p Fi(,)e Fh(b)g Fi(and)f Fh(c)p Fi(,)i(here)f(and)f(b)s(elo)m(w.)39 b(As)28 b(a)g(preliminary)118 4382 y(remark,)e(note)g(that)f(if)f Fh(k)963 4397 y Fg(abc)1091 4382 y Fi(is)g(symmetric)g(in)g Fh(b)h Fi(and)f Fh(c)h Fi(\(sa)m(y\))i(then)2438 4314 y Ff(P)2534 4409 y Fg(a;b;c)2691 4382 y Fh(k)2738 4397 y Fg(abc)2866 4382 y Fi(=)2962 4314 y Ff(P)3058 4409 y Fg(a;b;c)3215 4382 y Fh(k)3262 4397 y Fg(bac)3364 4382 y Fi(.)39 b(No)m(w)805 4591 y Fh(@)5 b(r)899 4606 y Fg(ab)p 805 4632 V 822 4715 a Fh(@)g(q)919 4689 y Fg(c)1006 4653 y Fi(=)1151 4591 y Fh(@)1204 4558 y Fd(2)1243 4591 y Fh(h)1295 4605 y Fg(a)p 1112 4632 264 4 v 1112 4715 a Fh(@)g(q)1209 4689 y Fg(b)1244 4715 y Fh(@)g(q)1341 4689 y Fg(c)1406 4653 y Fe(\000)1552 4591 y Fh(@)1605 4558 y Fd(2)1645 4591 y Fh(h)1697 4606 y Fg(b)p 1507 4632 271 4 v 1507 4715 a Fh(@)g(q)1604 4689 y Fg(a)1645 4715 y Fh(@)g(q)1742 4689 y Fg(c)1808 4653 y Fe(\000)1908 4617 y Fd(1)p 1908 4632 36 4 v 1908 4684 a(2)1996 4567 y Fi(\026)1980 4591 y Fh(d)p 1964 4632 81 4 v 1964 4715 a(dt)2069 4524 y Ff(\022)2184 4591 y Fh(@)2237 4558 y Fd(2)2277 4591 y Fh(h)2329 4605 y Fg(a)p 2146 4632 264 4 v 2146 4715 a Fh(@)g(q)2243 4689 y Fg(c)2278 4715 y Fh(@)22 b Fi(_)-42 b Fh(q)2375 4689 y Fg(b)2439 4653 y Fe(\000)2586 4591 y Fh(@)2639 4558 y Fd(2)2679 4591 y Fh(h)2731 4606 y Fg(b)p 2540 4632 271 4 v 2540 4715 a Fh(@)5 b(q)2637 4689 y Fg(c)2672 4715 y Fh(@)22 b Fi(_)-42 b Fh(q)2769 4689 y Fg(a)2821 4524 y Ff(\023)2903 4653 y Fh(;)118 4896 y Fi(and)30 b(so)g(b)m(y)h(the)f(preliminary)d(remark)186 5107 y Ff(X)183 5305 y Fg(a;b;c)345 5132 y Fh(@)5 b(r)439 5147 y Fg(ab)p 345 5172 167 4 v 363 5256 a Fh(@)g(q)460 5229 y Fg(c)547 5193 y Fi(=)25 b Fe(\000)724 5157 y Fd(1)p 724 5172 36 4 v 724 5225 a(2)811 5108 y Fi(\026)795 5132 y Fh(d)p 779 5172 81 4 v 779 5256 a(dt)884 5011 y Ff(0)884 5174 y(@)966 5107 y(X)963 5305 y Fg(a;b;c)1116 5065 y Ff(\022)1231 5132 y Fh(@)1284 5099 y Fd(2)1324 5132 y Fh(h)1376 5146 y Fg(a)p 1193 5172 264 4 v 1193 5256 a Fh(@)5 b(q)1290 5229 y Fg(c)1325 5256 y Fh(@)22 b Fi(_)-42 b Fh(q)1422 5229 y Fg(b)1486 5193 y Fe(\000)1633 5132 y Fh(@)1686 5099 y Fd(2)1726 5132 y Fh(h)1778 5147 y Fg(b)p 1587 5172 271 4 v 1587 5256 a Fh(@)5 b(q)1684 5229 y Fg(c)1719 5256 y Fh(@)22 b Fi(_)-42 b Fh(q)1816 5229 y Fg(a)1868 5065 y Ff(\023)1935 5011 y(1)1935 5174 y(A)2040 5193 y Fi(=)25 b Fe(\000)2217 5157 y Fd(1)p 2217 5172 36 4 v 2217 5225 a(2)2304 5108 y Fi(\026)2288 5132 y Fh(d)p 2271 5172 81 4 v 2271 5256 a(dt)2377 5011 y Ff(0)2377 5174 y(@)2459 5107 y(X)2456 5305 y Fg(a;b;c)2609 5065 y Ff(\022)2724 5132 y Fh(@)2777 5099 y Fd(2)2817 5132 y Fh(h)2869 5146 y Fg(a)p 2686 5172 264 4 v 2686 5256 a Fh(@)5 b(q)2783 5229 y Fg(c)2818 5256 y Fh(@)22 b Fi(_)-42 b Fh(q)2915 5229 y Fg(b)2979 5193 y Fe(\000)3118 5132 y Fh(@)3171 5099 y Fd(2)3211 5132 y Fh(h)3263 5146 y Fg(a)p 3080 5172 V 3080 5256 a Fh(@)5 b(q)3177 5229 y Fg(b)3212 5256 y Fh(@)22 b Fi(_)-42 b Fh(q)3309 5229 y Fg(c)3353 5065 y Ff(\023)3420 5011 y(1)3420 5174 y(A)3515 5193 y Fh(:)1839 5538 y Fi(5)p eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 448 -8 a SDict begin H.S end 448 -8 a 448 -8 a SDict begin H.R end 448 -8 a 448 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end 448 -8 a 118 399 a Fi(On)36 b(the)h(other)g(hand,)h(using)d(the)i(comm)m(utation)h (relation)e(\()2286 399 y SDict begin H.S end 2286 399 a Fi(13)2377 340 y SDict begin H.R end 2377 340 a 2377 399 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.13) cvn H.B /ANN pdfmark end 2377 399 a Fi(\))h(and)f(the)h(fact)h(that)f Fh(@)5 b(=@)g(q)3379 366 y Fg(a)3459 399 y Fi(and)134 487 y(\026)118 511 y Fh(d=dt)31 b Fi(comm)m(ute)h(it)e(is)f(easy)i(to)g (see)g(that)g(condition)e(\()1984 511 y SDict begin H.S end 1984 511 a Fi(11)2075 453 y SDict begin H.R end 2075 453 a 2075 511 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.11) cvn H.B /ANN pdfmark end 2075 511 a Fi(\))i(leads)f(to)1002 735 y Fd(1)p 1002 750 36 4 v 1002 802 a(2)1066 685 y Ff(X)1063 882 y Fg(a;b;c)1215 643 y Ff(\022)1330 710 y Fh(@)1383 677 y 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Fb(The)32 b(ne)-5 b(c)g(essary)33 b(and)f(su\016cient)f(c)-5 b(onditions)34 b(for)e(the)g(e)-5 b(quations)32 b Fh(f)3041 1415 y Fg(a)3082 1401 y Fi(\()7 b(\177)-52 b Fh(q)s(;)33 b Fi(_)-43 b Fh(q)t(;)15 b(q)s(;)g(t)p Fi(\))26 b(=)f(0)118 1514 y Fb(to)39 b(b)-5 b(e)38 b(of)g(L)-5 b(agr)g(angian)40 b(form)f(with)g(dissip)-5 b(ative)39 b(for)-5 b(c)g(es)39 b(of)f(gr)-5 b(adient)40 b(typ)-5 b(e)38 b(as)h(in)f(\()3014 1514 y SDict begin H.S end 3014 1514 a Fb(2)3060 1455 y SDict begin H.R end 3060 1455 a 3060 1514 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.2) cvn H.B /ANN pdfmark end 3060 1514 a Fb(\))g(ar)-5 b(e)39 b(that)h(the)118 1627 y(functions)33 b Fh(r)554 1642 y Fg(ab)658 1627 y Fb(and)h Fh(s)878 1642 y Fg(ab)982 1627 y Fb(ar)-5 b(e)33 b(of)g(\014rst)h(or)-5 b(der,)34 b(that)1639 1807 y Fh(@)5 b(f)1737 1821 y Fg(a)p 1639 1848 140 4 v 1643 1931 a Fh(@)12 b Fi(\177)-52 b Fh(q)1740 1905 y Fg(b)1814 1869 y Fi(=)1923 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3610 y Fd(2)1736 3643 y Fh(h)1788 3657 y Fg(a)p 1605 3684 V 1605 3767 a Fh(@)5 b(q)1702 3741 y Fg(c)1737 3767 y Fh(@)22 b Fi(_)-42 b Fh(q)1834 3741 y Fg(b)1878 3577 y Ff(\023)1970 3705 y Fi(=)25 b(0)p Fh(:)118 4001 y Fi(The)j(\014rst)f (of)i(these)f(is)g(one)g(of)h(the)f(classical)g(conditions.)38 b(The)28 b(second)g(is)f(the)i(condition)e Fh(\032)3311 4016 y Fg(abc)3438 4001 y Fi(=)e(0,)118 4114 y(whic)m(h)39 b(holds)g(in)h(the)g(classical)g(case)h(as)g(w)m(e)g(ha)m(v)m(e)h(sho)m (wn.)70 b(The)40 b(third)f(is)g(just)h(condition)f(\()3478 4114 y SDict begin H.S end 3478 4114 a Fi(19)3570 4055 y SDict begin H.R end 3570 4055 a 3570 4114 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.19) cvn H.B /ANN pdfmark end 3570 4114 a Fi(\))118 4227 y(ab)s(o)m(v)m(e)e(expressed)f(in)f (terms)h(of)g Fh(g)1318 4242 y Fg(ab)1426 4227 y Fi(and)g Fh(h)1661 4241 y Fg(a)1739 4227 y Fi(\(or)g(as)h(it)e(turns)g(out,)j (in)d(terms)h(of)g Fh(h)3075 4241 y Fg(a)3153 4227 y Fi(alone\),)j(and)118 4340 y Fh(r)159 4355 y Fg(ab)260 4340 y Fi(=)28 b Fh(s)402 4355 y Fg(ab)502 4340 y Fi(=)g(0)33 b(in)e(the)i(classical)e(case.)48 b(It)33 b(is)e(eviden)m(t)i (therefore)f(that)h(the)g(conditions)e(ab)s(o)m(v)m(e)j(are)118 4452 y(indeed)29 b(a)i(generalization)f(of)g(those)h(for)f(the)h (classical)e(case.)118 4599 y(W)-8 b(e)36 b(end)f(this)f(section)h(b)m (y)g(giving)f(an)h(alternativ)m(e)g(pro)s(of)f(of)i(the)f(su\016ciency) f(of)h(the)g(generalized)118 4711 y(Helmholtz)j(conditions,)h(based)f (on)g(this)f(form)m(ulation)g(of)i(them,)h(whic)m(h)d(is)h(shorter)g (and)f(in)g(our)118 4824 y(view)30 b(more)g(elegan)m(t)i(than)e(the)h (pro)s(of)e(in)g([)1603 4824 y SDict begin H.S end 1603 4824 a Fi(3)1649 4766 y SDict begin H.R end 1649 4766 a 1649 4824 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 1649 4824 a Fi(])h(\(necessit)m(y)i(is)d (an)h(easy)h(if)f(tedious)f(calculation\).)118 4971 y(W)-8 b(e)32 b(note)f(\014rst)e(that)i(if)f Fh(g)988 4986 y Fg(ab)1090 4971 y Fi(is)g(symmetric)f(and)h(satis\014es)g Fh(@)5 b(g)2232 4986 y Fg(ab)2304 4971 y Fh(=@)23 b Fi(_)-43 b Fh(q)2446 4938 y Fg(c)2507 4971 y Fi(=)25 b Fh(@)5 b(g)2699 4985 y Fg(ac)2772 4971 y Fh(=@)22 b Fi(_)-42 b Fh(q)2914 4938 y Fg(b)2979 4971 y Fi(then)1598 5230 y Fh(g)1641 5245 y Fg(ab)1739 