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\begin{center}
  {\bf University of Kentucky, Physics 404G \\
    Homework \#3, Rev. A, due Tuesday, 2020-09-15 }
\end{center}

\vspace{.3 in}

{\bf 1.} The {\bf Cartesian, cylindrical}, and {\bf spherical} coordinate systems are by far the most common.  Do parts a),b),c), and e) for each of these coordinate systems, d) for cylindrical only:

{\bf \quad a)} Determine the coordinate transformation functions $q^{i}(\bs r)$ in terms of each of the other two systems.  {\it Hint: combine the simpler Cartesian-cylindrical and cylindrical-spherical transformations to obtain Cartesian-spherical.}

{\bf \quad b)} Pick a point $\bs r_{0}$ and illustrate the three coordinate isosurfaces $q^{i}(\bs r)=q^{i}_{0}$ (constant) that pass through $\bs r_{0}$, labeling all lengths and angles in your diagram.   For each coordinate $q^{i}$, identify the curve $\bs s(q^{i}; q^{j}_{0},q^{k}_{0})$ at the intersection of two surfaces of constant $q^{j}=q^{j}_{0}$ and $q^{k}=q^{k}_{0}$.

{\bf \quad c)} Calculate $\bs b_{i}=\partial\bs s/\partial q^{i}$ using $\bs{ds}=\bh x dx + \bh y dy + \bh z dz$.  Normalize $\bs b_{i}=\bh e_{i}h_{i}$ to find the {\it unit vectors} and {\it line element} $\bs{ds}=\bh e_{i}h_{i}dq^{i}$, [bonus: {\it area element} $\bs{da} = \frac12 \bs{ds} \times \bs{ds} = \bh e_{k}\,h_{i}h_{j}\,dq^{i}dq^{j}$ for $i,j,k$ cyclic], and [bonus: {\it volume element} $d\tau = \frac13\bs{ds}\cdot\bs{da} = h_{1}h_{2}h_{3}\, dq^{1}dq^{2}dq^{3}$ (for orthogonal coordinates)].  Note that the scale factors $h_{\theta}$ and $h_{\phi}$ for angular coordinates are just radius of curvature, according to the arc length formulae $ds_{\theta} = r d\theta$ and $ds_{\phi} = \rho d\phi$.

{\bf \quad d)} Invert cylindrical coordinates to obtain $(\rho,\phi,z)=f^{-1}(x,y,z)$.  Calculate the covariant basis $\bs b^{i}=\nabla q^{i} = \bh e_{i}/h_{i}$ to verify $\bs b_{i}\cdot \bs b^{j}=\delta_{i}^{j}$. [bonus: Same for spherical coordinates].

{\bf \quad e)} Calculate $g_{ij}=\bs b_{i}\cdot \bs b_{j}={\rm diag}(h_{1}^{2}, h_{2}^{2}, h_{3}^{2})$ and [bonus: $\bs \Gamma_{ij} = \partial \bs b_{i}/\partial q^{j}$].

\begin{wrapfigure}[3]{r}{1in}
  \includegraphics[width=1.2in, trim=.13in 0 0 .42in]{H03_fig1}
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{\bf 2.} The position of the lower mass $m_{b}$ of a {\bf double pendulum} is
most naturally parametrized by the non-orthogonal coordinate system
$(\alpha,\beta)$, illustrated below.  The position of the upper mass $m_{a}$ is
trivially parametrized by $\alpha$, but that of the lower mass $m_{b}$ depends on both angles $(\alpha,\beta)$.

{\bf \quad a)} Calculate and draw the $\alpha$ and $\beta$ coordinate lines, the contravariant basis vectors $\bs b_{i}$ [bonus: and covariant basis $\bs b^{i}$].  Use the components of $\bs{ds}$ to obtain the \mbox{velocity $\bs v=\dot{\bs r} = \bs b_{i}\dot q^{i}$ of the lower mass}.

{\bf \quad b)} Calculate the metric $g_{ij}=\bs b_{i}\cdot \bs b_{j}$ and thus $ds^{2}=g_{ij}dq^{i}dq^{j}$ to obtain the kinetic energy $T_{b}=\frac12 m_{b} v^{2}=\frac12m_{b}g_{ij}\dot q^{i}\dot q^{j}$ of the lower mass.  Raise the indices of $\bs b_{i}$ with the inverse~metric~$g^{ij}$ to obtain the covariant basis $\bs b^{i}$. [bonus: Compare with the direct \mbox{calculation of $\bs b^{i}=\nabla q^{i}$ from a)]}.

{\bf \quad c)} Calculate the connection vectors $\bs \Gamma_{ij}=\partial\bs b_{i}/\partial q^{j}$.  Compare the covariant components $\Gamma_{ijk}=\bs \Gamma_{ij}\cdot \bs b_{k}$ of $\bs \Gamma_{\alpha\alpha}$ with those obtained directly from the metric $\Gamma_{ijk}=\frac12(g_{ik,j}+g_{jk,i}-g_{ij,k})$.  Use $\Gamma_{ijk}$ to obtain the covariant components of acceleration $\bs A= \dot{\bs v} = \bs b_{k}\ddot q^{k}+ \bs \Gamma_{ij} \dot q^{i}\dot q^{j}$ of the mass $m_{b}$.

{\bf \quad d)} Calculate the potential of gravity $V(\alpha,\beta)=mgh$ for both masses $m_{a}$ and $m_{b}$ in terms of $(\alpha,\beta)$ to construct the Lagrangian $\mathcal L=T-V$.  Compare the two equations of motion for $\alpha$ and $\beta$, in the covariant components of $\bs F=m\bs A$ using $F_{i}=\partial V/\partial q^{i}$ and $A_{i}$ from above, with the direct calculation using Lagrange's equations.

{\bf \quad e)} Computer exercise: numerically integrate the four first order equations of motion in the variables $\bs q=(\alpha, \beta, \dot\alpha, \dot\beta)$ for $a=2~\q{m}$ and $b=1~\q{m}$, $m_{a}=m_{b}=1~\q{kg}$, starting from initial conditions $\bs q_{0}=(.1,\; .2,\; 0,\; 0)$, in units of [rad] and [rad/s].  Vary the initial value $\beta_{0}$ to discover two {\it modes} of pure periodic motion, and calculate their [eigen]frequencies.  Increase $\alpha_{0}$ and $\beta_{0}$ until the motion becomes {\it chaotic} (unpredictable).  {\it Hint: start with the code for this \href{http://www.pa.uky.edu/~crawford/phy404_fa19/pendulum.m}{simple pendulum}.}

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