{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 } {CSTYLE "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 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 "M aple 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 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: wirf4.ms" }} {PARA 0 "" 0 "" {TEXT -1 103 "Autor: Komma \+ Datum: 28.3.94" }} {PARA 0 "" 0 "" {TEXT -1 81 "Thema: M\366glichkeit und Kausalit\344t, \+ Vergleich von gedachten und wirklichen Bahnen." }}{PARA 0 "" 0 "" {TEXT -1 202 "Approximation durch st\374ckweise gleichf\366rmige Beweg ung. Dazu wird die \"Bahn\" x(t) in gleiche Zeitschritte dt unterteilt und die zugeh\366rige \"mittlere Wirkung\" nach den xi-Werten variier t. " }}{PARA 0 "" 0 "" {TEXT -1 82 "Standard-Beispiele : gleichm\344\337ig beschleunigte Bewegung und harmonische Schwingung. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "M\366glichkeit und Kausalit\344t, Vergleich von gedachten und wirk lichen Bahnen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 710 "N\344herungsweise Berechnung der Orts-Zeit-Funktion x(t) zu vorgegebenem Potential mit Hilfe des Wirkungsprinzips. Der Bewegun gsablauf (von (0|0) nach (x1|t1)) wird in n Zeitschritte der Dauer dt= t1/n unterteilt, und in diesen Zeitschritten als gleichf\366rmig angen ommen. Daraus ergibt sich ein Gleichungssystem f\374r die Koordinaten \+ x[i] zu den Zeiten t[i]: die Summe der Wirkungen l\344ngs der Teilstre cken mu\337 minimal sein, d.h. die Ableitungen der \"Lagrangefunktion \" nach den Teilpunkten der Strecke m\374ssen verschwinden. Der Grenz \374bergang dt->0 bzw. n->oo mu\337 dann die \"wirkliche Bahn\" liefer n. Zum Vergleich mit dem kausalen Denken wird die \"exakte L\366sung\" der Newtonschen Bewegungsgleichung zur Verf\374gung gestellt." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "Sie Finde n in diesem Worksheet eine Reihe von Optionen:" }}{PARA 0 "" 0 "" {TEXT -1 190 "1.) Zwei Bewegungstypen: a) gleichm\344\337ig beschleuni gte Bewegung und b) harmonische Schwingung. (Weitere Bewegungstypen k \366nnen Sie durch eine entsprechende \304nderung des Potentials hinzu f\374gen.)" }}{PARA 0 "" 0 "" {TEXT -1 119 "2.) Die zugeh\366rigen \"p hysikalischen Parameter\" wie Masse, Fallbeschleunigung und Federkonst ante. (phys. Parameter-loop)" }}{PARA 0 "" 0 "" {TEXT -1 40 "3.) Die G \374te der Approximation (n-loop)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart; with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "\"Gemittelte\" Lagrangefunktion, d.h. sum((Ti-V i)dt,i=1..n) od. \"mittlere Wirkung\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "Lq:=Tq-Vq(); # Vq() wird als proc formuliert, damit \+ die \304nderungen aller Parameter weitergereicht werden" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#LqG,&%#TqG\"\"\"-%#VqG6\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "L\366schprozeduren" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "clear:= proc(x) local p;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "for p to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "x [p]:=evaln(x[p]); od; end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&clear GR6#%\"xG6#%\"pG6\"F*?(8$\"\"\"F-%\"nG%%trueG>&9$6#F,-%&evalnG6#F1F*F* F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 55 "\"mittlere kinetische Wirkung\". In Release 3 mit \+ Indizes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "n:='n': dt:='dt' : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Tq:=m/(2*dt)*sum((x[j ]-x[j+1])^2,j=0..n-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#TqG,$*&*& %\"mG\"\"\"-%$sumG6$*$),&&%\"xG6#%\"jGF)&F16#,&F3F)F)F)!\"\"\"\"#F)/F3 ;\"\"!,&%\"nGF)F)F7F)F)%#dtGF7#F)F8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Potentialtyp \+ (zum Umschalten, vorher zur\374ck zu restart)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "V:=proc(x)" }}{PARA 0 "" 0 "" {MPLTEXT 0 21 10 " 1/2*k*x^2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "m*g*x;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VGR 6#%\"xG6\"F(F(*(%\"mG\"\"\"%\"gGF+9$F+F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "g:=10: # nur fuer glm. beschl. Bew." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "#V(x);Vq();" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "\"mittlere potentielle Wirkung\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "if assigned(n) then clear(x) fi;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "n:='n': dt:='dt':" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#x[4];" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 37 "Vq:=proc() dt*sum(V(x[j]),j=0..n-1); " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#VqGR6\"F&F&F&*&%#dtG\"\"\"-%$sumG6$-%\"VG6#&%\"xG6#%\"jG/F3; \"\"!,&%\"nGF)F)!\"\"F)F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Kontrolle der \"Lagrange -Funktion\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "Lq;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&*&%\"mG\"\"\"-%$sumG6$*$),&&%\"xG6#%\"jG F'&F/6#,&F1F'F'F'!