{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 "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 0 1 } 1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }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 "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }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 "R 3 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 94 "c International Thomson Pu blishing 1995 filename: fourw.ms" }}{PARA 0 "" 0 "" {TEXT -1 103 "Autor: Komma \+ Datum: 28.3.94" }} {PARA 0 "" 0 "" {TEXT -1 22 "Index:Wirkungsfunktion" }}{PARA 0 "" 0 " " {TEXT -1 64 "Thema: Wirkungsprinzip, schwaches Extremum der Wirkungs funktion." }}{PARA 0 "" 0 "" {TEXT -1 34 "Wurf und harmonischer Oszill ator: " }}{PARA 0 "" 0 "" {TEXT -1 61 "N\344herungsl\366sung durch Bes timmung des schwachen Extremums der " }}{PARA 0 "" 0 "" {TEXT -1 75 "W irkungsfunktion, wenn die Ortsfunktion als \"Fourierreihe\" angesetzt \+ wird. " }}{PARA 0 "" 0 "" {TEXT -1 34 "Vergleich der Reihenentwicklung en." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "T:=m/2*v^2 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,$*&%\"mG\"\"\")%\"vG\"\"#F (#F(F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "v:=diff(x(t),t); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%%diffG6$-%\"xG6#%\"tGF+" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "L:=T-V(x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,&*&%\"mG\"\"\")-%%diffG6$-%\"xG6#%\"tGF0 \"\"#F(#F(F1-%\"VG6#F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "S:=int(L,t=t0..t1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG-%$in tG6$,&*&%\"mG\"\"\")-%%diffG6$-%\"xG6#%\"tGF3\"\"#F+#F+F4-%\"VG6#F0!\" \"/F3;%#t0G%#t1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "H:=T+V( x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG,&*&%\"mG\"\"\")-%%dif fG6$-%\"xG6#%\"tGF0\"\"#F(#F(F1-%\"VG6#F-F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "t0:=0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "\334 berlagerung von n Oberschwingungen (n>2) , Kurve durch (0|0) und (t1|x 1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "xx:=proc(t) local xl;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "xl: =0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "for i to n do" }}{PARA 0 "" 0 "" {TEXT -1 71 "mit dem Cosinus bekommt man die Randbed. ohne gleich f. Bewegung herein." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "xl:=xl+a||i* sin(i*Pi*t/t1)+b||i*cos(i*Pi*t/t1); " }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "RETURN(xl);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 7 "" 1 "" {TEXT -1 60 "Warning, `i` is implicitly declared local to pr ocedure `xx`\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#xxGR6#%\"tG6$%#xl G%\"iG6\"F+C%>8$\"\"!?