{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "M aple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 10 0 0 255 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "c International Thomson Pu blishing Bonn 1995 filename: wirf3.ms" }} {PARA 0 "" 0 "" {TEXT -1 103 "Autor: Komma \+ Datum: 28.3.94" }} {PARA 0 "" 0 "" {TEXT -1 54 "Thema: Wirkungsfunktion des harmonischen Oszillators:" }}{PARA 0 "" 0 "" {TEXT -1 75 " N\344herung sl\366sung durch schwaches Extremum der Wirkungsfunktion," }}{PARA 0 " " 0 "" {TEXT -1 75 " wenn die Ortsfunktion als Polynom n-t en Grades angesetzt wird." }}{PARA 0 "" 0 "" {TEXT -1 68 " \+ Vergleich mit der Reihenentwicklung der exakten L\366sung." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 194 "Die Bewegung des ha rmonischen Oszillators kann ebenfalls \374ber das schwache Minimum der Wirkungsfunktion n\344herungsweise bestimmt werden, wenn wir zum Bei spiel ein Polynom n-ten Grades ansetzen. " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "T:=m/2*v^2: v:=diff(x(t),t): L:=T-V: S:=int(L,t=t0..t1): t0:=0:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 131 "Das Polynom n-ten Grades durch die zwei Punkte (0|0) u nd (t1|x1) bauen wir zur Abwechslung mit Hilfe des Punkt-Operators (ca t) auf:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "xx:=proc(t) local xx,i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "xx:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "for i to \+ n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "xx:=xx+a||i*t^i;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "RETURN(xx);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxGR6#%\" tG6$F$%\"iG6\"F*C%>8$\"\"!?(8%\"\"\"F1%\"nG%%trueG>F-,&F-F1*&(%\"aGF0F 1)9$F0F1F1-%'RETURNG6#F-F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 101 "Ein Koeffizient (z.B. a1) l\344\337t sich durch die Bedingung x(t1)=x1 und die anderen Koef f. ausdr\374cken: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "as1: =solve(xx(t1)=x1,a1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$as1G,$*&,6 *&%#a5G\"\"\")%#t1G\"\"&F*F**&%#a2GF*)F,\"\"#F*F**&%#a3GF*)F,\"\"$F*F* *&%#a4GF*)F,\"\"%F*F**&%#a9GF*)F,\"\"*F*F**&%#a6GF*)F,\"\"'F*F**&%#a7G F*)F,\"\"(F*F**&%#a8GF*)F,\"\")F*F**&%$a10GF*)F,\"#5F*F*%#x1G!\"\"F*F, FOFO" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Unser CAS liefert uns auf Knopfdruck die Weg-Zeit -Funktion zu den aufgestellten Bedingungen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "x:=t->subs(a1=as1,xx(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x(t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,6*&*&,6 *&%#a5G\"\"\")%#t1G\"\"&F)F)*&%#a2GF))F+\"\"#F)F)*&%#a3GF))F+\"\"$F)F) *&%#a4GF))F+\"\"%F)F)*&%#a9GF))F+\"\"*F)F)*&%#a6GF))F+\"\"'F)F)*&%#a7G F))F+\"\"(F)F)*&%#a8GF))F+\"\")F)F)*&%$a10GF))F+\"#5F)F)%#x1G!