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\title{HW 4}
\author{5500-2018}
\date{}
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\begin{enumerate} 
\item Describe potential and kinetic energy of the following two-mass system: Mass $m_1$ is attached by a linear spring of the length $L$ with the spring constant $C$ to steady support, mass $M$ is attached by the similar spring to the mass $m$. 


Write the kinetic and potential energy of the system, find first integrals. Find the motion of both masses. Then consider the case when $m\to 0$. 

Consider two cases:
\newline
(a) Initially, the first spring is elongated; the elongation is equal to $\Delta$. Initial velocities are equal to zero. 
\newline
(b) Initially, both springs are elongated; the elongations are equal to $\frac{1}{2}\Delta$. Initial velocities are equal to zero. 

Compare the energy in both cases. Comment on the difference in the solutions.

\item Necessary conditions. \newline Consider the Lagrangian
$$L= \frac{1}{2} m \,x\,\dot{x}^2- x(1+x), \quad x\in [0, ~ A), ~~ A>>1$$
Write the equation for the stationary trajectory. Does it correspond to the minimum? Check Legendre and nonlocal Jacobi-type condition. 

\item Filtration (linear)\newline
The received signal $h(t)$ is the sum of the true response $s(t)$ and the noice $v(t) $
$$
s(t) = t (L-t), \quad v(t)= 2 \sin(10 \pi L t) + \cos(7 \pi Lt) , ~~ t\in[0, L], ~~ L= 2
$$
Filter the signal using the penalties
$ p_1=  \alpha (u')^2$
and $ p_2= \alpha (u'')^2$. 

Graph the received and filtered signals for several values of $\alpha$. Discuss. 
\end{enumerate}

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