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%----------------------------------------------------%
%%%%%%%%%%%%%%%%% CLASSICAL MECHANICS %%%%%%%%%%%%%%%%
%----------------------------------------------------%
\begin{center}
{\bf ADVANCED MECHANICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf HOMEWORK 2}\\%
\vspace {3mm} (\today) \\
\vspace {1mm}{Due on Monday, January 20, 2003} \\ 
\end{center}
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\centerline{\bf PROBLEM 4}
\medskip
Consider a particle of mass $m\!\equiv\!1$ moving, from 
$x_1\!=\!0$ at time $t_1\!=\!0$ to
$x_2\!=\!1$ at time $t_2\!=\!\pi/2$, under the
influence of a one-dimensional anharmonic potential 
of the form:
$$
  V(x)=\frac{1}{2}x^2+\frac{\lambda}{4}x^4 \;,
  \quad (\lambda\!\equiv\!\frac{1}{4})\;. 
$$
\begin{description}
\item{a)} Using Euler-Lagrange's equations of motion,
          obtain---but do not solve---the nonlinear
	  differential equation for $x(t)$.

\item{b)} Propose an approximate solution of the form
          $x(t)\!=\!\sin(t)+\alpha\sin(2t)$ and fix
	  the value of the ``variational'' parameter
	  $\alpha$ by minimizing the action. Compare
	  the action for the optimal path and for the
	  path having $\alpha\!=\!0$.

\item{c)} Using the $\alpha\!=\!0$ path and the optimal 
	  path obtained in part b), plot the energy of 
	  the system. Is the energy conserved? Explain.
\end{description}
\bigskip
\bigskip
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\centerline{\bf PROBLEM 5}
\medskip
A particle of mass $m$ is constrained to move on a
massless hoop of radius $a$ fixed in a vertical
plane that is rotating about the vertical axis with 
constant angular speed $\omega$. The particle can 
slide through the hoop under the action of gravity.

\begin{description}
\item{a)} Obtain the Lagrangian of the system using
	  the polar angle $\theta$ as the sole
	  generalized coordinate. 

\item{b)} Compute the Hamiltonian function of the
	  system and argue that it is a constant
	  of the motion. In particular show that 
	  it may be written as follows:
	  $$
	    h=\frac{1}{2}ma^{2}\dot{\theta}^{2}
             +V_{\rm eff}(\theta) \;.
	  $$
	  Compute explicitly the value of 
	  $V_{\rm eff}(\theta)$.

\item{c)} Make a plot of $V_{\rm eff}(\theta)$ and
	  use it to show that if $\omega$ is greater 
	  than a minimum value $\omega_{0}$, there can 
	  be a solution in which the particle remains
	  stationary at a point other than the bottom
	  of the hoop. Find this stationary point and
	  the value of $\omega_{0}$.
\end{description}
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\centerline{\bf PROBLEM 6}
\centerline{\small{(Problem 7.7 Marion and Thornton)}}

\medskip
A double pendulum consists of two simple pendula, with 
one pendulum suspended from the bob of the other. The 
two pendula have equal length $l$ and have bobs of equal 
mass $m$; both pendula are confined to move in the same 
plane.
\begin{description}
\item{a)} Identify clearly the number of independent
          generalized coordinates. In an effort to
	  maintain uniformity of notation, label the
	  generalized coordinates as $\theta_{1},
	  \theta_{2},\ldots,\theta_{N}$.
	  
\item{b)} Write the Lagrangian of the system in 
          terms of the generalized coordinates 
	  identified in part a).

\item{c)} Compute---without solving---the 
          Euler-Lagrange's equations of motion for
	  the system. Do not assume small angles.
\end{description}
\vfill\eject
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