Question: Phase PLane for the simple pendulum

Hi all,

> deq := diff(theta(t), t, t) = -g*sin(theta(t))/L;

                           2                              
                          d                g sin(theta(t))
                  deq := ---- theta(t) = - ---------------
                            2                     L       
                          dt                              
> diff(theta(t), t) = v(t);

                              d                 
                             --- theta(t) = v(t)
                              dt                
> diff(%, t);

                            2                     
                           d               d      
                          ---- theta(t) = --- v(t)
                             2             dt     
                           dt                     
so 
> diff(v(t), t) = -g*sin(theta(t))/L;
r
                         d           g sin(theta(t))
                        --- v(t) = - ---------------
                         dt                 L       
> (Diff(v, theta))*(Diff(theta, t)) = -g*sin(theta)/L;

                  /   d     \ / d       \     g sin(theta)
                  |------- v| |--- theta| = - ------------
                  \ dtheta  / \ dt      /          L      
> (Diff(v, theta))*v = -g*sin(theta)/L;

                       /   d     \       g sin(theta)
                       |------- v| v = - ------------
                       \ dtheta  /            L      
> Int(v, v = vo .. v) = -g*(Int(sin(theta), theta = `θo` .. theta))/L;

                                /  /theta                   \
                                | |                         |
                              g | |        sin(theta) dtheta|
                  /v            | |                         |
                 |              \/θo                  /
                 |   v dv = - -------------------------------
                 |                           L               
                /vo                                          
> eq1 := value(%);

                  1  2   1   2     g (cos(θo) - cos(theta))
           eq1 := - v  - - vo  = - ------------------------------
                  2      2                       L               
> eq2 := solve(%, v);

                                                              (1/2)  
              /  /  2                                       \\       
              \L \vo  L - 2 g cos(θo) + 2 g cos(theta)//       
       eq2 := -----------------------------------------------------, 
                                        L                            

                                                           (1/2)
           /  /  2                                       \\     
           \L \vo  L - 2 g cos(θo) + 2 g cos(theta)//     
         - -----------------------------------------------------
                                     L                          
> sol1 := vp = sqrt(L*(vo^2*L-2*g*cos(`θo`)+2*g*cos(theta)))/L;

                                                                  (1/2)
                  /  /  2                                       \\     
                  \L \vo  L - 2 g cos(θo) + 2 g cos(theta)//     
      sol1 := vp = -----------------------------------------------------
                                            L                          
> sol2 := vn = -sqrt(L*(vo^2*L-2*g*cos(`θo`)+2*g*cos(theta)))/L;

                    /  /  2                                       \\     
                    \L \vo  L - 2 g cos(θo) + 2 g cos(theta)//     
      sol2 := vn = -  -----------------------------------------------------
                                             L                          

Taking 0<&theta;o<Pi, for a sufficiant vo, there is the case witch the pendulum will go all the way around at a speed v at theta=2*Pi. I am not able to plot that situation.

Am I doing something wrong?

Mario