2-2

Large amplitude motion of a simple pendulum

Nicholas Wheeler
Reed College Physics Department
2 October 2009

A bob (mass m) moves on a circular arc of radius R. Let θ denote angular displacement from the bottom of the arc. Take the zero of gravitational potential to be at the center of the arc. Then

so the Lagrangian reads

and Lagrange's equations become

which can be written

with (ω^2)=g/R.

DSolve[θ''[t]+(ω^2)Sin[θ[t]]==0,θ[t],t]

We are interested in the solution with

but here Mathematica 7 is of no help (though Mathematica 4 did promptly supply a result):

DSolve[{θ''[t]+(ω^2)Sin[θ[t]]==0,θ[0]==0, θ'[0]=(Subscript[ω, 0])},θ[t],t]

So we tinker: graphic exploration

Manipulate[Plot[JacobiAmplitude[α t,β],{t,0,20}, PlotRange->{-5,5}],{α,1,2},{β,0,2}]

suggests that we can achieve

by setting

We do so, and looking to θdot

D[2 JacobiAmplitude[(1/2) (Sqrt[(2 (ω^2)+C[1]) (((t))^2)]),((4 (ω^2))/(2 (ω^2)+C[1]))],t]

find that to achieve θdot[0]=(Subscript[ω, 0]) we must

Limit[%,t->0,Direction->-1]

Solve[(Sqrt[2 (ω^2)+C[1]])==(Subscript[ω, 0]),C[1]]

set C[1]=(Subscript[ω, 0]^2)-2 (ω^2). We do so

2 JacobiAmplitude[(1/2) (Sqrt[(2 (ω^2)+C[1]) (((t+C[2]))^2)]),((4 (ω^2))/(2 (ω^2)+C[1]))]/.{C[1]->((Subscript[ω, 0])^2)-2(ω^2),C[2]->0}

and (adjusting notation (Subscript[ω, 0])->Ω for Mathematica's convenience) obtain

θ[t_,ω_,Ω_]:=2 JacobiAmplitude[(1/2) Ω t,(((2ω/Ω))^2)]

Manipulate[Plot[θ[t,1,Ω],{t,0,20},PlotRange->{-π,6π}],{Ω,1.9825,2.0175,0.005}]

As the figure demonstrates, the motion is vibrational (bounded oscillatory) if Ω > ω,  and librational (unbounded undulatory) if Ω > ω .

Here is a description of energy as a function of θ and θdot:

((2ℰ)/(m (R^2)))=((θdot)^2)-(g/R)Cos[θ]

I use that information to plot isoenergertic curves in  θ-θdot space:

Table[((θdot)^2)-1Cos[θ]==ℰ,{ℰ,-.4,2.4,.2}]//Chop

ContourPlot[{((θdot)^2)-Cos[θ]==-0.4`,((θdot)^2)-Cos[θ]==-0.2`,((θdot)^2)-Cos[θ]==0,((θdot)^2)-Cos[θ]==0.2`,((θdot)^2)-Cos[θ]==0.4`,((θdot)^2)-Cos[θ]==0.6000000000000001`,((θdot)^2)-Cos[θ]==0.8`,((θdot)^2)-Cos[θ]==1.`,((θdot)^2)-Cos[θ]==1.2000000000000002`,((θdot)^2)-Cos[θ]==1.4000000000000001`,((θdot)^2)-Cos[θ]==1.6`,((θdot)^2)-Cos[θ]==1.8`,((θdot)^2)-Cos[θ]==2.`,((θdot)^2)-Cos[θ]==2.2`,((θdot)^2)-Cos[θ]==2.4000000000000004`},{θ,-2π,2π},{θdot,-3,3},Frame->False,Axes->True,Ticks->{{-2π,-π,0,π,2π},{}}]