(8%\"\"\"F2%\"nG%%trueG>F.,(F.F2*&(%\"aGF1F2-%$s inG6#*&*(F1F2%#PiGF29$F2F2%#t1G!\"\"F2F2*&(%\"bGF1F2-%$cosGF " 0 "" {MPLTEXT 1 0 6 "xx(t);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#,:*&%#a1G\"\"\"-%$sinG6#*&*&%#PiGF&% \"tGF&F&%#t1G!\"\"F&F&*&%#b1GF&-%$cosGF)F&F&*&%#a2GF&-F(6#,$F*\"\"#F&F &*&%#b2GF&-F3F7F&F&*&%#a3GF&-F(6#,$F*\"\"$F&F&*&%#b3GF&-F3F@F&F&*&%#a4 GF&-F(6#,$F*\"\"%F&F&*&%#b4GF&-F3FIF&F&*&%#a5GF&-F(6#,$F*\"\"&F&F&*&%# b5GF&-F3FRF&F&*&%#a6GF&-F(6#,$F*\"\"'F&F&*&%#b6GF&-F3FenF&F&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 139 "Zwei Koeffizienten (z.B. b1 und b 2) lassen sich mit Hilfe der Bedingungen x(0)=0 und x(t1)=x1 durch die anderen Koeffizienten ausdr\374cken: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "solb:=s olve(\{xx(0)=0,xx(t1)=x1\},\{b1,b2\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solbG<$/%#b1G,(%#b3G!\"\"%#b5GF**&#\"\"\"\"\"#F.%#x1GF.F*/%#b 2G,(F0#F.F/%#b6GF*%#b4GF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "x:=t->subs(solb,xx(t)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xGR6#%\"tG6\"6$%)operatorG%&a rrowGF(-%%subsG6$%%solbG-%#xxG6#9$F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x(t);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#,:*&%#a1G\"\"\"-%$sinG6#*&*&%#PiGF&% \"tGF&F&%#t1G!\"\"F&F&*&,(%#b3GF/%#b5GF/*&#F&\"\"#F&%#x1GF&F/F&-%$cosG F)F&F&*&%#a2GF&-F(6#,$F*F6F&F&*&,(F7#F&F6%#b6GF/%#b4GF/F&-F9F=F&F&*&%# a3GF&-F(6#,$F*\"\"$F&F&*&F2F&-F9FHF&F&*&%#a4GF&-F(6#,$F*\"\"%F&F&*&FCF &-F9FPF&F&*&%#a5GF&-F(6#,$F*\"\"&F&F&*&F3F&-F9FXF&F&*&%#a6GF&-F(6#,$F* \"\"'F&F&*&FBF&-F9FjnF&F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "V(x); " }}{PARA 0 "" 0 "" {TEXT -1 5 "x(0);" }}{PARA 0 "" 0 "" {TEXT -1 44 " lineares Potential / qudratisches Potential:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "k:='k':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "V:=proc(x)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "m*g*x;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "#1/2*k*x^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VGR6#%\"xG 6\"F(F(*(%\"mG\"\"\"%\"gGF+9$F+F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#g:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#t 1:=2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "#S;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Ss:=simplify(S,power);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#SsG,$*&*&%\"mG\"\"\",ho*&)%#PiG\"\"$F))%#b5G \"\"#F)\"'g.O*(\"&Dt\"F)F,F))%#x1GF1F)F)*(\"'+WbF)F,F))%#b6GF1F)F)*(\" 'SZ7F)F,F))%#a3GF1F)F)*(\"'+'Q\"F)F,F))%#b3GF1F)F)*(\"'+lMF)F,F))%#a5G F1F)F)*(\"'g*)\\F)F,F))%#a6GF1F)F)*(\"&gQ\"F)F,F))%#a1GF1F)F)**\"&%y9F ))F-F1F)F6F)%#a4GF)!