\"\"F)% \"tGF)F)F+FNFN*&F.F))FOF0F)F)*&F2F))FOF4F)F)*&F6F))FOF8F)F)*&F(F))FOF, F)F)*&F>F))FOF@F)F)*&FBF))FOFDF)F)*&FFF))FOFHF)F)*&F:F))FOF " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Und mit dem qudratischen Potential" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k:='k':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "V:=1/2*k*x(t)^2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "erhalten wir f\374r die Wirkung einen etwas l\344ngeren A usdruck ..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Ss:=simplify (S);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#SsG,$*&,fw**%\"kG\"\"\"%#a9 GF*%$a10GF*)%#t1G\"#@F*!)3[n[*,\")C6F*%\"mGF*F+F*F,F*F**,\")C#)3YF*F9F*F)F*F+F*F5F*F6*, \")?_EXF*F9F*F)F*F,F*%#a7GF*F6*,\"*S=qi)F*)F.\"#=F*F;F*F,F*F5F*F**,\") ]FzUF*FCF*F)F*F,F*%#a6GF*F6*,\")+CEWF*FCF*F)F*F+F*F@F*F6*,\"*gRx9)F*)F .\"#+VF*FLF*F)F*F5F*F@F*F6*,\")&f\"\\RF*FLF*F)F*F,F*%#a 5GF*F6*,\"))pQ2%F*)F.\"#;F*F)F*F5F*FGF*F6*,\")%)zpQF*FYF*F)F*F+F*FVF*F 6*,\")%))=\\$F*FYF*F)F*F,F*%#a4GF*F6*,\"*!ox$)pF*FYF*F;F*F,F*FGF*F**, \"*#>O\\uF*FYF*F;F*F+F*F@F*F**,\"*+;7l'F*)F.\"#:F*F;F*F+F*FGF*F**,F[oF *F`oF*F;F*F5F*F@F*F**,\"*S%4')fF*F`oF*F;F*F,F*FVF*F**,\")C-pPF*F`oF*F) F*F5F*FVF*F6*,\")owEGF*F`oF*F)F*F,F*%#a3GF*F6*,\")]1ERF*F`oF*F)F*F@F*F GF*F6*,\")SQEMF*F`oF*F)F*F+F*FinF*F6*,\"*+#\\niF*)F.\"#9F*F;F*F5F*FGF* F**,\"*Sy-t&F*F`pF*F;F*F+F*FVF*F**,\"*SA\\$[F*F`pF*F;F*F,F*FinF*F**,\" )?r!z\"F*F`pF*F)F*F,F*%#a2GF*F6*,\")oJyFF*F`pF*F)F*F+F*FioF*F6*,\")?FQ OF*F`pF*F)F*FVF*F@F*F6*,\")CmULF*F`pF*F)F*F5F*FinF*F6*,\"*SE=V&F*)F.\" #8F*F;F*F5F*FVF*F**,\"*S))=\\$F*FaqF*F;F*F,F*FioF*F**,\"*+9)>eF*FaqF*F ;F*F@F*FGF*F**,\"*?^el%F*FaqF*F;F*F+F*FinF*F**,\")+ejeF*)F.\"#7F*F)F*F,F*%#x1GF*F**,\")+BLKF*FaqF*F)F*FinF*F@F*F6* ,\")%3$GF 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F[xF*F**,\")o([O#F*F_tF*F)F*FVF*FioF*F6*,\")!oJ<&F*)F.\"\"*F*F)F*F@F*F crF*F**,\")]h;;F*F_tF*F)F*FGF*FhpF*F6*,\"*?Wfu\"F*FfxF*F;F*F@F*FhpF*F* *,\"*+2*4HF*FfxF*F;F*FGF*FioF*F**,FdqF*FfxF*F;F*FVF*FinF*F**,\")qBC:F* FfxF*F)F*FVF*FhpF*F6*,\")=$R8#F*FfxF*F)F*FinF*FioF*F6*,\")]%)\\[F*FbwF *F)F*FGF*FcrF*F**,\")S9MWF*)F.\"\"(F*F)F*FVF*FcrF*F**,\")+n&Q\"F*FbwF* F)F*FinF*FhpF*F6*,\"*S'[gEF*FbwF*F;F*FVF*FioF*F**,\"*+/Gm\"F*FbwF*F;F* FGF*FhpF*F**,\"*gDzK#F*FgyF*F;F*FinF*FioF*F**,\"*S]>b\"F*FgyF*F;F*FVF* FhpF*F**,F]xF*FiwF*F)F*FinF*FcrF*F**,\")G'R;\"F*FgyF*F)F*FioF*FhpF*F6* ,\"*ObnR\"F*FiwF*F;F*FinF*FhpF*F**,\")3!R5$F*)F.\"\"&F*F)F*FioF*FcrF*F **,F`uF*FjzF*F;F*FioF*FhpF*F**,\")!Q*R>F*F`xF*F)F*FhpF*FcrF*F***\"*?n# p?F*F_tF*F;F*FjvF*F***\")g?%\\#F*F)F*FeuF*)F.\"#AF*F6**\")Cq6$*F*FiwF* F;F*FfwF*F*F*F.F6#F*F`z" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Wir st ellen wieder unser Gleichungssystem auf und lassen es l\366sen." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 6 "#x(t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sys:=seq(diff(Ss,a||j), j=2..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "#sys;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 10 "#a.(2..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "# Probleme in R4!erst Werte einsetzen!