5230 y Fi(=)1892 5169 y Fh(@)1945 5136 y Fd(2)1984 5169 y Fh(K)p 1845 5209 271 4 v 1845 5293 a(@)22 b Fi(_)-42 b Fh(q)1942 5266 y Fg(a)1983 5293 y Fh(@)23 b Fi(_)-43 b Fh(q)2080 5266 y Fg(b)1839 5538 y Fi(6)p eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 110 -8 a SDict begin H.S end 110 -8 a 110 -8 a SDict begin H.R end 110 -8 a 110 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end 110 -8 a 118 399 a Fi(for)38 b(some)g(function)e Fh(K)45 b Fi(=)37 b Fh(K)7 b Fi(\()17 b(_)-42 b Fh(q)s(;)15 b(q)s(;)g(t)p Fi(\))39 b(\(a)f(w)m(ell-kno)m(wn)f(result,)i(whic)m(h)d(also)i(app)s(ears)f(in) f([)3312 399 y SDict begin H.S end 3312 399 a Fi(3)3358 340 y SDict begin H.R end 3358 340 a 3358 399 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 3358 399 a Fi(]\).)64 b(Of)118 511 y(course)30 b Fh(K)37 b Fi(is)29 b(not)h(determined)f(b)m(y)h(this)f(relation;)g (in)g(fact)i(if)e(\003)c(=)g Fh(K)i Fi(+)19 b Fh(P)2702 525 y Fg(a)2761 511 y Fi(_)-42 b Fh(q)2788 478 y Fg(a)2849 511 y Fi(+)19 b Fh(Q)p Fi(,)30 b(where)g Fh(P)3387 525 y Fg(a)3459 511 y Fi(and)118 624 y Fh(Q)c Fi(are)h(an)m(y)f(functions)f (of)h Fh(q)j Fi(and)d Fh(t)p Fi(,)h(then)f(\003)g(has)g(the)h(same)f (Hessian)g(as)g Fh(K)33 b Fi(\(the)27 b(same)f Fh(g)3171 639 y Fg(ab)3244 624 y Fi(,)h(in)e(other)118 737 y(w)m(ords\).)54 b(Our)34 b(aim)g(is)g(to)h(c)m(ho)s(ose)h Fh(P)1398 751 y Fg(a)1475 737 y Fi(and)e Fh(Q)h Fi(so)g(that)g(the)g(giv)m(en)g (equations)g(are)g(of)g(Lagrangian)118 850 y(form)26 b(with)f(dissipativ)m(e)g(forces)i(of)f(gradien)m(t)g(t)m(yp)s(e)h(as)g (in)e(\()2123 850 y SDict begin H.S end 2123 850 a Fi(2)2169 792 y SDict begin H.R end 2169 792 a 2169 850 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.2) cvn H.B /ANN pdfmark end 2169 850 a Fi(\),)i(with)f(Lagrangian)g(\003,)h(assuming)e (that)118 963 y(the)35 b(generalized)g(Helmholtz)f(conditions)f(ab)s(o) m(v)m(e)j(hold.)53 b(In)34 b(fact)i(w)m(e)g(w)m(on't)f(need)g(to)g (consider)f Fh(Q)118 1076 y Fi(b)s(ecause)e(it)g(can)g(b)s(e)g(absorb)s (ed:)f(if)g(\003)h(is)f(a)i(Lagrangian)f(and)f Fh(D)k Fi(a)e(dissipation)c(function)i(for)h(some)118 1189 y(functions)d Fh(f)555 1203 y Fg(a)596 1189 y Fi(,)i(so)f(are)h(\003)20 b(+)g Fh(Q)31 b Fi(and)e Fh(D)23 b Fi(+)37 b(_)-42 b Fh(q)1601 1156 y Fg(a)1643 1189 y Fh(@)5 b(Q=@)g(q)1910 1156 y Fg(a)1952 1189 y Fi(.)41 b(W)-8 b(e)31 b(shall)e(therefore)i (tak)m(e)h Fh(Q)25 b Fi(=)g(0)31 b(b)s(elo)m(w.)118 1335 y(Let)36 b Fh(E)353 1349 y Fg(a)430 1335 y Fi(b)s(e)f(the)g (Euler-Lagrange)h(expressions)e(of)h Fh(K)7 b Fi(.)55 b(Then)35 b Fh(E)2426 1349 y Fg(a)2501 1335 y Fi(=)e Fh(g)2648 1350 y Fg(ab)2727 1335 y Fi(\177)-52 b Fh(q)2764 1302 y Fg(b)2822 1335 y Fi(+)23 b Fh(k)2963 1349 y Fg(a)3041 1335 y Fi(for)35 b(some)g(\014rst-)118 1448 y(order)30 b Fh(k)403 1462 y Fg(a)445 1448 y Fi(,)g(b)m(y)g(construction,)g(so)g Fh(f)1328 1462 y Fg(a)1389 1448 y Fe(\000)19 b Fh(E)1546 1462 y Fg(a)1613 1448 y Fi(=)25 b Fh(h)1761 1462 y Fg(a)1823 1448 y Fe(\000)19 b Fh(k)1960 1462 y Fg(a)2027 1448 y Fi(=)25 b Fh(\024)2175 1462 y Fg(a)2247 1448 y Fi(sa)m(y)-8 b(,)32 b(where)d Fh(\024)2736 1462 y Fg(a)2808 1448 y Fi(is)h(also)g(of)g(\014rst)f(order.)118 1561 y(Moreo)m(v)m(er,)j Fh(f)587 1575 y Fg(a)658 1561 y Fi(satis\014es)c(the)i(generalized)f (Helmholtz)g(conditions)f(b)m(y)h(assumption,)f(and)h Fh(E)3359 1575 y Fg(a)3430 1561 y Fi(do)s(es)118 1674 y(so)i(b)m(y)f(construction)g(\(it)g(satis\014es)g(the)h(classical)e (conditions)g(after)i(all\),)f(whence)g Fh(\024)3051 1688 y Fg(a)3123 1674 y Fi(satis\014es)1611 1872 y Fh(@)1664 1839 y Fd(2)1704 1872 y Fh(\024)1756 1886 y Fg(a)p 1573 1912 264 4 v 1573 1996 a Fh(@)22 b Fi(_)-42 b Fh(q)1670 1969 y