Content-Type: multipart/mixed; boundary="-------------0105040630216" This is a multi-part message in MIME format. ---------------0105040630216 Content-Type: text/plain; name="01-166.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-166.keywords" Mechanics, Hamilton, Euler, Lagrange, integrator, Runge Kutta, Vermaseren, FORM, action principle, non-conservative system, holonomic ---------------0105040630216 Content-Type: application/postscript; name="pcmdap3.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="pcmdap3.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: mvm8.dvi %%CreationDate: Fri May 04 13:08:04 2001 %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: NewCenturySchlbk-Roman NewCenturySchlbk-Bold Symbol %%+ Times-Italic Times-Roman NewCenturySchlbk-Italic %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips mvm8 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2001.05.04:1307 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: 8r.enc % @@psencodingfile@{ % author = "S. 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y(*)i(1.)g(Declarations:)38 b(starting)i(with)i(keywords)e (Symbol\(s\),)g(Function\(s\),)e(...)432 5255 y(*)43 b(2.)g(Specifications:)37 b(e.g.statistics)h(Off)k(...)432 5355 y(*)h(3.)g(Definitions:)38 b(starting)j(with)g(keywords)g(Local,)g (...)p Black 1821 5637 a FD(12)p Black eop %%Page: 13 13 13 12 bop Black Black 432 573 a Fa(*)43 b(4.)g(Executable)c (Statements:)g(starting)h(with)i(keywords)f(id)h(...)432 672 y(*)h(5.)g(Output)e(control:)f(such)i(as)h(Print)e(and)i(Bracket) 432 872 y(Symbols)d(tau,dh,n;)432 971 y(Local)h(G0tau=dh*G0;)432 1071 y(*)i(without)e(this)h(marking)e(technique)g(also)i(the)g(h)i(in)e (expressions)d(G1)k(would)f(be)432 1171 y(*)h(changed)e(to)h(tau)h (despite)d(the)j(encapsulation)38 b(of)43 b(the)f(id)h(statement)d(in)i (a)i(module:)432 1270 y(id)e(dh*h^n?=tau^n;)432 1370 y(.sort)432 1569 y(Symbols)e(dy;)432 1669 y(*doing)h(the)h(midpoint)e (derivative)g(under)h(the)i(integral)d(sign)i(of)h(the)f(L)h(term)432 1768 y(*)g(the)f(expression)e(is)i(calculated)e(by)j(hand)432 1868 y(Local)e(deri1=-tau^-2*\(h*)o(\(h)o(-2*)o(ta)o(u\))o(*dx)o(\(i)o (\)+2)o(*\()o(h-t)o(au)o(\)*)o(dv\()o(i\))o(\)*L)o(;)432 1968 y(Local)g(deri2=)432 2067 y(tau^-2*\(2*h*\(2*t)o(au)o(-h)o(\)*a)o (\(k)o(\)*d)o(x\()o(i\))o(*F\()o(k\))o(+2*)o(\(h)o(-ta)o(u\))o(*a)o (\(k\))o(*d)o(v\(i)o(\)*)o(F\()o(k\)-)o(2*)o(F\(i)o(\)\))o(;)432 2167 y(Local)g(Si1=dy*deri1*G1;)432 2267 y(Local)g (Si2=dy*\(h/2\)*\(h-2)o(*t)o(au\))o(*d)o(er)o(i2*)o(G1)o(;)432 2366 y(*)i(doing)e(the)i(integral)432 2466 y(id)f(dy*h^n?