\"\"**\"&?R(F)FQF)FNF)F6F)FS**\"&#zLF)FQF)FNF)F:F) F)**FLF)F,F)FBF)F6F)F)**\"&Sa&F)F,F)F6F)%#b4GF)FS**\"&?x#F)F,F)FBF)F0F )F)**\"')3t%F)FQF)%#a2GF)FBF)FS**\"'_F?F)FQF)FBF)FJF)F)**\"'K'4(F)FQF) F:F)F>F)FS**\"'s1aF)FQF)FBF)FRF)F)**\"(Kk-\"F)FQF)FenF)F>F)FS**\"(++#> F)FQF)FFF)F:F)FS**\"&o&HF)FQF)FNF)FenF)F)**\"'++))F)FQF)FenF)FFF)F)** \"(#*H\\\"F)FQF)F0F)FJF)F)**\"(o:E\"F)FQF)F0F)FRF)FS**\"'?zLF)FQF)F0F) FjnF)FS**FZF)F,F)F6F)F:F)FS**\"'!)36F)F,F)FenF)F:F)F)**FLF)F,F)F0F)F6F )F)**\"'cI8F)FQF)F6F)F>F)F)**\"%/&*F)FQF)F6F)FJF)FS**\"&gp$F)FQF)F6F)F jnF)FS**\"&+G&F)FQF)F6F)FFF)F)*(\"'gF)FS*(\"'+sFF)F,F))FenF1F)F)F)F)*&FcqF)F-F)FS#F)FZ" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x(t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,:*&%#a1G\"\"\"-%$sinG6#*&*&%#PiGF&%\"tGF&F&%#t1G!\"\"F&F&*&,(%# b3GF/%#b5GF/*&#F&\"\"#F&%#x1GF&F/F&-%$cosGF)F&F&*&%#a2GF&-F(6#,$F*F6F& F&*&,(F7#F&F6%#b6GF/%#b4GF/F&-F9F=F&F&*&%#a3GF&-F(6#,$F*\"\"$F&F&*&F2F &-F9FHF&F&*&%#a4GF&-F(6#,$F*\"\"%F&F&*&FCF&-F9FPF&F&*&%#a5GF&-F(6#,$F* \"\"&F&F&*&F3F&-F9FXF&F&*&%#a6GF&-F(6#,$F*\"\"'F&F&*&FBF&-F9FjnF&F&" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Notwendige Bedingung f\374r sch waches Extremum: die partiellen Ableitungen der Wirkung nach den Formv ariablen m\374ssen verschwinden." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sys:=seq(diff(Ss,a||j),j=1.. n),seq(diff(Ss,b||j),j=3..n); # n>2" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%$sysG6,,$*&*&%\"mG\"\"\",,*&)%#PiG\"\"$F*%#a1GF*\"&?x#*(\"&?R(F*)F. \"\"#F*%#x1GF*!\"\"*(\"&#zLF*F4F*%#b6GF*F**(\"&o&HF*F4F*%#b4GF*F**(\"' !)36F*)%#t1GF5F*%\"gGF*F7F*F**&FAF*F.F*F7#F*\"&Sa&,$*&*&F)F*,**&F4F*%# b3GF*!')3t%*(\"'?zLF*F4F*%#b5GF*F7*(\"&gp$F*F4F*F6F*F7*(F?F*F-F*%#a2GF *F*F*F**&FAF*F.F*F7FD,$*&*&F)F*,,*&F-F*%#a3GF*\"'![\\#*(\"'K'4(F*F4F*F :F*F7*(\"(Kk-\"F*F4F*F=F*F7*(\"'cI8F*F4F*F6F*F**(FQF*F@F*FBF*F7F*F**&F AF*F.F*F7FD,$*&*&F)F*,**&F4F*F6F*!&%y9*(\"'s1aF*F4F*FKF*F**(\"(o:E\"F* F4F*FOF*F7*(\"'?NWF*F-F*%#a4GF*F*F*F**&FAF*F.F*F7FD,$*&*&F)F*,,*&F-F*% #a5GF*\"'+Ip*(\"(++#>F*F4F*F:F*F7*(\"'++))F*F4F*F=F*F**(\"&+G&F*F4F*F6 F*F**(\"&w@#F*F@F*FBF*F7F*F**&FAF*F.F*F7FD,$*&*&F)F*,**&F-F*%#a6GF*\"' ?z***(\"'_F?F*F4F*FKF*F**(\"(#*H\\\"F*F4F*FOF*F**(\"%/&*F*F4F*F6F*F7F* F**&FAF*F.F*F7FD,$*&*&F)F*,.*&F-F*FKF*\"'+sF*(\"&gQ\"F*F-F*F6F*F**(F1F *F-F*FOF*F**(\"')3t%F*F4F*FSF*F7*(FdqF*F4F*FaqF*F**(FeoF*F4F*FjoF*F*F* F**&FAF*F.F*F7FD,$*&*&F)F*,.*&F-F*F6F*!&Sa&*(FinF*F4F*FZF*F7*(F " 0 "" {MPLTEXT 1 0 37 "#sys:=seq(diff(Ss,a.j),j=1.. n); # n=2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "sol:=solve(\{s ys\},\{a||(1..n),b||(3..n)\}); #n>2" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #>%$solG<,/%#a1G,$*&,.*&)%#PiG\"\"%\"\"\"%#x1GF/!0s9oKeQo\"**\"0kg$4f_ 'p#F/)F-\"\"#F/)%#t1GF5F/%\"gGF/!