Home » OCN-ED » The Double Pendulum # The Double Pendulum

Just to give you an idea about what inter­ac­tive learn­ing will be like on OCN Acad­e­my — here's an exam­ple applet illus­trat­ing the sub­space of allow­able motions of a dou­ble pen­du­lum. Sub­space of allow­able motions? What's that sup­posed to be?! Well, stay tuned to learn what this is all about as the actu­al primer on the mat­ter will be pub­lished dur­ing the upcom­ing weeks.

### Positions

$latex \boldsymbol{r} = \left[\begin{array}{c} \boldsymbol{r}_1 \\ \hline \boldsymbol{r}_2 \end{array}\right] = \left[\begin{array}{c} l_1 \sin \alpha_1 \\ ‑l_1 \cos \alpha_1 \\ \hline l_1 \sin \alpha_1 + l_2 \sin \alpha_2 \\ ‑l_1 \cos \alpha_1 — l_2 \cos \alpha_2 \end{array}\right]$

### Velocities

$latex \boldsymbol{v} = \left[\begin{array}{c} \boldsymbol{v}_1 \\ \hline \boldsymbol{v}_2 \end{array}\right] = \left[\begin{array}{c} l_1 \cos \alpha_1 \\ l_1 \sin \alpha_1 \\ \hline l_1 \cos \alpha_1 \\ l_1 \sin \alpha_1 \end{array}\right] \dot{\alpha_1} + \left[\begin{array}{c} 0 \\ 0 \\ \hline l_2 \cos \alpha_2 \\ l_2 \sin \alpha_2 \end{array}\right] \dot{\alpha_2}$

$latex \boldsymbol{v} = \left[\begin{array}{c} \boldsymbol{v}_1 \\ \hline \boldsymbol{v}_2 \end{array}\right] = \left[\begin{array}{cc} \boldsymbol{J_{11}} & \boldsymbol{J_{12}} \\ \boldsymbol{J_{21}} & \boldsymbol{J_{22}} \end{array}\right] \begin{bmatrix} \dot{\alpha_1} \\ \dot{\alpha_2} \end{bmatrix}$

### Subspace of Allowable Motions

The Sub­space of allow­able motions is spec­i­fied by the columns of $latex \boldsymbol{J}$.