=\(2*tau\)^\() o(n+1)o(\)/)o(\(n+)o(1\))o(;)432 2565 y(.sort)432 2765 y(*)h(expressions)c(for)j(the)h(motive)e(force)g(from)h(midpoint)e (derivative)g(of)j(the)432 2864 y(*)g(action)e(integral;)f(1)j(refers)e (to)i(the)f(contribution)d(of)k(the)f(Lagrangian,)432 2964 y(*)h(2)g(to)g(the)f(additional)e(forces:)432 3064 y(Local)h(MiAction=-3/\(4*ta)o(u\))o(*\(S)o(i1)o(+S)o(i2\))o(;)432 3263 y(*)i(expressions)c(for)j(the)h(motive)e(force)g(from)h (Lagrange's)e(differential)e(expression.)432 3362 y(Local)j (MiTau=\(\(dv\(i\)*\(dv)o(\(k)o(\)*a)o(\(k)o(\)+)o(dx\()o(k\))o(*\(v)o (\(k)o(\)+t)o(au)o(*a)o(\(k\))o(\)+)o(dt\))o(-d)o(x\()o(i\)\))o(*L)o (-F\()o(i\))o(\)*G)o(0t)o(au)o(;)432 3562 y(*)i(Comparing)d(the)i (expressions:)432 3661 y(Local)f(diff=MiAction-MiT)o(au)o(;)432 3861 y(Bracket)f(tau;)432 3960 y(Print)h(diff;)432 4060 y(*)i(development)c(should)i(start)h(with)f(tau^2,)h(notice)f(that)h (constant)e(terms)h(\(that)h(are)432 4159 y(*)h(claimed)e(not)h(to)h (exist\))e(would)g(appear)g(at)i(the)g(end)f(of)h(the)f(development)432 4359 y(.end;)388 4558 y(diff)g(=)562 4658 y(+)i(tau^2)d(*)i(\()87 b(-)43 b(1/10*dx\(i\)*dt^2*L)37 b(-)43 b(1/5*dx\(i\)*dx.v*d)o(v.a)o(*L) 37 b(-)43 b(1/5*dx\(i\)*)650 4757 y(dx.v*dt*L)d(-)j(1/10*dx\(i\)*dx.v)o (^2*)o(L)37 b(-)44 b(3/5*dx\(i\)*dx.a*)o(L)38 b(-)43 b(1/5*dx\(i\)*dv.a*)650 4857 y(dt*L)f(-)h(1/10*dx\(i\)*dv.a^)o(2*)o(L) 37 b(+)44 b(4/5*dx\(i\)*a.F)38 b(+)43 b(1/10*dv\(i\)*dt^3*)o(L)38 b(+)43 b(4/5*)650 4956 y(dv\(i\)*dx.v*dx.a)o(*L)37 b(+)43 b(3/5*dv\(i\)*dx.v*dv)o(.a*)o(dt)o(*L)37 b(+)43 b(3/10*dv\(i\)*dx.v*d)o (v.)o(a^2)650 5056 y(*L)f(+)i(1/5*dv\(i\)*dx.v*)o(a.F)37 b(+)43 b(3/10*dv\(i\)*dx.v*)o(dt)o(^2*)o(L)37 b(+)44 b(3/10*dv\(i\)*dx.v)o(^2*)650 5156 y(dv.a*L)d(+)i(3/10*dv\(i\)*dx.v^)o (2*)o(dt*)o(L)37 b(+)44 b(1/10*dv\(i\)*dx.v)o(^3*)o(L)37 b(+)43 b(4/5*dv\(i\)*)650 5255 y(dx.a*dv.a*L)c(+)k(4/5*dv\(i\)*dx.a*d)o (t*)o(L)38 b(+)43 b(1/5*dv\(i\)*dv.a*a)o(.F)37 b(+)43 b(3/10*dv\(i\)*)650 5355 y(dv.a*dt^2*L)c(+)k(3/5*dv\(i\)*dv.a*L)37 b(+)43 b(3/10*dv\(i\)*dv.a^)o(2*d)o(t*)o(L)37 b(+)44 b(1/10*dv\(i\)*)p Black 1821 5637 a FD(13)p Black eop %%Page: 14 14 14 13 bop Black