MTH-696A: Topics in Geometric Mechanics Assignment 3 (by Dr. Leo Butler)
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Let $M$ be the configuration space of the planar double pendulum. See figure 1.
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The configuration space of the planar double pendulum is the set of pairs of angles $(x, y) \mod 2 \pi$. What is the standard name for this surface?
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Determine the kinetic energy of the system depicted. The bobs of masses $m$ and $M$ move on the inflexible rods of lengths $l$ and $L$ and the suspension point $P$ is fixed.
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| Planar double pendulum. |
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Let $M$ be an $n$-manifold and let $g$ be a Riemannian metric on $M$.
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Suppose that $x_i$ and $y_j$ are smooth coordinate systems on $M$, and $x = \phi(y)$ for a smooth diffeomorphism $\phi$. Show that
\[ \D{x_1} \wedge \cdots \wedge \D{x_n} = \det \Phi \cdot \D{y_1} \wedge \cdots \wedge \D{y_n}, \] where
\[ \Phi = \begin{bmatrix} \displaystyle\didi{\phi_i}{y_j} \end{bmatrix} \] is the Jacobian matrix of the coordinate transformation.
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In a local system of coordinates $x_i$ on $M$, one can define an $n$-form
\[ \Omega = \sqrt{\det \mathbf{g}} \D{x_1} \wedge \cdots \wedge \D{x_n}, \] where
\[ \mathbf{g} = [ g_{ij} ] \]
is the matrix of $g = \sum_{i, j=1}^n g_{ij} \D{x_i} \cdot \D{x_j}$ in the local coordinates. Use the first part to show that if $M$ is orientable, then $\Omega$ is a tensor field on $M$. What goes wrong if $M$ is not orientable?
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The $n$-form defined above is non-degenerate. Prove this.
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The $n$-form $\Omega$ is called a Riemannian volume form. Compute the Riemannian $2$-form for the sphere in $\R^3$ of radius $r$. [Hint: it is easiest to use spherical coordinates on $\R^3$.]
