midterm1 - \item Show that Hamilton's equations give...

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% BB Mar 10 2010 \documentclass[12pt]{article} \textheight 9.5in \textwidth 6.0in \topmargin -0.7in \headsep 0.4in \oddsidemargin 0.25in \evensidemargin 0.25in \def\pmbf#1{{\mbox{\boldmath${#1}$}}} \def\undertext#1{$\underline{\smash{\hbox{#1}}}$} \pagestyle{empty} \begin{document} %----------------------------------------------------% %%%%%%%%%%%%%%%%% ADVANCED DYNAMICS %%%%%%%%%%%%%%%% %----------------------------------------------------% \begin{center} {\bf MIDTERM ADVANCED DYNAMICS --- PHY--4241/5227}\\ \vspace {1mm}{March 17, 2010, 9:05-9:55 am, HCB 317} \\ \end{center} \setcounter{equation}{0} \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%% \centerline{\bf PROBLEM 1 (20 points)} \medskip \noindent \begin{enumerate} \item Write down the Euler-Lagrange equations and the definition of the generalized momentum. \item Write down the Lagrangian of the 1D harmonic oscillator and continue with this system. \item Find the momentum $p$ from the Lagrangian. \item Write down the Hamiltonian $H(p,x)$. \item Write down Hamilton's equations.
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Unformatted text preview: \item Show that Hamilton's equations give Newton's force law. \end{enumerate} \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%% \centerline{\bf PROBLEM 2 (20 points)} \begin{enumerate} \item Use spherical coordinates $(\theta,\phi,r)$ to write down the Lagrangian for a particle of mass $m$ moving in the presence of a spherically symmetric potential $V(r)$, $r=|\vec{r}|$, $\theta$ polar and $\phi$ azimuth angle. \item Write down the Euler-Lagrange equations for the two angles. \item Use the Lagrangian to identify two conserved quantities. \item Find $\phi(t)$ for the solution with the initial condition $\dot{\phi}(0)=0$ where $\phi$ is the azimuth angle. \end{enumerate} \hfil\break %%%%%%%%%%%%%%%%%%%%%%%%%% \centerline{\bf PROBLEM 3 (10 points)} \medskip \noindent Use SI units and write down the four Maxwell equations (in vacuum) in the presence of a charge density $\rho$ and a current density $\vec{J}$. \bigskip \bigskip %%%%%%%%%%%%%%%%%%%%%%%%%% \vfill \end{document}...
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midterm1 - \item Show that Hamilton's equations give...

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