Black 650 573 a Fa(dv.a^3*L)40 b(+)j (1/5*dv\(i\)*a.F*dt)37 b(-)43 b(1/10*F\(i\)*dt^2)38 b(-)43 b(1/5*F\(i\)*dx.v*dv.)o(a)37 b(-)44 b(1/)650 672 y(5*F\(i\)*dx.v*dt)37 b(-)44 b(1/10*F\(i\)*dx.v^)o(2)38 b(-)43 b(3/5*F\(i\)*dx.a)38 b(-)43 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y(dv\(i\)*dx.v*dv.a)o(*a.)o(F)37 b(-)44 b(2*dv\(i\)*dx.v*dv)o(.a*) o(dt)o(^2)o(*L)37 b(-)43 b(2*dv\(i\)*dx.v*dv.a)o(^2*)650 1768 y(dt*L)f(-)h(2/3*dv\(i\)*dx.v*d)o(v.)o(a^)o(3*L)37 b(+)43 b(2/5*dv\(i\)*dx.v*a)o(.F*)o(dt)37 b(-)43 b(2/3*dv\(i\)*)650 1868 y(dx.v*dt^3*L)c(-)k(1/2*dv\(i\)*dx.v^2)o(*d)o(x.a)o(*L)37 b(-)43 b(2*dv\(i\)*dx.v^2*d)o(v.a)o(*d)o(t*L)37 b(-)650 1968 y(dv\(i\)*dx.v^2*dv)o(.a^)o(2*)o(L)h(+)43 b(1/5*dv\(i\)*dx.v^2)o (*a)o(.F)37 b(-)43 b(dv\(i\)*dx.v^2*dt^2)o(*L)37 b(-)650 2067 y(2/3*dv\(i\)*dx.v^)o(3*d)o(v.)o(a*L)g(-)43 b(2/3*dv\(i\)*dx.v^3)o (*d)o(t*L)37 b(-)43 b(1/6*dv\(i\)*dx.v^4)o(*L)693 2167 y(-)g(dv\(i\)*dx.a*dv.a*d)o(t*L)37 b(-)43 b(1/2*dv\(i\)*dx.a*d)o(v.)o (a^2)o(*L)37 b(+)43 b(1/5*dv\(i\)*dx.a*)650 2267 y(a.F)f(-)h (1/2*dv\(i\)*dx.a*dt)o(^2)o(*L)37 b(+)43 b(2/5*dv\(i\)*dv.a*a.)o(F*)o (dt)37 b(-)43 b(2/3*dv\(i\)*dv.a*)650 2366 y(dt^3*L)e(+)i (1/5*dv\(i\)*dv.a^2)o(*a)o(.F)37 b(-)43 b(dv\(i\)*dv.a^2*dt^2)o(*L)37 b(-)43 b(2/3*dv\(i\)*)650 2466 y(dv.a^3*dt*L)c(-)k(1/6*dv\(i\)*dv.a^4)o (*L)37 b(+)43 b(1/5*dv\(i\)*a.F*dt^)o(2)37 b(-)43 b(1/10*F\(i\)*)650 2565 y(dt^3)f(-)h(4/5*F\(i\)*dx.v*dx)o(.a)37 b(-)43 b (3/5*F\(i\)*dx.v*dv)o(.a*)o(dt)37 b(-)43 b(3/10*F\(i\)*dx.v*)650 2665 y(dv.a^2)e(-)i(3/10*F\(i\)*dx.v*d)o(t^)o(2)38 b(-)43 b(3/10*F\(i\)*dx.v^2)o(*d)o(v.a)37 b(-)43 b(3/10*F\(i\)*)650 2765 y(dx.v^2*dt)d(-)j(1/10*F\(i\)*dx.v^)o(3)38 b(-)43 b(4/5*F\(i\)*dx.a*dv)o(.a)37 b(-)43 b(4/5*F\(i\)*dx.a*dt)693 2864 y(-)g(4/15*F\(i\)*dv.a)38 b(-)43 b(3/10*F\(i\)*dv.a*dt)o(^2)37 b(-)43 b(3/10*F\(i\)*dv.a^2*)o(dt)37 b(-)43 b(1/10*)650 2964 y(F\(i\)*dv.a^3)c(\))562 3163 y(+)44 b(tau^4)d(*)i(\()h (32/35*dx\(i\)*dx.)o(v*)o(dv.)o(a*)o(a.F)37 b(+)43 b (32/35*dx\(i\)*dx.v)o(*a)o(.F*)o(dt)37 b(+)43 b(16/35)650 3263 y(*dx\(i\)*dx.v^2*a)o(.F)37 b(+)43 b(16/35*dx\(i\)*dx.a*)o(a.F)37 b(+)43 b(32/35*dx\(i\)*dv.a)o(*a)o(.F*)o(dt)37 b(+)650 3362 y(16/35*dx\(i\)*dv.)o(a^2)o(*a)o(.F)g(+)43 b(16/35*dx\(i\)*a.F*d)o (t^)o(2)38 b(-)43 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%%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0105040630216--