Primer neharmoničnega periodičnega gibanja

Obravnavali bomo gibanje materialne točke z maso m pod vplivom potenciala

                                                                                                      V(x) = A((e^(-2αx))-2(e^(-αx))),
                                                                                                      
kjer sta A in α pozitivni konstanti. Izkaže se, da je gibanje omejeno za energije -A≤E<0. Točko v začetnem položaju (Subscript[x, 1]) (za katerega je E = V((Subscript[x, 1])) < 0) spustimo, da se začne
gibati pod vplivom sile, dane s potencialom V.

A=2;alpha=1;x1=-0.64;

V[x_]=A(E^(-2*alpha*x)-2E^(-alpha*x));

graf=Plot[V[x],{x,-3,5},PlotStyle->Blue,PlotRange->{{-2,5},{-3,6}},Ticks->None];

tekst=Graphics[{
{Dashed,Line[{{-0.53,0},{-0.53,V[-0.53]}}],Line[{{1.23,0},{1.23,V[1.23]}}]},
{Red,Dashed,Line[{{-0.53,-1},{1.23,-1}}]},
{Red,PointSize[0.015],Point[{-0.53,V[-0.53]}],Point[{1.23,V[1.23]}]},
{PointSize[0.015],Point[{-0.53,0}],Point[{1.23,0}],Point[{0,-1}]},
Text[Style["V(x)",18],{-1.5,5.5}],
Text[Style["(Subscript[x, 1])",18],{-0.53,0.6}],Text[Style["(Subscript[x, 2])",18],{1.23,0.6}],Text[Style["E",18],{0.25,-0.5}]
},
AspectRatio->Automatic];

Show[graf,tekst,PlotRange->{{-2,5},{-3,6}},AxesOrigin->{0,0},AxesStyle->Arrowheads[{0.04}]]

Med gibanjem se ohranja skupna energija točke

							1/2m (((dx/dt))^2) + V(x) = E.                                                                                 
                                                                                                                                                                                  
To enačbo je težko rešiti analitično, pa tudi z numeričnim reševanjem je problem v točkah, kjer se smer hitrosti obrne. To je v točkah (Subscript[x, 1]) in (Subscript[x, 2]), za kateri je V((Subscript[x, 1])) = V((Subscript[x, 2])) = E.
Lažje je numerično rešiti odvajano verzijo te enačbe

                                                                             m(d^2)x/(dt^2) = -V'(x).

x2=x/.Solve[V[x]==V[x1],x][[2]];

NDSolve[{x''[t]==-V'[x[t]],x'[0]==0,x[0]==x1},x,{t,0,60}];

resitev[t_]=First[Evaluate[x[t]/.%]];

Plot[resitev[t],{t,0,60},PlotStyle->Blue,PlotRange->{{-2,60},{-1.5,3}},AxesStyle->Arrowheads[{0.04}],AxesLabel->{"t","x"},LabelStyle->{12,Bold}]

Z grafa lahko preberemo, da je gibanje periodično, ni pa harmonično. Poglejmo si še simulacijo gibanja točke.

Animate[
graf=Plot[V[x],{x,-3,5},PlotStyle->Blue];
tekst=Graphics[{
{Dashed,Line[{{x1,0},{x1,V[x1]}}],Line[{{x2,0},{x2,V[x2]}}]},
{Red,Dashed,Line[{{x1,V[x1]},{x2,V[x1]}}]},
{Red,PointSize[0.01],Point[{x1,V[x1]}],Point[{x2,V[x2]}]},
{Blue,PointSize[0.015],Point[{resitev[t],V[resitev[t]]}]},
{Black,PointSize[0.015],Point[{resitev[t],0}]},
{PointSize[0.01],Point[{x1,0}],Point[{x2,0}],Point[{0,V[x1]}]},
Text[Style["V(x)",18],{-1.5,5.5}],
Text[Style["(Subscript[x, 1])",18],{x1,0.4}],Text[Style["(Subscript[x, 2])",18],{x2,0.4}],Text[Style["E",18],{0.15,V[x1]-0.3}]
},
AspectRatio->Automatic];
Show[graf,tekst,PlotRange->{{-2,5},{-3,6}},ImageSize->{700,450},AxesOrigin->{0,0},AxesStyle->Arrowheads[{0.04}]],
{{t,0,"Čas (s)"},0,60,ControlType->Trigger,AnimationRate->1}]

Pri majhnih odmikih je gibanje skoraj harmonično.

A=2;alpha=1;x1=-0.1;

V[x_]=A(E^(-2*alpha*x)-2E^(-alpha*x));

x2=x/.Solve[V[x]==V[x1],x][[2]];

NDSolve[{x''[t]==-V'[x[t]],x'[0]==0,x[0]==x1},x,{t,0,60}];

resitev[t_]=First[Evaluate[x[t]/.%]];

Plot[{resitev[t],x1*Cos[Sqrt[V''[0]]*t]},{t,0,60},PlotStyle->{Blue,{Blue,Dashed}},PlotRange->{{-2,15},{-0.12,0.15}},AxesStyle->Arrowheads[{0.04}],AxesLabel->{"t","x"},LabelStyle->{12,Bold}]

Animate[
graf=Plot[V[x],{x,-3,5},PlotStyle->Blue];
tekst=Graphics[{
{Dashed,Line[{{x1,0},{x1,V[x1]}}],Line[{{x2,0},{x2,V[x2]}}]},
{Red,Dashed,Line[{{x1,V[x1]},{x2,V[x1]}}]},
{Red,PointSize[0.01],Point[{x1,V[x1]}],Point[{x2,V[x2]}]},
{Blue,PointSize[0.015],Point[{resitev[t],V[resitev[t]]}]},
{Black,PointSize[0.015],Point[{resitev[t],0}]},
{PointSize[0.01],Point[{x1,0}],Point[{x2,0}],Point[{0,V[x1]}]},
Text[Style["V(x)",18],{-1.5,5.5}],
Text[Style["(Subscript[x, 1])",18],{x1,0.4}],Text[Style["(Subscript[x, 2])",18],{x2,0.4}],Text[Style["E",18],{0.15,V[x1]-0.3}]
},
AspectRatio->Automatic];
Show[graf,tekst,PlotRange->{{-1,2},{-3,6}},ImageSize->{700,450},AxesOrigin->{0,0},AxesStyle->Arrowheads[{0.04}]],
{{t,0,"Čas (s)"},0,60,ControlType->Trigger,AnimationRate->1}]