Das Bunimowitsch-Stadion, benannt nach Leonid Abramowitsch Bunimowitsch, ist ein chaotisches Billard mit zwei Freiheitsgraden in Form eines Stadions. In rot und gelb sind zwei Trajektorien eines Teilchens eingezeichnet. Eine Eigenschaft von chaotischen Systemen ist, dass die Trajektorien mit nahezu identischen Startbedingungen stark auseinanderlaufen.[1]
English: billiards in a Bunimovich stadium, initial deviation is an angle of one degree Mathematica source code In[403]:= NN[v_]:=Sqrt[v[[1]]^2+v[[2]]^2]; Ang[v0_,va_,vb_]:=(va-v0).(vb-v0)/NN[va-v0]/NN[vb-v0]; 1st trajectory p0={0,0}; q0=\[Pi]/9; In[334]:= NSolve[(p0[[1]]+t Cos[q0]-1)^2+(p0[[2]]+t Sin[q0])^2==1,t] Out[334]= {{t\[Rule]0.},{t\[Rule]1.87939}} In[335]:= t0=1.8793852415718169`; p1=p0+t0{Cos[q0],Sin[q0]}; q1=-\[Pi]+(ArcCos[p1[[1]]-1]+q0); NSolve[p1[[2]]+t Sin[q1]\[Equal]-1,t] Out[338]= {{t\[Rule]1.89693}} In[180]:= t1=1.896927737347811; p2=p1+t1{Cos[q1],Sin[q1]}; q2=2\[Pi]-q1; NSolve[p2[[2]]+t Sin[q2]\[Equal]1,t] Out[183]= {{t\[Rule]2.3094}} In[202]:= t2=2.3094010767585043; p3=p2+t2{Cos[q2],Sin[q2]}; q3=2\[Pi]-q2; NSolve[(p3[[1]]+t Cos[q3]+1)^2+(p3[[2]]+t Sin[q3])^2==1,t] Out[205]= {{t\[Rule]0.200212},{t\[Rule]2.19472}} In[405]:= t3=2.194718395858327; p4=p3+t3{Cos[q3],Sin[q3]}; Solve[Ang[p4,p3,{-1,0}]\[Equal]Ang[p4,({Cos[t],Sin[t]}+p4),{-1,0}],t] From In[405]:= \!\(\* RowBox[{\(Power::"infy"\), \(\(:\)\(\ \)\), "\<\"Infinite expression \ \\!\\(1\\/0\\^2\\) encountered. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"Power::infy\\\"]\\)\"\>"}]\) From In[405]:= \!\(\* RowBox[{\(Solve::"ifun"\), \(\(:\)\(\ \)\), "\<\"Inverse functions are \ being used by \\!\\(Solve\\), so some solutions may not be found; use Reduce \ for complete solution information. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"Solve::ifun\\\"]\\)\"\>"}]\) Out[407]= {{t\[Rule]1.0472},{t\[Rule]1.19548}} In[328]:= q4=1.1954752520981573; NSolve[p4[[2]]+t Sin[q4]\[Equal]1,t] Out[329]= {{t\[Rule]2.04289}} In[440]:= t4=2.0428873267106815`; p5=p4+t4{Cos[q4],Sin[q4]}; q5=2\[Pi]-q4; 2 nd trajectory In[384]:= P0={0,0}; Q0=\[Pi]/9+\[Pi]/180; In[386]:= NSolve[(P0[[1]]+t Cos[Q0]-1)^2+(P0[[2]]+t Sin[Q0])^2==1,t] Out[386]= {{t\[Rule]0.},{t\[Rule]1.86716}} In[387]:= T0=1.8671608529944035`; P1=P0+T0{Cos[Q0],Sin[Q0]}; Q1=-\[Pi]+(ArcCos[P1[[1]]-1]+Q0); NSolve[P1[[2]]+t Sin[Q1]\[Equal]-1,t] Out[390]= {{t\[Rule]1.87331}} In[391]:= T1=1.8733090735550966`; P2=P1+T1{Cos[Q1],Sin[Q1]}; Q2=2\[Pi]-Q1; NSolve[P2[[2]]+t Sin[Q2]\[Equal]1,t] Out[394]= {{t\[Rule]2.24465}} In[395]:= T2=2.2446524752687225`; P3=P2+T2{Cos[Q2],Sin[Q2]}; Q3=2\[Pi]-Q2; NSolve[(P3[[1]]+t Cos[Q3]+1)^2+(P3[[2]]+t Sin[Q3])^2==1,t] Out[398]= {{t\[Rule]0.341712},{t\[Rule]2.23354}} In[419]:= T3=2.233539454680641`; P4=P3+T3{Cos[Q3],Sin[Q3]}; Solve[Ang[P4,P3,{-1,0}]\[Equal]Ang[P4,({Cos[t],Sin[t]}+P4),{-1,0}],t] From In[419]:= \!