\"\"*(\"/+ur%)HH5F/)F-\"\"'F/F0F/F/** \"/v&4&y[M;F/F6F/F8F/F,F/F/*(\"0+cmwpqf'F/F4F/F0F/F/*(\"1[g#Q\"[#)p5F/ F6F/F8F/F/F/*&,(\"/7Xuv&f0'F/*&\"/+#>oUx_\"F/F4F/F9*&\"-vVT!eE*F/F,F/F /F/)F-\"\"$F/F9#\"$+\"\"$T%/%#b5G,$*&*&F0F/,(*$F,F/\",Dc`+h$*&\"-+;s4, tF/F4F/F9\".ks!pP*o$F/F/F/,(FV\".DJLRw*[*&\"/+/JI;o%*F/F4F/F9\"0;[:PoR d%F/F9#!#L\"#9/%#b4G,$*&,,*&F\"*(\"/+3AKb2?F/F,F/F0F/F 9*(\"/7*y'ov+#)F/F4F/F0F/F/**\",+Sm#\\fF/F4F/F6F/F8F/F/*(\"-!ovR#\\aF/ F6F/F8F/F9F/*&F4F/FEF/F9#F/F\\o/%#a6G,$*&*&F0F/,(FV\"-v.;tfN*&\".gh*fB yqF/F4F/F9\"/K))3sV=NF/F/F/*&F-F/FenF/F9#\"$v#FO/%#b3G,$*&*&F0F/,(FV\" ,Dhstk#*&\"-gdwJn^F/F4F/F9\"._F!e9@DF/F/F/FenF9#!$D\"F\\o/%#a5G,$*&,.* &F4F/F0F/!0+GL)[`)H$*(\"0CI\"pS7\\`F/F6F/F8F/F9*(\"0++_NEIr#F/F,F/F0F/ F9**\"0+%Q?bWlP[#Q7 F/F6F/F8F/F,F/F9F/*&FEF/FKF/F9#!#W\"&v$=/%#a4G,$*&*&F0F/,(FV\"-v)4bo?) *&\"/!3!fC(H_\"F/F4F/F9\"/kw " 0 "" {MPLTEXT 1 0 35 "#sol:=solve(\{sys\},\{a.(1..n)\}); \+ #n=2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "x(t);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#,:*&%#a1G\"\"\"-%$sinG6#*&*&%#PiGF&%\"tGF&F&%#t1 G!\"\"F&F&*&,(%#b3GF/%#b5GF/*&#F&\"\"#F&%#x1GF&F/F&-%$cosGF)F&F&*&%#a2 GF&-F(6#,$F*F6F&F&*&,(F7#F&F6%#b6GF/%#b4GF/F&-F9F=F&F&*&%#a3GF&-F(6#,$ F*\"\"$F&F&*&F2F&-F9FHF&F&*&%#a4GF&-F(6#,$F*\"\"%F&F&*&FCF&-F9FPF&F&*& %#a5GF&-F(6#,$F*\"\"&F&F&*&F3F&-F9FXF&F&*&%#a6GF&-F(6#,$F*\"\"'F&F&*&F BF&-F9FjnF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "L\366sung des Gleichungssystems in x(t) einsetzen:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "xs:=pro c() subs(sol,x(t)); end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xsGR6\" F&F&F&-%%subsG6$%$solG-%\"xG6#%\"tGF&F&F&" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 5 "xs();" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,:*&*&,.*&) %#PiG\"\"%\"\"\"%#x1GF+!0s9oKeQo\"**\"0kg$4f_'p#F+)F)\"\"#F+)%#t1GF1F+ %\"gGF+!\"\"*(\"/+ur%)HH5F+)F)\"\"'F+F,F+F+**\"/v&4&y[M;F+F2F+F4F+F(F+ F+*(\"0+cmwpqf'F+F0F+F,F+F+*(\"1[g#Q\"[#)p5F+F2F+F4F+F+F+-%$sinG6#*&*& F)F+%\"tGF+F+F3F5F+F+*&,(\"/7Xuv&f0'F+*&\"/+#>oUx_\"F+F0F+F5*&\"-vVT!e E*F+F(F+F+F+)F)\"\"$F+F5#\"$+\"\"$T%*&,(*&*&F,F+,(*$F(F+\",Dhstk#*&\"- gdwJn^F+F0F+F5\"._F!e9@DF+F+F+,(FW\".DJLRw*[*&\"/+/JI;o%*F+F0F+F5\"0;[ :PoRd%F+F5#\"$D\"\"#9*&*(#\"#LF]oF+F,F+,(FW\",Dc`+h$*&\"-+;s4,tF+F0F+F 5\".ks!pP*o$F+F+F+FfnF5F+*&#F+F1F+F,F+F5F+-%$cosGFBF+F+*&**#F\\o\"$Z\" F+,(FW\"-DKOu)G%*&\"._v/\\wz(F+F0F+F5\"/K))3sV=NF+F+F,F+-FA6#,$FCF1F+F +*&F)F+FfnF+F5F+*&,(F,#F+F1*&#\"#A\"#@F+*&,,*&F0F+F,F+\".%Q7XbCW*(\"-? 6Y&RV#F+F2F+F4F+F5*(\"-+;;oX!)F+F(F+F,F+F5**\",+5_VY#F+F0F+F2F+F4F+F+* (FcoF+F8F+F,F+F+F+*&F0F+FGF+F5F+F5*&#F+F]oF+*&,,*&F8F+F,F+\".Dcn<8>\"* (\"/+3AKb2?F+F(F+F,F+F5*(\"/7*y'ov+#)F+F0F+F,F+F+**\",+Sm#\\fF+F0F+F2F +F4F+F+*(\"-!ovR#\\aF+F2F+F4F+F5F+*&F0F+FGF+F5F+F5F+-FjoFepF+F+*&#\"#? \"%B8F+*&*&,.F'\".ob?/s()**(\".