\"\"\"\"#F'/F1;\"\"!,&%\"nGF'F'F5F'F'%#dtGF5#F'F6*&F " 0 "" {MPLTEXT 1 0 46 "n:=10; t1:='t1': j:='j': k:='k':g:='g':m:='m':" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"nG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "clear(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"xG6#\"#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dt:=t1/n;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#dtG,$%#t1G#\"\"\"\"#5" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 4 "#Lq;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Aufstel len des Gleichungssystems (bis zu quadratischem Potential ist es linea r):" }}{PARA 0 "" 0 "" {TEXT -1 101 "(ggf. Formulierung mit Matrizen, \+ EW ... es ist eine Matrix mit 2 i.d. Diagonale und -1 daneben, o.\344. )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "kk:='kk': sys:=seq(dif f(Lq,x[kk])=0,kk=1..n-1):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "L \366sung des LGS (dauert f\374r n=53 etwas l\344nger als f\374r n=35 . ..):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sol:=solve(\{sys\}, \{seq(x[j],j=1..n-1)\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " #sol;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "\334bertragen der L\366s ung auf die x[i]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assign( sol);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "#x[2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "*********** phys. Parameter-loop ******** ***" }}{PARA 0 "" 0 "" {TEXT -1 36 "Parameter und Plot der Approximati on" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "m:=1/5: g:=2.5: k:=1/ 2: # m,g und k duerfen nicht float sein, " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 96 "#sonst Fehlermeldung von soly (s.u.), bzw. dsolve i ndirekt?\n# In R5 und R6 wieder floats erlaubt" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "t1:=5: #fuer grosse t1 problematisch" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "dt:=t1/n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "x[0]:=0: x[n]:=3:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 32 "Aufbau eines arrays f\374r den Plot" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 75 "#pkte:=array(1..n): # kann entfallen: implizite Dek laration in pkte[i]:=..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "for i to n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "pkte[i]:=x[i] :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "#pkte();" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Darst ellung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plota:=plot([[0,0 ],seq([i*dt,pkte[i]],i=1..n-1),[n*dt,x[n]]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "plota;" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Ausgabe der Zahlen zum Vergleich m it der exakten L\366sung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 " j:='j':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ITq:=evalf(Tq); \+ IVq:=evalf(Vq()); Sq:=evalf(Lq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $ITqG$\"++DJDm!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$IVqG$\"++DcE; !\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#SqG$!++DJS'*!\"*" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "exakte L\366sung" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "soly:=proc() local F;\nF:=-diff(V(y),y):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "rhs(dsolve(\{diff(y(t),t$2)=subs(y=y(t),F)/m,y(0)=0,y (t1)=x[n]\},y(t)));\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solyGR 6\"6#%\"FGF&F&C$>8$,$-%%diffG6$-%\"VG6#%\"yGF3!\"\"-%$rhsG6#-%'dsolveG 6$<%/-F36#\"\"!F?/-F36#%#t1G&%\"xG6#%\"nG/-F.6$-F36#%\"tG-%\"$G6$FM\" \"#*&-%%subsG6$/F3FKF+\"\"\"%\"mGF4FKF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "soly();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\" tG\"\"#\"\"\"#!\"&\"\"%*&#\"$P\"\"#?F(F&F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "plote:=plot(soly(),t=0..t1,color=black):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "#plote;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "display(\{plota,plote\});" }}{PARA 13 "" 1 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 2 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 20 "Zahlen zum Vergleich" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "IT:=evalf(int(m/2*diff(sol y(),t)^2,t=0..t1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ITG$\"+nmT!p '!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "IV:=evalf(int(V(so ly()),t=0..t1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#IVG$\"+LL3x;!\" )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "S:=IT-IV;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"SG$!+m;/35!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "komma@oe.uni- 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 }