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\title{5500-18 HW 2 }
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\item {\bf "Inverse brachistochrone":}  Assume that the fastest trajectory is a part of a circle
$$ y(x)= \sqrt {A^2 - x^2} $$
Find the  speed dependence  $v(y)$.

\item The astronauts arrives at planet EasterEgg with a viscous atmosphere and find that the speed of travel depends on the distance $r$ from the center as $v(r)$. They need to find out the brachistochrone trajectory to travel form point $(r_1, \theta_1)$ to point $(r_2, \theta_2)$.

Derive the equation of brachistochrone in polar coordinates. 

Find the trajectory, if $v(r)= r$.


\item Find the boundary conditions for the point  in which %of the brachistochrone where  
the brachistochrone that started from the point $(0, 0)$ meets the circle $ x^2 + y^2 =R^2$.

\item The thermal equilibrium in a bar $ 0 \le x \le 1$ is described by the boundary value problem for the temperature $T(x)$
$$  T''= \gamma(x) \in( 0 \le x \le 1), \quad  T'+ \alpha (T-T_0)^4=0~~\mbox{at } x=1, ~~ T'=0 ~\mbox{at } x=1.
$$
where $\gamma$ is the density of heat sources, $\alpha $ is the radiation constant, $T_0$ is a constant outside temperature. 

Write the variational problem which minimizer describes the equilibrium. 



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