\documentclass[12pt]{article} % BB Jan 22 2009.
\usepackage{epsfig}
\textheight 9.0in \textwidth  6.0in \topmargin -0.4in \headsep 0.4in  
\oddsidemargin  0.25in  \evensidemargin 0.25in  

\def\sqr#1#2{{\vcenter{\hrule height.#2pt
   \hbox{\vrule width.#2pt height#1pt \kern#1pt
   \vrule width.#2pt} \hrule height.#2pt}}}
\def\square{{\mathchoice\sqr64\sqr64\sqr{4.2}3\sqr{3.0}3}}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\pagestyle{empty}

\begin{document}

\begin{center}
{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 4}\\%
\vspace {3mm} (January 27, 2009) \\
\end{center}
\setcounter{equation}{0}

\begin{description}

%%%%%%%%%%%%%%%%%%%%%%%%%%
\item{(15)} A homogeneous cylinder of mass $M$ roles down and 
inclined plane. Use a Lagrange multiplier to find the constraint
force, which prevent the cylinder from sliding (suitable coordinates
will be given in class). Due January 26 in class (5 points).

\item{(16)} Derive (due January 28 in class, 8 points)
$$ \frac{d}{dt}\,\left( L - \sum_j \dot{q}_j\,
   \frac{\partial L}{\partial \dot{q}_j} \right) = 0~~{\rm from}~~
   \delta_t L = \frac{\partial L}{\partial t}\,\delta t = 0\ .$$
Assume a bilinear kinetic Energy
$$ T = \sum_{j,k} a_{ij}\,\dot{q}_j\dot{q}_k~~{\rm and\ prove}~~
   \sum_i \frac{T}{\dot{q}_i} = 2 T\ .$$

Read M\&T chapter 7.9 (due January 28). 

\item{(17)} Legendre transformation: Define the Hamiltonian by
$$ H = \left(\sum_j\dot{q}_j\,\frac{\partial L}{\partial\dot{q}_j} 
  -L \right)\ {\rm and\ the\ generalized\ momentum\ by}\
   p_j=\frac{\partial L}{\partial\dot{q}_j}\ .                  $$
Show that the Hamiltonian is a function of $q_j$ and $p_j$ only:
$H=H(q_j,p_j)$. Then derive Hamilton's equations of motion. 
Hint: Calculate $dH$. Due January 30 in class (6 points).

\item{(18)} Make a table of the nine scalar products of the cartesian unit 
vectors $\widehat{x},\, \widehat{y},\, \widehat{z}$ with the spherical 
unit vectors $\widehat{r},\, \widehat{\theta},\, \widehat{\phi}$.
Due February 2 before class (6 points).

\item{(19)} Calculate the velocity 
$$ \vec{v}={d\ \over dt}\vec{r}={d\ \over dt}(r\widehat{r}) $$
algebraically along the following lines: 

(a) Calculate $${d\ \over dt} \widehat{r}$$ as linear combination in 
    the basis $\widehat{x},\, \widehat{y}$ and $\widehat{z}$
    (5 points).

(b) Expand $\widehat{x},\, \widehat{y}$ and $\widehat{z}$ in terms of 
    $\widehat{r},\, \widehat{\theta}$ and $\widehat{\phi}$. Use 
    trigonometric relations to simplify the result (5 points).

Due February 4 before class.
\end{description}

% \begin{figure}[tb] \vspace{5pc}
% \centerline{\hbox{ \psfig{figure=figs/ladder.eps,width=7.5cm} }}
% \caption{Ladder for assignment~14.} \label{fig_lader} \end{figure}
\end{document}
