MTH-696A: Topics in Geometric Mechanics Assignment 6 (by Dr. Leo Butler)
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-
Let $L : T\Torus{2} \to \R$ be the kinetic energy of the double pendulum (see assignment 3).
-
Compute the Euler-Lagrange equations
\begin{align*}
\ddt{t}{}\didi{L}{\dot{x}_i} &= \didi{L}{x_i}, && x_1=x, x_2=y,
\end{align*}
for this Lagrangian.
-
Introduce the coordinates $x=r, y=r+s \pmod 2\pi$. Show that
\begin{align*}
\ddt{t}{}\didi{L}{\dot{r}} &= 0.
\end{align*}
 |
| Planar double pendulum. |
-
Let $\Liegp{G}$ be a Lie group and $\liealg{g}$ be its Lie
algebra. Suppose that $\Liegp{G}$ acts linearly on $\R^n$;
suppose also that $G$ leaves invariant the open subset $M$. Let
$L : TM \to \R$ be a smooth Lagrangian function that is constant
on the $\Liegp{G}$ orbits:
\begin{align*}
L(gx, g\dot{x}) &= L(x, \dot{x}), && \forall g \in \Liegp{G}, (x, \dot{x}) \in TM.
\end{align*}
For each $\xi \in \liealg{g}$, define
\begin{align*}
f_{\xi}(x, \dot{x}) &= \ip{\didi{L}{\dot{x}}}{\xi x}.
\end{align*}
Use the Euler-Lagrange equations for $L$ to show that
\begin{align*}
\ddt{t}{} f_{\xi} & = 0.
\end{align*}
-
Let $\Liegp{G} = \Orth{3}$, $M=\R^3-\set{0}$ and
\begin{align*}
L &= \frac{1}{2} \norm{\dot{x}}^2 - \frac{1}{\norm{x}}.
\end{align*}
Show that the functions $f_{\xi}$ you obtain can be regarded as the components of angular momentum.
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