Pendulum. Non Linear solution.

Phase Space Plots

Renan Cabrera
March 2002
renanbo@hotmail.com

The simple pendulum problem in mechanics can be solved using the Sn Jacobi Elliptic function. In this notebook we make some nice plots of the phase space of the system using this solution. The nicer plot is the phase space plotted over the surface of a cylinder.

Implementation.

First we define the dimensionless parameter K, as a function of the energy ϵ , Pendulum Length L and mass m. (In this case we also chose gravity = 10)

K[ϵ_,L_,m_] = Sqrt[2*m*g*L/ϵ]/. g->10;

The solution of the angle as a function of time is:

Theta[t_,ϵ_,L_,m_] =
2*ArcSin[ JacobiSN[ t /Sqrt[2*m*L^2/ϵ],K[ϵ,L,m]^2 ] ];

The generalized momentum of the angle is:

DTheta[t_,ϵ_,L_,m_] = m*L^2*D[Theta[t,ϵ,L,m],t] ;

Now we can make some plots with the  angle and its momentum. Each trajectory 
for a different energy.

2D Plot

Show[Map[
ParametricPlot[
{Theta[t,#,1,1],DTheta[t,#,1,1]} ,{t,0,2*Pi},PlotPoints->40]&
,Range[0.2,26,4]
]]

3D Plot

Some small changes and a little bit of art.

Show[
Show[Map[
ParametricPlot3D[
{4*Cos[c=1.005*Theta[t,#,1,1]],
4*Sin[1.01*Theta[t,#,1,1]],Abs[DTheta[t,#,1,1]],Hue[c/Pi]}
,{t,0,1.2*Pi},PlotPoints->60
,ViewPoint->{6.160, -4.556, 4.722},Boxed->False,Axes->False]&
,Range[0.1,30,2]
]]
,
Show[Map[
ParametricPlot3D[
{4*Cos[c=1.005*Theta[t,#,1,1]],
4*Sin[1.01*Theta[t,#,1,1]],-Abs[DTheta[t,#,1,1]],Hue[c/Pi]}
,{t,0,1.2*Pi},PlotPoints->60
,ViewPoint->{6.160, -4.556, 4.722},Boxed->False,Axes->False]&
,Range[0.1,30,2]
]]
]