91Ó°ÊÓ

The speed of light in a medium of index of refraction \(n\) is \(v=d s / d t=c / n .\) Then the time of transit from \(A\) to \(B\) is \(t=\int_{A}^{B} d t=c^{-1} \int_{A}^{B} n d s .\) By Fermat's principle above, \(t\) is stationary. If the path consists of two straight line segments with \(n\) constant over each segment, then \(\int_{A}^{B} n d s=n_{1} d_{1}+n_{2} d_{2}\) and the problem can be done by ordinary calculus. Thus solve the following problems: Derive the optical law of reflection. Hint: Let light go from the point \(A=\left(x_{1}, y_{1}\right)\) to \(B=\left(x_{2}, y_{2}\right)\) via an arbitrary point \(P=\) \((x, 0)\) on a mirror along the \(x\) axis. Set \(d t / d x=(n / c) d D / d x=\) \(0,\) where \(D=\) distance \(A P B,\) and show that then \(\theta=\phi\).

Set up Lagrange's equations in cylindrical coordinates for a particle of mass \(m\) in a potential field \(V(r, \theta, z) .\) Hint: \(v=d s / d t ;\) write \(d s\) in cylindrical coordinates.

In the brachistochrone problem, show that if the particle is given an initial velocity \(v_{0} \neq 0,\) the path of minimum time is still a cycloid.

A uniform flexible chain of given length is suspended at given points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right) .\) Find the curve in which it hangs. Hint: It will hang so that its center of gravity is as low as possible.

Find the equation of motion of a particle moving along the \(x\) axis if the potential energy is \(V=\frac{1}{2} k x^{2} .\) (This is a simple harmonic oscillator.)

Write and solve the Euler equations to make the following integrals stationary. Change the independent variable, if needed, to make the Euler equation simpler. \(\int_{\phi_{1}}^{\phi_{2}} \sqrt{\theta^{\prime 2}+\sin ^{2} \theta} d \phi, \quad \theta^{\prime}=d \theta / d \varphi\)

Find the geodesics on the parabolic cylinder \(y=x^{2}\).

A yo-yo (as shown) falls under gravity. Assume that it falls straight down, unwinding as it goes. Find the Lagrange equation of motion. Hints: The kinetic energy is the sum of the translational energy \(\frac{1}{2} m \dot{z}^{2}\) and the rotational energy \(\frac{1}{2} I \dot{\theta}^{2}\) where \(I\) is the moment of inertia. What is the relation between \(\dot{z}\) and \(\dot{\theta}\) ? Assume the yo-yo is a solid cylinder with inner radius \(a\) and outer radius \(b\).

A particle moves without friction under gravity on the surface of the paraboloid \(z=x^{2}+y^{2} .\) Find the Lagrangian and the Lagrange equations of motion. Show that motion in a horizontal circle is possible and find the angular velocity of this motion. Use cylindrical coordinates.

A hoop of mass \(M\) and radius \(a\) rolls without slipping down an inclined plane of angle \(\alpha .\) Find the Lagrangian and the Lagrange equation of motion. Hint: The kinetic energy of a body which is both translating and rotating is a sum of two terms: the translational kinetic energy \(\frac{1}{2} M v^{2}\) where \(v\) is the velocity of the center of mass, and the rotational kinetic energy \(\frac{1}{2} I \omega^{2}\) where \(\omega\) is the angular velocity and \(I\) is the moment of inertia around the rotation axis through the center of mass.