Classical Mechanics: Problem Set 1 and 2

1. Invariance of the equations of motion to coordinate changes

Consider two coordinate systems $q_a$ and $\tilde{q}_a$, related by $\tilde{q}_a = f_a(q_b, t)$. Show that the velocoties $v_a \equiv dq_a/dt$ and $\tilde{v}_a \equiv d\tilde{q}_a/dt$ are related by \[ \tilde{v}_a = \pd{\tilde{q}_a}{t} + \sum_b \pd{\tilde{q}_a}{q_b} v_b. \]

For any function, define $L = L(q_a, v_a, t)$, define \[ S_a \equiv \dd{}{t} \left(\pd{L}{v_a}\right) - \pd{L}{q_a}, \]

with the corresponding definition for $\tilde{S}_a$. Show that \[ S_a = \sum_b \pd{\tilde{q}_b}{q_a} \tilde{S}_b. \]

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2. Coupled oscillators and strings

A light string is stretched to a tension $\tau$ between two fixed points $A$ and $B$, a distance $3a$ apart, on a smooth horizontal table. Two point masses, each of mass $m$ are attached to the string at the points $P_1$ and $P_2$. In equilibrium, the four points $A$, $P_1$, $P_2$, and $B$ are equal distances apart. The system is set to perform small trasversal oscillations by displacing transversely the two masses.

Assume the potential energy is $\tau \Delta$ where $\Delta$ is the extension of the string from equilibrium, and that the tension $\tau$ of the string remains constant throughout the motion. Find the normal frequencies and the normal modes of vibration. Sketch the normal modes.

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3. A weight, a rod, and two springs

A bead of mass $m_1$ slides without friction on a fixed horizontal wire which occupies the interval $[-a, a]$ of the $x$-axis. A light spring, of spring constant $k$, connects the bead to the point $-a$ and a second light spring with the same constant connects the bead to the point $a$. A massless rod of length $l$ hands freely from the bead and its other end carries a particle of mass $m_2$. Show that the Lagrangian is \[ L = \frac{m_1}{2} \dot{x}^2 + \frac{m_2}{2} \left(\dot{x}^2 + 2l\cos\theta\dot{x}\dot{\theta} + l^2\dot{\theta}^2\right) - kx^2 + m_2 gl \cos\theta, \]

where $x$ is the position of the bead on the wire, $\theta$ is the angle between the rod and the downward vertical, and $g$ is gravity.

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Find a constant of the motion.

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Write down the equations of motion and show that there are two equilibrium positions. Write down the approximate Lagrangian for small oscillations about the position of stable equilibrium. Find the normal frequencies when $g = l = k = m_1 = m_2 = 1$.

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To find the equilibrium solutions, we set $\dot{x} = \ddot{x} = \dot{\theta} = \ddot{\theta} = 0$. This gives the two points \[ (x, \theta) = (0, 0), \qquad (x, \theta) = (0, \pi), \]

for which the second is expected to be unstable. If we did not want to use this method, we could have equally applied the result that since $T$ is quadratic in the velocities (but not necessarily in the positions), and $V$ is only a function of the positions, then the equilibrium solutions is given by solving $\pd{V}{x} = 0 = \pd{V}{\theta}$.

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4. Moments of inertia

Find the moments of inertia at the centroid of an ellipsoid of mass $M$ and uniform density, bounded by \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1. \]

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