+E#QS9CF+F8F+F,F+F+**\"0![!e9XqV\"F+F0F +F2F+F4F+F+**\".v31)[!3*F+F2F+F4F+F(F+F5*(\"0+GL)[`)H$F+F0F+F,F+F5*(\" 0CI\"pS7\\`F+F2F+F4F+F5F+-FA6#,$FCFNF+F+*&FGF+FMF+F5F+F5*&#F\\oF]oF+*& *(F,F+FVF+-FjoF^tF+F+FfnF5F+F5*&**#\"#D\"#)*F+F,F+,(FW\"-v)4bo?)*&\"/! 3!fC(H_\"F+F0F+F5\"/kwP[#Q7F+F2F+F4F+F(F+F5F+-FA 6#,$FC\"\"&F+F+*&FGF+FMF+F5F+F5*&#FaoF]oF+*&*(F,F+FboF+-FjoF\\wF+F+Ffn F5F+F5*&**#\"$v#FQF+F,F+,(FW\"-v.;tfN*&\".gh*fByqF+F0F+F5FcpF+F+-FA6#, $FCF9F+F+*&F)F+FfnF+F5F+*&*(#F]qF^qF+F`qF+-FjoF^xF+F+*&F0F+FGF+F5F+" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 21 "simplify(xs(),power);" }}{PARA 0 "" 0 "" {TEXT -1 25 "F ederkonstante und Masse:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "k:=2: m:=1/4: g:=10:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Endpunk t:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "t1:=2:x1:=3:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "xs();" }}{PARA 0 "" 0 "" {TEXT -1 37 "N \344herungsl\366sung (schwaches Extremum) " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "plot(xs(),t);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Exakte L\366sung der Newton-DGL:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "soly:=proc() rhs(dsolve( \{diff(y(t),t$2)=-diff(V(y),y)/m,y(0)=0,y(t1)=x1\},y(t))); end;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solyGR6\"F&F&F&-%$rhsG6#-%'dsolveG6 $<%/-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F5\"\"#,$*&-F06$-%\"VG6#F3F3\"\"\"% \"mG!\"\"FC/-F36#\"\"!FG/-F36#%#t1G%#x1GF2F&F&F&" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 10 "Vergleich:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(\{soly(),xs()\},t=0..t1+1);" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "soly();" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*$)%\"tG\"\"#\"\"\"!\"&*&#\"#BF'F(F&F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "diff(V(y),y);" }}{PARA 0 "" 0 "" {TEXT -1 8 "V(x(t));" }}{PARA 0 "" 0 "" {TEXT -1 48 "Vergleich des Pol ynoms mit der Reihenentwicklung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf(series(soly(),t,10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #+'%\"tG$\"++++]6!\")\"\"\"$!\"&\"\"!\"\"#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "evalf(series(xs(),t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"tG$\"+9%G/:\"!\")\"\"\"$!+[=y'3&!\"*\"\"#$\"*(=a< \\F+\"\"$$!*CX>/(F+\"\"%$!*U8BB#F'\"\"&-%\"OG6#F(\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 285 "Anmerkungen: Der L\366sungsansatz wird m it der Periode t1 gemacht. Deshalb erh\344lt man auch bei quadratische m Potential (Oszillator) nicht die exakte L\366sung, man kann aber die Reihenentwicklungen vergleichen. Nat\374rlich kann man mit t1 (im Arg ument der Winkelfunktionen) experimentieren ..." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "komma@ oe.uni-tuebingen.de" }}}}{MARK "0 3 0" 7 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }