Z x2 x1 x py2 + x2 dx

Derive the optical law of reflection. Hint: Let light go from the point A = (x1, y1) to B = (x2, y2) via an arbitrary point P = (x, 0) on a mirror along the x axis. Set dt/dx = (n/c)dD/dx = 0, where D = distance AP B, and show that then = .

Derive Snells law of refraction: n1 sin 1 = n2 sin 2 (see figure).

Show that the actual path is not necessarily one of minimum time. Hint: In the diagram, A is a source of light; CD is a cross section of a reflecting surface, and B is a point to which a light ray is to be reflected. AP B is to be the actual path and AP
B, AP

B represent varied paths. Then show that the varied paths: (a) Are the same length as the actual path if CD is an ellipse with A and B as foci. (b) Are longer than the actual path if CD is a line tangent at P to the ellipse in (a). (c) Are shorter than the actual path if CD is an arc of a curve tangent to the ellipse at P and lying inside it. Note that in this case the time is a maximum! (d) Are longer on one side and shorter on the other if CD crosses the ellipse at P but is tangent to it (that is, CD has a point of inflection at P).

Z x2 x1 x ds

Z x2 x1 (y
2 + y2 ) dx

Z x2 x1 (y
2 + y ) dx

R x2 x1 exp1 + y
2 dx Hint: In the last integration, let u = ex and see Chapter 5, Problem 1.6.

Z x2 x1 x py
2 + x2 dx

x2 x1 (1 + yy
) 2 dx

Z x2 x1 x2 dx xy
+ 1

x + 1

y1

y

r1

Find the geodesics on a plane using polar coordinates.

Show that the geodesics on a circular cylinder (with elements parallel to the z axis) are helices az+b = c, where a, b, c are constants depending on the given endpoints. (Hint: Use cylindrical coordinates.) Note that the equation az +b = c includes the circles z = const. (for b = 0), straight lines = const. (for a = 0), and the special helices az + b = 0.

Find the geodesics on the cone x2 + y2 = z2. Hint: Use cylindrical coordinates.

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 . To recognize your result as a great circle, find, in terms of spherical coordinates and , the equation of intersection of the sphere with a plane through the origin.

For small vibrations, find the characteristic frequencies and the characteristic modes of vibration of the coupled pendulums shown. All motion takes place in a single vertical plane. Assume the spring unstretched when both pendulums hang vertically, and take the spring constant as k = mg/l to simplify the algebra. Hints: Write the kinetic and potential energies in terms of the rectangular coordinates of the masses relative to their positions hanging at rest. Dont forget the gravitational potential energies. Then write the rectangular coordinates x and y in terms of and , and for small vibrations approximate sin = , cos = 12/2, and similar equations for .

Do Problem 19 if the spring constant is k = 3mg/l.

Find the Lagrangian and Lagranges equations for the double pendulum shown. All motion takes place in a single vertical plane. Hint: See the hint in Problem 19.

Do Problem 21 if the two masses are different. Let m be the lower mass and let M be the sum of the two masses.

For small oscillations of the double pendulum in Problem 22, let M = 4m and find the characteristic frequencies and characteristic modes of vibration.

Do Problem 23 if M/m = 9/4

Do Problem 23 in general, that is, in terms of the ratio M/m. Hint: You may find it helpful to use a single letter to represent pm/M, say 2 = m/M.

A wire carrying a uniform distribution of positive charge lies in the (x, y) plane and joins two given points. Find its shape to minimize the electrostatic potential at the origin.

Find a first integral of the Euler equation for Problem 26 if the length of the wire is given.

Write the Lagrange equation for a particle moving in a plane if V = V (r) (that is, a central force). Use the equation to show that: (a) The angular momentum r mv is constant. (b) The vector r sweeps out equal areas in equal times (Keplers second law).