91Ó°ÊÓ

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.)

In Problems 5 to 7, use Fermat's principle to find the path followed by a light ray if the index of refraction is proportional to the given function. $$ x^{-1 / 2} $$

A particle moves on the surface of a sphere of radius \(a\) under the action of the earth's gravitational field. Find the \(\theta, \phi\) equations of motion. (Comment: This is called a spherical pendulum.)

Find the geodesics on a sphere. Hints: Use spherical coordinates with constant \(r=a\). Choose your integration variable so that you can write a first integral of the Euler equation. For the second integration, make the change of variable \(w=\cot \theta .\) To recognize your result as a great circle, find, in terms of spherical coordinates \(\theta\) and \(\phi\), the equation of intersection of the sphere with a plane through the origin.

Write Lagrange's equations in cylindrical coordinates for a particle in the gravitational field \(V=m g z\)