Fg(b)1704 1996 y Fh(@)22 b Fi(_)-42 b Fh(q)1801 1969 y Fg(c)1866 1933 y Fe(\000)2013 1872 y Fh(@)2066 1839 y Fd(2)2105 1872 y Fh(\024)2157 1887 y Fg(b)p 1967 1912 271 4 v 1967 1996 a Fh(@)22 b Fi(_)-42 b Fh(q)2064 1969 y Fg(a)2106 1996 y Fh(@)22 b Fi(_)-42 b Fh(q)2203 1969 y Fg(c)2273 1933 y Fi(=)25 b(0)3443 1933 y SDict begin H.S end 3443 1933 a 3443 1933 a SDict begin 16.6 H.A end 3443 1933 a 3443 1933 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.20) cvn /DEST pdfmark end 3443 1933 a Fi(\(20\))1287 2098 y Ff(X)1284 2295 y Fg(a;b;c)1436 2056 y Ff(\022)1551 2122 y Fh(@)1604 2089 y Fd(2)1644 2122 y Fh(\024)1696 2136 y Fg(a)p 1513 2163 264 4 v 1513 2246 a Fh(@)5 b(q)1610 2220 y Fg(b)1645 2246 y Fh(@)22 b Fi(_)-42 b Fh(q)1742 2220 y Fg(c)1807 2184 y Fe(\000)1946 2122 y Fh(@)1999 2089 y Fd(2)2038 2122 y Fh(\024)2090 2136 y Fg(a)p 1907 2163 V 1907 2246 a Fh(@)5 b(q)2004 2220 y Fg(c)2039 2246 y Fh(@)22 b Fi(_)-42 b Fh(q)2136 2220 y Fg(b)2181 2056 y Ff(\023)2273 2184 y Fi(=)25 b(0)p Fh(:)3443 2184 y SDict begin H.S end 3443 2184 a 3443 2184 a SDict begin 16.6 H.A end 3443 2184 a 3443 2184 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.21) cvn /DEST pdfmark end 3443 2184 a Fi(\(21\))118 2496 y(Let)30 b(us)e(set)i Fh(@)5 b(\024)641 2510 y Fg(a)683 2496 y Fh(=@)23 b Fi(_)-43 b Fh(q)825 2463 y Fg(b)877 2496 y Fe(\000)18 b Fh(@)5 b(\024)1071 2511 y Fg(b)1106 2496 y Fh(=@)22 b Fi(_)-42 b Fh(q)1248 2463 y Fg(a)1315 2496 y Fi(=)25 b Fh(R)1480 2511 y Fg(ab)1552 2496 y Fi(.)40 b(Then)28 b(b)m(y)h(\()2013 2496 y SDict begin H.S end 2013 2496 a Fi(20)2104 2437 y SDict begin H.R end 2104 2437 a 2104 2496 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.20) cvn H.B /ANN pdfmark end 2104 2496 a Fi(\))h Fh(R)2238 2511 y Fg(ab)2339 2496 y Fi(is)e(indep)s(enden)m(t)f(of)46 b(_)-42 b Fh(q)s Fi(,)29 b(and)f(b)m(y)h(\()3478 2496 y SDict begin H.S end 3478 2496 a Fi(21)3570 2437 y SDict begin H.R end 3570 2437 a 3570 2496 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.21) cvn H.B /ANN pdfmark end 3570 2496 a Fi(\))1585 2657 y Ff(X)1582 2854 y Fg(a;b;c)1745 2681 y Fh(@)5 b(R)1867 2696 y Fg(ab)p 1745 2722 195 4 v 1776 2805 a Fh(@)g(q)1873 2779 y Fg(c)1974 2743 y Fi(=)25 b(0)p Fh(:)118 3044 y Fi(There)30 b(are)h(therefore)g(functions)d Fh(P)1364 3058 y Fg(a)1406 3044 y Fi(\()p Fh(q)s(;)15 b(t)p Fi(\))32 b(suc)m(h)e(that)1116 3298 y Fh(R)1185 3313 y Fg(ab)1282 3298 y Fi(=)25 b(2)1438 3170 y Ff(\022)1515 3237 y Fh(@)5 b(P)1626 3251 y Fg(a)p 1515 3277 154 4 v 1526 3361 a Fh(@)g(q)1623 3334 y Fg(b)1699 3298 y Fe(\000)1800 3237 y Fh(@)g(P)1911 3252 y Fg(b)p 1800 3277 147 4 v 1804 3361 a Fh(@)g(q)1901 3334 y Fg(a)1956 3170 y Ff(\023)2048 3298 y Fi(=)2154 3237 y Fh(@)g(\024)2259 3251 y Fg(a)p 2154 3277 148 4 v 2162 3361 a Fh(@)22 b Fi(_)-42 b Fh(q)2259 3334 y Fg(b)2331 3298 y Fe(\000)2432 3237 y Fh(@)5 b(\024)2537 3252 y Fg(b)p 2432 3277 141 4 v 2433 3361 a Fh(@)22 b Fi(_)-42 b Fh(q)2530 3334 y Fg(a)2582 3298 y Fh(;)118 3547 y Fi(whic)m(h)29 b(is)h(to)h(sa)m(y)g(that)g(if)e(w)m(e)i(set)1308 3796 y Fh(\031)1360 3811 y Fg(ab)1457 3796 y Fi(=)1563 3735 y Fh(@)5 b(\024)1668 3749 y Fg(a)p 1563 3776 148 4 v 1571 3859 a Fh(@)22 b Fi(_)-42 b Fh(q)1668 3833 y Fg(b)1740 3796 y Fe(\000)1831 3668 y Ff(\022)1908 3735 y Fh(@)5 b(P)2019 3749 y Fg(a)p 1908 3776 154 4 v 1919 3859 a Fh(@)g(q)2016 3833 y Fg(b)2091 3796 y Fe(\000)2192 3735 y Fh(@)g(P)2303 3750 y Fg(b)p 2192 3776 147 4 v 2196 3859 a Fh(@)g(q)2293 3833 y Fg(a)2348 3668 y Ff(\023)118 4046 y Fi(then)30 b Fh(\031)377 4061 y Fg(ba)474 4046 y Fi(=)25 b Fh(\031)622 4061 y Fg(ab)694 4046 y Fi(.)40 b(Moreo)m(v)m(er,)1399 4147 y Fh(@)5 b(\031)1504 4162 y Fg(ab)p 1399 4188 177 4 v 1422 