Energy Method

Konstantinos Efstathiou <k.efstathiou@dukekunshan.edu.cn>

This work by Konstantinos Efstathiou is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

In the following examples use the horizontal slider to change the value of the energy and see how the shape of the corresponding energy level set changes.

Example 1

Potential .

Manipulate[GraphicsColumn[{Plot[Evaluate[{((x^2)/2),h}],{x,-2,2},PlotRange->{{-2,2},{-2,2}},PlotStyle->{Directive[Thick,Red],Directive[Thick,Blue]},Frame->True,Axes->False,FrameTicks->None,FrameStyle->Directive[Black,18],FrameLabel->{None,U[x]},RotateLabel->False],Plot[Evaluate[{(Sqrt[2 h-(x^2)]),-(Sqrt[2 h-(x^2)])}],{x,-2,2},PlotRange->{{-2,2},{-2,2}},PlotStyle->Directive[Thick,Blue],Frame->True,Axes->True,FrameTicks->None,FrameStyle->Directive[Black,18],FrameLabel->{x,y},RotateLabel->False]},ImageSize->Large],{h,-2,2}]

Example 2

Potential .

Manipulate[GraphicsColumn[{
Plot[Evaluate[{-1/2x^2,h}],{x,-2,2},
PlotRange->{{-2,2},{-2,2}},
PlotStyle->{Directive[Thick,Red],Directive[Thick,Blue]},
Frame->True,Axes->False,FrameTicks->None,FrameStyle->Directive[Black,18],
FrameLabel->{None,},RotateLabel->False],
Plot[Evaluate[{Sqrt[2h+x^2],-Sqrt[2h+x^2]}],
{x,-2,2},PlotRange->{{-2,2},{-2,2}},
PlotStyle->Directive[Thick,Blue],
Frame->True,Axes->True,FrameTicks->None,FrameStyle->Directive[Black,18],
FrameLabel->{x,y},RotateLabel->False]
},ImageSize->Large],{h,-2,2}]

Example 3 (pendulum)

Potential .

Manipulate[GraphicsColumn[{
Plot[Evaluate[{-Cos[x],h}],{x,-Pi,Pi},
PlotRange->{{-Pi,Pi},{-2,2}},
PlotStyle->{Directive[Thick,Red],Directive[Thick,Blue]},
Frame->True,Axes->False,FrameTicks->None,FrameStyle->Directive[Black,18],
FrameLabel->{None,},RotateLabel->False],
Plot[Evaluate[{Sqrt[2h+2Cos[x]],-Sqrt[2h+2Cos[x]]}],
{x,-Pi,Pi},PlotRange->{{-Pi,Pi},{-4,4}},
PlotStyle->Directive[Thick,Blue],
Frame->True,Axes->True,FrameTicks->None,FrameStyle->Directive[Black,18],
FrameLabel->{x,y},RotateLabel->False]
},ImageSize->Large],{h,-2,2}]

Example 4

Potential .

Manipulate[Module[{U1},
U1=(-1/2x^2+1/4x^4+1/10x);
GraphicsColumn[{
Plot[Evaluate[{U1,h}],{x,-2,2},
PlotRange->{{-1.8,1.8},{-0.4,0.6}},
PlotStyle->{Directive[Thick,Red],Directive[Thick,Blue]},
Frame->True,Axes->False,FrameTicks->None,FrameStyle->Directive[Black,18],
FrameLabel->{None,},RotateLabel->False],
Plot[Evaluate[{Sqrt[2h-2U1],-Sqrt[2h-2U1]}],
{x,-2,2},PlotRange->{{-1.8,1.8},{-1.4,1.4}},
PlotStyle->Directive[Thick,Blue],
Frame->True,Axes->True,FrameTicks->None,FrameStyle->Directive[Black,18],
FrameLabel->{x,y},RotateLabel->False]
},ImageSize->Large]],{h,-0.4,0.6}]

Example 5

Potential .

Manipulate[Module[{U1},
U1=-(-1/2x^2+1/4x^4+1/10x);
GraphicsColumn[{
Plot[Evaluate[{U1,h}],{x,-2,2},
PlotRange->{{-1.8,1.8},{-0.4,0.6}},
PlotStyle->{Directive[Thick,Red],Directive[Thick,Blue]},
Frame->True,Axes->False,FrameTicks->None,FrameStyle->Directive[Black,18],
FrameLabel->{None,},RotateLabel->False],
Plot[Evaluate[{Sqrt[2h-2U1],-Sqrt[2h-2U1]}],
{x,-2,2},PlotRange->{{-1.8,1.8},{-1.4,1.4}},
PlotStyle->Directive[Thick,Blue],
Frame->True,Axes->True,FrameTicks->None,FrameStyle->Directive[Black,18],
FrameLabel->{x,y},RotateLabel->False]
},ImageSize->Large]],{h,-0.4,0.6}]