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%----------------------------------------------------%
%%%%%%%%%%%%%%%%% CLASSICAL MECHANICS %%%%%%%%%%%%%%%%
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\begin{center}
{\bf ADVANCED MECHANICS --- PHY--4241/5227}\\\vspace {2mm} 
{\bf HOMEWORK 1}\\%
\vspace {3mm} (\today) \\
\vspace {1mm}{Due on Monday, January 14, 2002} \\ 
\end{center}
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\centerline{\bf PROBLEM 1} 
\medskip
\noindent
Read, and read again and again, 
{\bf ``The Principle of Least Action''}
from Chapter 19 of 
{\it ``The Feynman Lectures on Physics'', Vol. II.}
A true jewel written by the best physicist of the
second half of the twentieth century; Enjoy! 
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\centerline{\bf PROBLEM 2}
\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 harmonic potential 
of the form:
$$
  V(x)=\frac{1}{2}x^2 \;.
$$
\begin{description}
\item{a)} Using Euler-Lagrange's equations of motion,
          obtain the time-dependent motion of the 
          system; {\it i.e.,} solve for $x(t)$.
	  Compute the action for this exact path.

\item{b)} Using an approximate linear path of the
          form $x(t)\!=\!a+bt$, compute the action
	  for this path and compare it with the 
	  exact value obtained in part a).

\item{c)} Using an approximate quadratic path of the
          form $x(t)\!=\!a+bt+ct^2$, compute the action
	  for this path and compare it with the exact 
	  value obtained in part a).
\end{description}
{\bf Hint:} For parts b) and c) make sure that the paths
	    are consistent with the boundary conditions.
	    If any constant remains undetermined, fix it
	    by minimizing the action. 
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\centerline{\bf PROBLEM 3}
\medskip
A projectile of mass $M$ is fired at $t\!=\!0$ from 
a cannon located at the origin of the coordinate system 
($x(0)\!=\!y(0)\!=\!0$) with a velocity $v_{0}$ directed 
at an angle of $\theta_{0}\ne 0$ relative to the horizontal 
direction. The projectile lands after a time $T$ at a
horizontal distance $L$ away from the origin, {\it i.e.,}
$x(T)\!=\!L$ and $y(T)\!=\!0$.
\begin{description}
\item{a)} Write the Lagrangian for the system and,
	  using Euler-Lagrange's equations of motion,
          solve for the trajectory of the projectile.
	  In particular, obtain the values of the
	  initial velocity $v_{0}$ and $\theta_{0}$
	  necessary for the projectile to follow the 
	  prescribed trajectory.

\item{b)} Compute the action for the exact trajectory 
	  obtained in part a).

\item{c)} Using the exact value of $x(t)$ obtained in 
	  part a) but now approximating $y(t)$ by a 
	  sinusoidal function of the form
	  $$
	    y(t)=A\sin(\pi\frac{t}{T}) \;,
	  $$
	  compute the action for this approximate path.
	  Determine the amplitude $A$ by demanding that
	  the action be minimized with respect to $A$.

\item{d)} Make a plot of $y(t)$ for the exact and for 
	  the (optimal) sinusoidal paths (using the 
	  values of $g\!=\!9.81{\rm m/s}^{2}$ and 
	  $T=10$~s. Compute the exact and approximate
	  actions for these values of the parameters.
\end{description}
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