4271 a Fh(@)22 b Fi(_)-42 b Fh(q)1519 4245 y Fg(c)1611 4209 y Fi(=)1755 4147 y Fh(@)1808 4114 y Fd(2)1848 4147 y Fh(\024)1900 4161 y Fg(a)p 1717 4188 264 4 v 1717 4271 a Fh(@)22 b Fi(_)-42 b Fh(q)1814 4245 y Fg(b)1849 4271 y Fh(@)22 b Fi(_)-42 b Fh(q)1946 4245 y Fg(c)2016 4209 y Fi(=)2122 4147 y Fh(@)5 b(\031)2227 4161 y Fg(ac)p 2122 4188 178 4 v 2144 4271 a Fh(@)23 b Fi(_)-43 b Fh(q)2241 4245 y Fg(b)2309 4209 y Fh(:)118 4421 y Fi(It)31 b(follo)m(ws)e(\(just)h(as)h(is)e(the)i (case)g(for)f Fh(g)1464 4436 y Fg(ab)1537 4421 y Fi(\))g(that)h(there)g (is)e(a)i(\014rst-order)e(function)h Fh(D)3059 4388 y Fa(0)3112 4421 y Fi(suc)m(h)g(that)1581 4681 y Fh(\031)1633 4696 y Fg(ab)1730 4681 y Fi(=)1874 4619 y Fh(@)1927 4586 y Fd(2)1967 4619 y Fh(D)2045 4586 y Fa(0)p 1836 4660 271 4 v 1836 4743 a Fh(@)22 b Fi(_)-42 b Fh(q)1933 4717 y Fg(a)1975 4743 y Fh(@)22 b Fi(_)-42 b Fh(q)2072 4717 y Fg(b)2116 4681 y Fh(;)118 4924 y Fi(from)30 b(whic)m(h)f(w)m(e)i (obtain)1168 5064 y Fh(\024)1220 5078 y Fg(a)1288 5064 y Fi(=)1383 4935 y Ff(\022)1460 5002 y Fh(@)5 b(P)1571 5016 y Fg(a)p 1460 5043 154 4 v 1471 5126 a Fh(@)g(q)1568 5100 y Fg(b)1644 5064 y Fe(\000)1745 5002 y Fh(@)g(P)1856 5017 y Fg(b)p 1745 5043 147 4 v 1749 5126 a Fh(@)g(q)1846 5100 y Fg(a)1901 4935 y Ff(\023)2000 5064 y Fi(_)-42 b Fh(q)2027 5026 y Fg(b)2081 5064 y Fi(+)2182 5002 y Fh(@)5 b(D)2313 4969 y Fa(0)p 2182 5043 155 4 v 2190 5126 a Fh(@)22 b Fi(_)-42 b Fh(q)2287 5100 y Fg(a)2367 5064 y Fi(+)19 b Fh(S)2513 5078 y Fg(a)1839 5538 y Fi(7)p eop end end %%Page: 8 8 TeXDict begin HPSdict begin 8 7 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 448 -8 a SDict begin H.S end 448 -8 a 448 -8 a SDict begin H.R end 448 -8 a 448 -8 a SDict begin [/View [/XYZ H.V]/Dest (page.8) cvn /DEST pdfmark end 448 -8 a 118 399 a Fi(where)34 b Fh(S)441 413 y Fg(a)517 399 y Fi(is)g(indep)s(enden)m (t)e(of)52 b(_)-42 b Fh(q)s Fi(.)53 b(No)m(w)36 b(tak)m(e)g(\003)d(=)f Fh(K)d Fi(+)23 b Fh(P)2232 413 y Fg(a)2291 399 y Fi(_)-42 b Fh(q)2318 366 y Fg(a)2360 399 y Fi(.)53 b(Denoting)35 b(the)g(Euler-Lagrange)118 511 y(expressions)29 b(of)i Fh(K)37 b Fi(b)m(y)30 b Fh(E)1003 525 y Fg(a)1075 511 y Fi(as)h(b)s(efore,)f(the)g(Euler-Lagrange)h(expressions)e(for)h(\003) g(are)651 757 y Fh(E)718 771 y Fg(a)780 757 y Fi(+)871 629 y Ff(\022)948 696 y Fh(@)5 b(P)1059 710 y Fg(a)p 948 736 154 4 v 959 820 a Fh(@)g(q)1056 793 y Fg(b)1132 757 y Fe(\000)1233 696 y Fh(@)g(P)1344 711 y Fg(b)p 1233 736 147 4 v 1237 819 a Fh(@)g(q)1334 793 y Fg(a)1389 629 y Ff(\023)1487 757 y Fi(_)-41 b Fh(q)1515 720 y Fg(b)1569 757 y Fi(+)1670 696 y Fh(@)5 b(P)1781 710 y Fg(a)p 1670 736 154 4 v 1703 819 a Fh(@)g(t)1858 757 y Fi(=)25 b Fh(E)2021 771 y Fg(a)2083 757 y Fi(+)20 b Fh(\024)2226 771 y Fg(a)2288 757 y Fe(\000)2389 696 y Fh(@)5 b(D)2520 663 y Fa(0)p 2389 736 155 4 v 2397 819 a Fh(@)22 b Fi(_)-42 b Fh(q)2494 793 y Fg(a)2574 757 y Fe(\000)20 b Fh(S)2721 771 y Fg(a)2782 757 y Fi(+)2883 696 y Fh(@)5 b(P)2994 710 y Fg(a)p 2883 736 154 4 v 2917 819 a Fh(@)g(t)3046 757 y(:)118 1003 y Fi(Th)m(us,)30 b(putting)1335 1161 y Fh(D)e Fi(=)d Fh(D)1612 1123 y Fa(0)1656 1161 y Fi(+)1746 1032 y Ff(\022)1813 1161 y Fh(S)1869 1175 y Fg(a)1931 1161 y Fe(\000)2032 1099 y Fh(@)5 b(P)2143 1113 y Fg(a)p 2032 1140 V 2065 1223 a Fh(@)g(t)2195 1032 y Ff(\023)2294 1161 y Fi(_)-42 b Fh(q)2321 1123 y Fg(a)2363 1161 y Fh(;)118 1363 y Fi(w)m(e)31 b(get)1284 1521 y Fh(f)1329 1535 y Fg(a)1396 1521 y Fi(=)1518 1460 y Fh(d)p 1501 1500 81 4 v 1501 1584 a(dt)1607 1393 y Ff(\022)1695 1460 y Fh(@)5 b Fi(\003)p 1684 1500 139 4 v 1684 1584 a Fh(@)22 b Fi(_)-42 b Fh(q)1781 1557 y Fg(a)1832 1393 y Ff(\023)1920 1521 y Fe(\000)2032 1460 y Fh(@)5 b Fi(\003)p 