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\title{Calculus of Variations \\
HW 2}
\author{Andrej Cherkaev}
\date{February 3, 2012}
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\begin{enumerate}
\item
 Two masses $m_1$ and $m_2$ are sequentially jointed to a wall $(x=0)$ by two springs. At equilibrium, the distance from the wall is equal to one for the first mass, and equal to two for the second one. The kinetic energy $K_i$ of each mass is $K_i=\frac{1}{2}m_i \dot{x}_i^2$, the potential energy of the first and second springs are
$$
\Pi_1= \frac{C}{2} (x_1-1)^2, \quad \Pi_2= \frac{C}{2} (x_2-x_1-1)^2
$$
where $C>0$ is a spring constant. 

a. Compose the variational problem of minimal action of the whole system, find the Euler-Lagrange equations for the motion.

b. Numerically integrate the equations for the following parameters
$$
C=1, ~m_1=0.1, ~m_2=1
$$
with initial conditions $ x_1= 1.2, ~ x_2 = 2.2, ~ \dot{x}_1=0, ~\dot{x}_2=0 $. Graph the motion of the first and second mass. 

c. Is it possible to approximate the motion of $m_2$ with one-mass system? Write the equations, compare the energy of the two systems. 

\item
Assume that the speed of motion in a $x, y$-plane depends on $y$ coordinate as $v_1=e^y$, $ v_2=\frac{1}{1+ y^2}$.

a. Find the fastest path between points $(0, 0)$ and $(1, 1)$. Use Maple, if needed. 

b. Compute the Hamiltonian. 

\item Perform the Legendre and Weierstrass tests for the Lagrangian
$$ F= [(u')^2- u^2]^2 $$
\end{enumerate}


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