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  {\Large{\bf {Practice Problems- Classical Mechanics}}}\\
  
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%\rightline{Due date: 06-9-16, before 6 PM.}
\begin{enumerate}
\item A particle is acted upon a force ${\mathbf{F}}=F(r)\mathbf{{\hat{r}}}$ which is directed towards a fixed point. Show that the particle obeys Kepler's second law of planetary motion. {\bf Hint:} Use polar coordinates. Show the motion is planar and angular momentum about the fixed point (origin) is conserved.\\

Suppose $F(r)=\frac{k}{r^2}$, then write down the Lagrangian of the particle and deduce equation of motion. Write down the Hamiltonian of the system and write down Hamilton's equations of motion.

\item A block of mass $m$ is sliding down an inclined plane under the influence of gravity. Find the equation of motion of the block using d' Alembert's principle.

\item Use d' Alembert's principle to obtain equation of motion of the blob of a pendulum.

\item A simple pendulum of mass m whose point of support moves in (a) circular motion on the vertical plane with frequency $\omega$, (b) oscillates horizontally in the plane of it's motion according to\\ $x=a cos\,\omega t$. Find out the Lagrangians of the mass for both the cases. Find the equations of motion.

\item Lagrangian of a particle of mass m is \[L=\frac{1}{2}m\dot{x}^2 + \dot{x}(\ddot{x}x+\frac{1}{2}\dot{x}^2)  - V(x).\]
Find out the equation of motion of the particle. 

\item Find out the Hamiltonian of a charged particle of mass m and charge q in an electromagnetic field and write down the equations of motion. 

\item A mass m is suspended from the ceiling by a string of length l. The mass is free to move in all directions. This arrangement is called spherical pendulum. Obtain the Lagrangian and equation of motion of the mass. Identify the cyclic coordinate and corresponding conserved quantity. \\ {\bf Hint:} Use spherical polar coordinate system and deduce the velocity of the mass in that coordinate. 

\item A double plane pendulum consists of two simple pendulums, with one pendulum
suspended from the bob of the other. The "upper" pendulum has mass $m_1$ and length $l_1$,
the "lower" pendulum has mass $m_2$ and length $l_2$, and  both pendulums move in the
same vertical plane. Find the Lagrangian and deduce the Lagrange's equations of motion. Find out the Hamiltonian of the system.

\item Find the shortest curve that connects two points on a sphere.

\item Consider a modified brachistochrone problem in which the particle has non-zero initial
speed $v_0$. Show that the brachistochrone is again a cycloid, but with cusp $h=v_0/2g$
higher than the initial point.

\item Let a particle's motion is examined with respect to a frame K which is moving with velocity $\mathbf{u}(t)$ with respect to (w.r.t.) an inertial frame $K_0$. The velocity of particle w.r.t. frame $K_0$ is $\mathbf{v_0}(t)$ and w.r.t. frame K is $ \mathbf{v}(t)$. Show that the Lagrangian of the particle w.r.t. the frame K is:
\[L= \frac{1}{2}m \mathbf{v}^2- m\frac{d\mathbf{u}}{dt}.\mathbf{r} +\frac{1}{2}m \mathbf{u}^2-V,\]
where $\mathbf{r}$ is the position vector of the particle w.r.t. frame K and $V$ is the potential. What should be the Lagrangian if the particle's motion is studied w.r.t. a frame $K'$ that is rotating with angular velocity $\mathbf{\omega}(t)$ w.r.t. frame K? 

\end{enumerate}

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