2020 1500 V 2020 1584 a Fh(@)g(q)2117 1557 y Fg(a)2189 1521 y Fi(+)2294 1460 y Fh(@)g(D)p 2290 1500 V 2290 1584 a(@)22 b Fi(_)-42 b Fh(q)2387 1557 y Fg(a)118 1731 y Fi(as)31 b(required.)118 1877 y(This)d(metho)s(d)h(of)h(pro)s(of)f(w)m(orks)g(equally)g(w)m(ell) f(in)g(the)i(classical)f(case.)42 b(The)29 b(pro)s(of)g(is)f (constructiv)m(e,)118 1990 y(in)35 b(the)i(same)g(sense)f(that)h(the)g (one)g(in)e([)1542 1990 y SDict begin H.S end 1542 1990 a Fi(3)1587 1932 y SDict begin H.R end 1587 1932 a 1587 1990 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 1587 1990 a Fi(])i(is,)g(in)e(either)h(case.) 60 b(It)37 b(is)e(particularly)f(w)m(ell)i(adapted)118 2103 y(to)31 b(the)f(familiar)e(situation)g(in)h(whic)m(h)g Fh(g)1508 2118 y Fg(ab)1610 2103 y Fi(is)g(indep)s(enden)m(t)f(of)47 b(_)-42 b Fh(q)s Fi(,)30 b(when)f(one)h(can)g(tak)m(e)i(the)e(kinetic) 118 2216 y(energy)419 2180 y Fd(1)p 419 2195 36 4 v 419 2248 a(2)464 2216 y Fh(g)507 2231 y Fg(ab)596 2216 y Fi(_)-42 b Fh(q)623 2183 y Fg(a)682 2216 y Fi(_)g Fh(q)709 2183 y Fg(b)773 2216 y Fi(for)30 b Fh(K)7 b Fi(.)118 2384 y SDict begin H.S end 118 2384 a 118 2384 a SDict begin 13.6 H.A end 118 2384 a 118 2384 a SDict begin [/View [/XYZ H.V]/Dest (section.3) cvn /DEST pdfmark end 118 2384 a 151 x Fj(3)135 b(Concluding)45 b(remarks)118 2771 y Fi(W)-8 b(e)32 b(wish)c(to)j(mak)m(e)h(four)d(remarks)h(in)f (conclusion.)118 2918 y(The)38 b(\014rst)f(remark)g(concerns)h(the)g (nature)g(of)g(conditions)e(\()2251 2918 y SDict begin H.S end 2251 2918 a Fi(10)2343 2859 y SDict begin H.R end 2343 2859 a 2343 2918 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 2343 2918 a Fi(\){\()2458 2918 y SDict begin H.S end 2458 2918 a Fi(12)2550 2859 y SDict begin H.R end 2550 2859 a 2550 2918 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 2550 2918 a Fi(\))i(on)g(the)g(deriv)-5 b(ativ)m(es)37 b(of)h Fh(r)3533 2933 y Fg(ab)118 3030 y Fi(and)31 b Fh(s)339 3045 y Fg(ab)411 3030 y Fi(,)h(as)g(originally)d (expressed)i(in)f([)1525 3030 y SDict begin H.S end 1525 3030 a Fi(3)1571 2972 y SDict begin H.R end 1571 2972 a 1571 3030 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 1571 3030 a Fi(])i(\(that)g(is,)f (ignoring)f(the)i(question)f(of)h(dep)s(endence\).)44 b(In)118 3143 y(particular,)29 b(b)s(earing)g(in)g(mind)f(the)j(fact)g (that)g Fh(r)1801 3158 y Fg(ab)1903 3143 y Fi(is)f(sk)m(ew)h(in)e(its)g (indices,)g(the)i(condition)1612 3296 y Ff(X)1609 3493 y Fg(a;b;c)1771 3320 y Fh(@)5 b(r)1865 3335 y Fg(ab)p 1771 3361 167 4 v 1789 3444 a Fh(@)g(q)1886 3418 y Fg(c)1973 3382 y Fi(=)25 b(0)118 3675 y(is)34 b(suggestiv)m(e:)i(if)d(p)s(erc)m (hance)i(the)g Fh(r)1392 3690 y Fg(ab)1499 3675 y Fi(w)m(ere)g (functions)e(of)i(the)g Fh(q)j Fi(alone)c(this)g(w)m(ould)f(ha)m(v)m(e) j(a)f(nat-)118 3787 y(ural)d(in)m(terpretation)h(in)f(terms)h(of)h(the) f(exterior)h(calculus,)f(b)s(eing)f(the)h(condition)f(for)h(the)h (2-form)118 3900 y Fh(r)159 3915 y Fg(ab)231 3900 y Fh(dq)322 3867 y Fg(a)389 3900 y Fe(^)25 b Fh(dq)566 3867 y Fg(b)638 3900 y Fi(to)38 b(b)s(e)f(closed,)j(that)e(is,)h(to)g(satisfy)e Fh(d)p Fi(\()p Fh(r)2053 3915 y Fg(ab)2126 3900 y Fh(dq)2217 3867 y Fg(a)2283 3900 y Fe(^)25 b Fh(dq)2460 3867 y Fg(b)2495 3900 y Fi(\))38 b(=)f(0.)63 b(This)36 b(p)s(oin)m(t)h(is)g(made,)118 4013 y(in)c(somewhat)i(di\013eren)m(t)e(terms,)j(in)d([)1438 4013 y SDict begin H.S end 1438 4013 a Fi(3)1484 3955 y SDict begin H.R end 1484 3955 a 1484 4013 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 1484 4013 a Fi(])h(\(and)g(w)m(e)h(app)s(ealed)e(to)i(the)f (same)h(general)f(result)g(in)e(our)118 4126 y(pro)s(of)25 b(of)i(su\016ciency)e(of)h(the)g(generalized)g(Helmholtz)f(conditions)g (in)g(Section)g(2\).)41 b(The)25 b(authors)h(of)118 4239 y([)143 4239 y SDict begin H.S end 143 4239 a Fi(3)189 4181 y SDict begin H.R end 189 4181 a 189 4239 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.germ1) cvn H.B /ANN pdfmark end 189 4239 a Fi(])32 b(go)g(on)g(to)g(sa)m(y)-8 b(,)33 b(ho)m(w)m(ev)m(er,)h(that)e(the)g(condition)e(ab)s(o)m(v)m(e)j (`can)g(b)s(e)e(in)m(terpreted)g(as)g(the)h(v)-5 b(anishing)118 4352 y(curv)g(ature)38 b(of)f(a)i(symplectic)d(space',)41 b(whic)m(h)c(seems)h(to)g(us)f(not)h(to)h(b)s(e)e(en)m(tirely)g(con)m (vincing.)62 b(In)118 4465 y(fact)28 b(it)f(is)f(p)s(ossible)f(to)j(in) m(terpret)e(the)h(three)h(conditions)d(\()2139 4465 y SDict begin H.S end 2139 4465 a Fi(10)2231 4406 y SDict begin H.R end 2231 4406 a 2231 4465 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.10) cvn H.B /ANN pdfmark end 2231 4465 a Fi(\){\()2346 4465 y SDict begin H.S end 2346 4465 a Fi(12)2438 4406 y SDict begin H.R end 2438 4406 a 2438 4465 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.2.12) cvn H.B /ANN pdfmark end 2438 4465 a Fi(\))i(collectiv)m(ely)g (as)g(signifying)e(the)118 4578 y(v)-5 b(anishing)28 b(of)i(a)g(certain)g(exterior)g(deriv)-5 b(ativ)m(e)29 b(of)h(a)h(certain)f(2-form)g(on)g(the)g(space)g(of)g(co)s(ordinates) 118 4691 y Fh(t;)15 b(q)s(;)32 b Fi(_)-42 b Fh(q)s(;)23 b Fi(\177)-53 b Fh(q)t(;)15 b(:)g(:)g(:)h Fi(,)29 b(a)g(2-form)f(whose) g(co)s(e\016cien)m(ts)h(in)m(v)m(olv)m(e)g(b)s(oth)e Fh(r)2262 4706 y Fg(ab)2363 4691 y Fi(and)g Fh(s)2580 4706 y Fg(ab)2652 4691 y Fi(.)40 b(This)26 b(in)m(terpretation)i(re-) 118 4804 y(ally)h(arises)h(from)g(seeing)h(the)f(problem)f(in)g(the)i (con)m(text)h(of)f(the)g(so-called)f(v)-5 b(ariational)30 b(bicomplex)118 4917 y(\(see)h([)325 4917 y SDict begin H.S end 325 4917 a Fi(7)371 4858 y SDict begin H.R end 371 4858 a 371 4917 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.vitolo) cvn H.B /ANN pdfmark end 371 4917 a Fi(])g(for)f(a)h (recen)m(t)g(review\).)118 5063 y(Secondly)-8 b(,)39 b(w)m(e)f(con)m(tend)g(that)g(the)g(problem)d(w)m(e)j(are)g(dealing)e (with)g(should)g(really)g(b)s(e)h(regarded)118 5176 y(as)d(one)f(ab)s (out)g(\(second-order\))h(dynamical)e(systems.)49 b(The)33 b(p)s(oin)m(t)f(is)h(that)g(a)h(dynamical)e(system)118 5289 y(ma)m(y)39 b(b)s(e)f(represen)m(ted)h(as)g(a)g(system)g(of)f 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0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.doug) cvn H.B /ANN pdfmark end 1632 2092 a Fi(],)h(whic)m(h)c(analyses)h(in)e(great)k(detail)d (the)h(case)h(of)g(t)m(w)m(o)118 2205 y(degrees)30 b(of)g(freedom.)41 b(F)-8 b(or)30 b(a)h(recen)m(t)g(review)e(of)h(dev)m(elopmen)m(ts)g (since)f(then)g(see)i(Sections)e(5)h(and)g(6)118 2318 y(of)k([)250 2318 y SDict begin H.S end 250 2318 a Fi(4)296 2259 y SDict begin H.R end 296 2259 a 296 2318 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.olgageoff) cvn H.B /ANN pdfmark end 296 2318 a Fi(])g(and)g(references)g(therein.)51 b(One)34 b(can)g(in)f(fact)i(also)f(form)m(ulate)g(conditions)f(on)h(a) g(m)m(ultiplier)118 2431 y(for)c(a)g(second-order)g(dynamical)e (system,)i(expressible)e(in)h(normal)g(form,)g(to)i(b)s(e)e(represen)m (table)h(as)118 2544 y(equations)22 b(of)h(Lagrangian)f(form)h(with)e 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pdfmark end 448 -8 a 118 399 a SDict begin H.S end 118 399 a 118 399 a SDict begin 18 H.A end 118 399 a 118 399 a SDict begin [/View [/XYZ H.V]/Dest (section*.2) cvn /DEST pdfmark end 118 399 a Fj(References)118 484 y SDict begin H.S end 118 484 a 118 484 a SDict begin 13.6 H.A end 118 484 a 118 484 a SDict begin [/View [/XYZ H.V]/Dest (cite.doug) cvn /DEST