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\title{HW 5 }
\author{Hamiltonian and Legendre Transform}
\date{}
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\begin{enumerate}
\item Consider a plane double pendulum: \newline
Mass $m_1$ is hanged on a rod of the length $l_1$ that is attached to the base point with coordinated $(0, 0)$. Mass $m_2$ is hanged on a rod of the length $l_2$ that is attached to mass $m_1$. 

Derive the kinetic and potential energy of the system via angles between the rods and the vertical, find Lagrangian and Hamiltonian, find first integrals, write the differential equation of motion of the masses. 



\item 
Find the Legendre transform of the functions:
$$ f_1(x)= - \log (x), \quad f_2(x)= \frac{1}{2}a x^2 +  b\, x + c 
$$
$$ f_3(x_1, x_2) = \frac{1}{2} \left(x_1^2 + x_2^2\right) + x_1 x_2 $$
Find the Legendre transform of the obtained transforms (second transform). 

\end{enumerate}

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