Description
Question 1 (4 marks)
Consider a double pendulum, as illustrated in Fig. ??.
(a) Find expressions for the potential and kinetic energies in terms of coordinates ϕ1 and ϕ2.
(b) Using your answer to (a), write down the Lagrangian and derive the corresponding Lagrange’s equations
of motion when l1 = l2 = l and m1 = m2 = m.
(c) Compare and relate your results in (a) and (b) to those obtained in the paper at https://doi.org/10.
1119/1.16860 (look at Appendix B).
In the remainder of the question we will look into regimes of different dynamics for the double pendulum.
You may assume l1 = l2 = l and m1 = m2 = m throughout.
(d) Consider the case where the upper pendulum is pinned in place, i.e., ˙ϕ1 = 0. Comment on what the
equations of motion [from (b)] reduce to (i.e., what they physically correspond to).
(e) Alternatively, we can assume that the motion of each pendulum remains small and invoke a small-angle
approximation. This generically entails expanding trigonometric functions to their lowest order, e.g.,
sin(θ) ≈ θ and cos θ ≈ 1. Under this approximation, solve for the possible frequencies assuming an
ansatz ϕj (t) = Aj e
iωt for j = 1, 2 (i.e., solve for the eigenfrequencies of the motion).
(f) Your answer to part (e) will suggest that in a certain limit the motion of the double pendulum is
exactly solvable. In this light, contrast to the results reported in Fig. 9 of the paper at https://doi.org/
10.1119/1.16860. What is the key result being communicated via this figure? What does it suggest
about the general motion of the double pendulum, when we drop the prior approximations?
Question 2 (2 marks)
The following is a recap problem for variational calculus.
Consider a bead constrained to slide (without friction) along a wire in 2D. The wire follows an arbitrary
path between the start point (x, y) = (x1, y1) to the end point (x2, y2). We will use variational calculus to
determine what the fastest path between these points is assuming the bead is subject only to the force of
gravity along +y.
Figure 1: Double pendulum for Q1
(a) Assuming the bead is initially at rest, show that the time taken to travel between the start and end
points is given by the integral,
t =
1
√
2g
Z y2
y1
s
1 + x
02
y
, (1)
where x
0 = dy/dx.
(b) To find the optimal path one should determine the stationary value of the integral (δt = 0) by using
the Euler equation,
∂F
∂x −
d
dy
∂F
∂x0
= 0, (2)
for F =
p
(1 + x
02)/y/√
2g. Do this and show that the optimal path is a cycloid,
x =
c
2
4g
(θ − sin θ),
y =
c
2
4g
(1 − cos θ)
(3)
where θ simply defines the parametrization of the path and we have assumed that (x1, y1) = (0, 0).
Note: You may find it useful to invoke the latter expression for y(θ) as a variable change when solving
an integral that arises.
Question 3 (2 marks)
Consider a generalized mechanics where the Lagrangian can be written as a function L = L(q, q˙ , q¨, t) so
that it additionally involves terms dependent on q¨.
Use Hamilton’s principle to show that the corresponding
Euler-Lagrange equation for such a system will be,
d
2
dt2
∂L
∂q¨i
−
d
dt
∂L
∂q˙i
+
∂L
∂qi
= 0. (4)
Question 4 (2 marks)
Consider a small bead of mass m that is constrained to slide frictionlessly around a hoop of radius R. The
hoop rotates at fixed angular frequency ω about its vertical axis of symmetry, but is otherwise motionless
(e.g., fixed in space).
(a) Assuming the system is subject to gravity, derive the equations describing the motion of the bead.
(b) Examine the equations obtained in (a) and obtain the possible scenarios for which the bead can be
stationary (with respect to its angular motion about the hoop). Compare and contrast the different
scenarios in terms of the value of ω.