pdfmark end 118 484 a 164 635 a Fi([1])46 b(J.)c(Douglas,)k(Solution)41 b(of)h(the)h(in)m(v)m(erse)f(problem)f(of)i(the)f(calculus)f(of)i(v)-5 b(ariations,)44 b(T)-8 b(rans.)305 748 y(Amer.)30 b(Math.)i(So)s(c.)e Fc(50)p Fi(,)h(71{128)i(\(1941\).)118 808 y SDict begin H.S end 118 808 a 118 808 a SDict begin 13.6 H.A end 118 808 a 118 808 a SDict begin [/View [/XYZ H.V]/Dest (cite.germ2) cvn /DEST pdfmark end 118 808 a 164 935 a Fi([2])46 b(U.)54 b(Jungnic)m(k)m(el,)k(G.)c(Kielau,)k(P)-8 b(.)54 b(Maisser)g(and)f(A.)g(M)s(\177)-48 b(uller,)58 b(A)c(generalization)f (of)h(the)305 1048 y(Helmholtz)38 b(conditions)f(for)i(the)g(existence) g(of)g(a)g(\014rst-order)f(Lagrangian)g(using)g(nonholo-)305 1161 y(nomic)30 b(v)m(elo)s(cities,)g(Z.)g(Angew.)h(Math.)g(Mec)m(h.)h Fc(89)p Fi(,)f(44{53)h(\(2009\).)118 1221 y SDict begin H.S end 118 1221 a 118 1221 a SDict begin 13.6 H.A end 118 1221 a 118 1221 a SDict begin [/View [/XYZ H.V]/Dest (cite.germ1) cvn /DEST pdfmark end 118 1221 a 164 1349 a Fi([3])46 b(G.)e(Kielau)f(and)g(P)-8 b(.)44 b(Maisser,)j(A)d (generalization)g(of)g(the)g(Helmholtz)f(conditions)f(for)i(the)305 1462 y(existence)31 b(of)f(a)h(\014rst-order)f(Lagrangian,)g(Z.)g (Angew.)h(Math.)g(Mec)m(h.)h Fc(86)p Fi(,)f(722{735)i(\(2006\).)118 1522 y SDict begin H.S end 118 1522 a 118 1522 a SDict begin 13.6 H.A end 118 1522 a 118 1522 a SDict begin [/View [/XYZ H.V]/Dest (cite.olgageoff) cvn /DEST pdfmark end 118 1522 a 164 1649 a Fi([4])46 b(O.)35 b(Krupk)m(o)m(v\023)-45 b(a)35 b(and)g(G.)16 b(E.)35 b(Prince,)h(Second)f(order)f(ordinary)g (di\013eren)m(tial)g(equations)h(in)e(jet)305 1762 y(bundles)i(and)i (the)i(in)m(v)m(erse)e(problem)f(of)i(the)g(calculus)f(of)h(v)-5 b(ariations,)39 b(in:)e(D.)h(Krupk)-5 b(a)37 b(and)305 1875 y(D.)16 b(J.)25 b(Saunders)e(\(eds.\))j(Handb)s(o)s(ok)e(of)h (Global)f(Analysis)f(\(Elsevier,)j(Amsterdam,)g(2008\))h(pp.)305 1988 y(837{904.)118 2025 y SDict begin H.S end 118 2025 a 118 2025 a SDict begin 13.6 H.A end 118 2025 a 118 2025 a SDict begin [/View [/XYZ H.V]/Dest (cite.MWC) cvn /DEST pdfmark end 118 2025 a 164 2176 a Fi([5])46 b(T.)34 b(Mestdag,)j(W.)e(Sarlet)e(and)h(M.)h(Crampin,)e(The)g(in)m(v)m(erse)h (problem)f(for)h(Lagrangian)g(sys-)305 2288 y(tems)h(with)e(certain)h (non-conserv)-5 b(ativ)m(e)35 b(forces,)h(preprin)m(t,)e(Ghen)m(t)g (Univ)m(ersit)m(y)-8 b(,)36 b(a)m(v)-5 b(ailable)33 b(at)305 2424 y SDict begin H.S end 305 2424 a -23 x Fi(h)m(ttp://users.ugen)m(t.b)s(e/)p Fe(\030)p Fi(tmestdag)1633 2333 y SDict begin H.R end 1633 2333 a 1633 2401 a SDict begin [/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [0 1 1]/Action << /Subtype /URI /URI (http://users.ugent.be/~tmestdag) >>/Subtype /Link H.B /ANN pdfmark end 1633 2401 a Fi(.)118 2461 y SDict begin H.S end 118 2461 a 118 2461 a SDict begin 13.6 H.A end 118 2461 a 118 2461 a SDict begin [/View [/XYZ H.V]/Dest (cite.Sant) cvn /DEST pdfmark end 118 2461 a 164 2589 a Fi([6])46 b(R.)15 b(M.)46 b(San)m(tilli,)h(F)-8 b(oundations)44 b(of)i(Theoretical)f(Mec) m(hanics)g(I.)h(The)f(In)m(v)m(erse)g(Problem)f(in)305 2702 y(Newtonian)30 b(Mec)m(hanics)h(\(Spinger,)e(New)i(Y)-8 b(ork,)31 b(1978\).)118 2762 y SDict begin H.S end 118 2762 a 118 2762 a SDict begin 13.6 H.A end 118 2762 a 118 2762 a SDict begin [/View [/XYZ H.V]/Dest (cite.vitolo) cvn /DEST pdfmark end 118 2762 a 164 2890 a Fi([7])46 b(R.)23 b(Vitolo,)h(V)-8 b(ariational)23 b(sequences,)h(in:)e(D.)i (Krupk)-5 b(a)21 b(and)h(D.)16 b(J.)23 b(Saunders)e(\(eds.\),)k(Handb)s (o)s(ok)305 3002 y(of)31 b(Global)e(Analysis)g(\(Elsevier,)g (Amsterdam,)i(2008\))h(pp.)e(1115{1163.)1816 5538 y(10)p eop end end %%Trailer end userdict /end-hook known{end-hook}if %%EOF