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Let $M$ be the configuration space of the spherical pendulum.
- Let $c : [0,1] \to M$ be a smooth curve. Show directly that the work done by the stiff rod along this curve is zero.
- Determine the kinetic and potential energies of a bob of mass $m$, assuming the stiff rod has zero mass.
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Let $V$ be an $n$-dimensional vector space, and let $(\tenalg{*}{V},\tensorproduct)$, $(\symalg{*}{V},\symmetricproduct)$ and $(\extalg{*}{V},\extproduct)$ be the tensor, symmetric and exterior algebras of $V$.
- Let $v_1, \ldots, v_n$ be a basis of $V$. Show that, for each natural number $k$, the set $$\set{ v_{i_1} \tensorproduct \cdots \tensorproduct v_{i_k} \st 1 \leq i_1, \ldots, i_k \leq n}$$ is a basis of $\tenalg{k}{V}$.
- Prove the analogous facts for $\extalg{k}{V}$ and $\symalg{k}{V}$.
- Let us say that a $k$-tensor $x$ is irreducible if there are $a_1, \ldots, a_k \in V$ such that $x = a_1 \tensorproduct \cdots \tensorproduct a_k$. Show that for $k = 2$, there are reducible (= not irreducible) tensors. [Remark: this is true for all $k \geq 2$.]
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Show that the previous fact is true for both symmetric and skew-symmetric tensors, too.
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Let us continue with the notation of the previous question. Say that a linear transformation $L : \tenalg{*}{V} \to \tenalg{*}{V}$ is a derivation if $$L(x \tensorproduct y) = L(x) \tensorproduct y + x \tensorproduct L(y)$$ for all $x, y \in \tenalg{*}{V}$.
- Let $A : V \to V$ be a linear transformation and let $\exp(tA) = I + tA + \frac{1}{2} t^2 A^2 + \cdots$ be the exponential. Show that $\ddt{t}{\exp(t A)}|_{t=0}$ induces a derivation of $\tenalg{*}{V}$.
- Show that there is a bijection between linear transformations $V \to V$ and derivations of $\tenalg{*}{V}$. [Hint: show that a derivation is uniquely determined by its action on $V$.]
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Let $I$ be the $n \times n$ identity matrix and
\begin{align*}
J &=
\begin{bmatrix}
0 & -I\\I & 0
\end{bmatrix}
&&
J : \R^n \oplus \R^n \to \R^n \oplus \R^n,
\\
\Symp{\R^{2n}} &= \set{ X \in \Mat{2n}{\R} \st X'JX=J }.
\end{align*}
Prove that $\Symp{\R^{2n}}$ is a submanifold of $\Mat{2n}{\R}$. [Bonus: show it is a group, too.]
The spherical pendulum. The bob (in green) moves freely about the pivot $P$.