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "# R5 packt es wieder, aber langsamer als R3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# schnelle L\366sung in R6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sol:=solve(\{sys\},\{a||( 2..n)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "#sol;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 413 "Wenn Sie sich die L\366sung xs im Worksheet ausgeben lassen, werden Sie sicher meine Begeisterung f\374 r dieses CAS teilen: Probleme, die von der Struktur her einfach sind, \+ jedoch einen immensen Rechenaufwand bedeuten w\374rden, wollte man sie von Hand l\366sen, sind ein gefundenes Fressen f\374r Maple. Hier kan n man zuschauen, wie Quantit\344t in Qualit\344t umschl\344gt, und - w as ebenso wichtig ist - man kann damit weiterarbeiten." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "xs:=subs(sol,x(t)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Nun k\366nnen wir Zahlen f\374r die Feder konstante, die Masse und den Endpunkt einsetzen" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 29 "k:=2: m:=1/4: t1:=2: x1:=3:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "und f\374r eine vergleichende Darstellung die exakte L\366sung der Newton-DGL parat stellen." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "exakt:=rhs(dsolve(\{diff(y(t),t$2)=-k/m*y (t),y(0)=0,y(t1)=3\},y(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&exa ktG,$*&-%$sinG6#,$*&-%%sqrtG6#\"\"#\"\"\"%\"tGF0F/F0*(-F(6#*$F,F0F0-%$ cosGF4F0,&*$)F6F/F0F/F0!\"\"F0F;#\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "simplify(exakt,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&-%$sinG6#,$*&-%%sqrtG6#\"\"#\"\"\"%\"tGF.F-F.*(-F&6 #*$F*F.F.-%$cosGF2F.,&*$)F4F-F.F-F.!\"\"F.F9#\"\"$\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "#plot(\{exakt\},t=-1..t1+1,-10..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{exakt,xs\},t=-1..t1+1,-10..1 0);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 273 "Der Vergleic h unseres N\344herungspolynoms mit der exakten L\366sung ist aber nich t \"nur\" graphisch m\366glich. Man kann mit Hilfe der Reihenentwicklu ng der exakten L\366sung auch untersuchen, welche Koeffizienten eine A bweichung verursachen bzw. was eine Erh\366hung der Ordnung bewirkt." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(series(exakt,t,10)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"tG$!+\"*\\cZ9!\")\"\"\"$\"+b m3I>F'\"\"$$!+?mM?x!\"*\"\"&$\"+9Aaq9F.\"\"($!+#zNRj\"!#5\"\"*-%\"OG6# F(\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(xs);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,6%\"tG$!+&y6vW\"!\")*&$\"+6kf\\9!#6\" \"\")F$\"\"#F,!\"\"*&$\"+\\?\"H%>F'F,)F$\"\"$F,F,*&$\"+Pf]5a!#5F,)F$\" \"%F,F/*&$\"+cl*zW'!\"*F,)F$\"\"&F,F/*&$\"+)>****y\"F>F,)F$\"\"'F,F/*& $\"+#\\'4-IF>F,)F$\"\"(F,F,*&$\"+NYJ\"f(F8F,)F$\"\")F,F/*&$\"+F&H^9)!# 7F,)F$\"\"*F,F,*&$\"+R?]u6F+F,)F$\"#5F,F," }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "komma@oe.un i-tuebingen.de" }}}}{MARK "0 2 0" 7 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }