Description
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
d
dt
∂T
∂q˙i
−
∂T
∂qi
= Qi (1)
can also be written as the following form (known as the Nielsen form)
∂T˙
∂q˙i
− 2
∂T
∂qi
= Qi
. (2)
4. (30 points) A constraint of the form
Xn
i=1
gi(x1, x2, . . . , xn)dxi = 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 (x
2 +y
2 +z
2
)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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