1. (15 points) Show that ~a × (~b ×~c) = (~a ·~c)~b − (~a ·~b)~c. What if ~b is a differential operation
∇?
2. (20 points) Show that spherical coordinates are orthogonal coordinates. In Cartesian coordinates the line element is defined as ds2 = dx2 + dy2 + dz2, derive its expression in spherical coordinates.
3. (15 points) Show that Lagrange’s equations
(1)
can also be written as the following form (known as the Nielsen form)
. (2)
4. (30 points) A constraint of the form
= 0 (3)
is holonomic only if an integrating function f(x1,x2,...,xn) can be found that turns it into an exact differential.
a) What condition shall f fulfill to turn Eq. (3) to a holonomic constraint?
b) Are the constraints (2x+y+z)dx+(x+2y+z)dy+(x+y+2z)dz = 0 and (x2+y2+z2)dx+ 2(xdx + ydy + zdz) = 0 holonomic?
5. (20 points) Consider a pendulum made of a spring with a mass m on the end. The spring is arranged to lie in a straight line with the equilibrium length of the spring being l. Let the spring have length l + x(t), and its angle with the vertical be θ(t). Assuming that the motion takes place in a vertical plane, find the equations of motion for x and θ.
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