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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 $latex \boldsymbol{J}$.