\(\* RowBox[{\(Power::"infy"\), \(\(:\)\(\ \)\), "\<\"Infinite expression \ \\!\\(1\\/0\\^2\\) encountered. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"Power::infy\\\"]\\)\"\>"}]\) From In[419]:= \!\(\* RowBox[{\(Solve::"ifun"\), \(\(:\)\(\ \)\), "\<\"Inverse functions are \ being used by \\!\\(Solve\\), so some solutions may not be found; use Reduce \ for complete solution information. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"Solve::ifun\\\"]\\)\"\>"}]\) Out[421]= {{t\[Rule]1.09956},{t\[Rule]1.76035}} In[423]:= Q4=1.786499618850784`; NSolve[(P4[[1]]+t Cos[Q4]+1)^2+(P4[[2]]+t Sin[Q4])^2==1,t] Out[424]= \!\({{t \[Rule] \(-2.961831812996791`*^-16\)}, {t \[Rule] 1.874216860919306`}}\) In[428]:= T4=1.874216860919306`; P5=P4+T4{Cos[Q4],Sin[Q4]}; Solve[Ang[P5,P4,{-1,0}]\[Equal]Ang[P5,({Cos[t],Sin[t]}+P5),{-1,0}],t] From In[428]:= \!\(\* RowBox[{\(Power::"infy"\), \(\(:\)\(\ \)\), "\<\"Infinite expression \ \\!\\(1\\/0\\^2\\) encountered. \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", \ ButtonStyle->\\\"RefGuideLinkText\\\", ButtonFrame->None, \ ButtonData:>\\\"Power::infy\\\"]\\)\"\>"}]\) From In[428]:= \!\(\* RowBox[{\(Solve::"ifun"\), \(\(:\)\(\ \)\), "\<\"Inverse functions are \ being used by \\!\\(Solve\\), so some solutions may not be found; use Reduce \ for complete solution information. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"Solve::ifun\\\"]\\)\"\>"}]\) Out[430]= {{t\[Rule]-1.35509},{t\[Rule]-0.642004}} In[432]:= Q5=-0.6420035368814776`; Illustration In[451]:= Show[Graphics[{ Thickness[.003], Line[{{-1,-1},{1,-1}}], Line[{{-1,1},{1,1}}], Circle[{1,0},1,{-\[Pi]/2,\[Pi]/2}], Circle[{-1,0},1,{\[Pi]/2,3\[Pi]/2}], RGBColor[254/256,194/256,0], Thickness[.0051],PointSize[.03], Line[{p0,p0+t0{Cos[q0],Sin[q0]}}], Line[{p1,p1+t1{Cos[q1],Sin[q1]}}], Line[{p2,p2+t2{Cos[q2],Sin[q2]}}], Line[{p3,p3+t3{Cos[q3],Sin[q3]}}], Line[{p4,p4+t4{Cos[q4],Sin[q4]}}], Line[{p5,p5+1.9{Cos[q5],Sin[q5]}}], Point[p5+1.9{Cos[q5],Sin[q5]}], RGBColor[188/256,30/256,71/256], Line[{P0,P0+T0{Cos[Q0],Sin[Q0]}}], Line[{P1,P1+T1{Cos[Q1],Sin[Q1]}}], Line[{P2,P2+T2{Cos[Q2],Sin[Q2]}}], Line[{P3,P3+T3{Cos[Q3],Sin[Q3]}}], Line[{P4,P4+T4{Cos[Q4],Sin[Q4]}}], Line[{P5,P5+1.9{Cos[Q5],Sin[Q5]}}], Point[P5+1.9{Cos[Q5],Sin[Q5]}] }],AspectRatio\[Rule]Automatic]