University of Toronto, Faculty of Arts and Science, April 2021 Examinations
PHY 354H1 Advanced Classical Mechanics Final Exam
24-h take-home exam, starting 7pm Tuesday April 20 and ending 7pm, Wednesday April 21 (Toronto time).
This exam can be completed in several hours, so you should have plenty of time. Don’t be intimidated by
its length – the explanations are verbose to make your life easier. The ample time allowance makes it your
responsibility to submit your finished exam answers to Quercus with time to spare, in order to guard against
unforeseen technical difficulties or delays. If all else fails, email your work to [email protected],
the email timestamp serves as proof of your on-time submission. (Please only do that if necessary, it will
make our life more difficult and make us grumpier graders if everyone does it.)
Late submissions will not be accepted and result in a zero mark on the final exam, so leave
yourself enough time to ensure your upload is done by the deadline.
You may submit via scanned handwritten work on pdf, or latex, or other digital document, as long as it is
clear to the grader. You must show all your work, you will not receive marks for correct solutions
if your work does not fully justify your answers in a way that proves your understanding. Use
of mathematical software or a calculator is not necessary to complete this exam. The exam is open-book,
so feel free to use any course material (textbooks, lecture notes, homework solutions) or look up whatever
purely mathematical definitions you need, but any collaboration, or use of online resources like solutions
manuals or answer websites, is strictly forbidden. We have spies everywhere.
There are 6 questions on 5 pages. The exam is out of 140 Points.
1. [ 20 Points ] Slidey boi. A bead of mass m is free to slide on a horizontal frictionless wire. It is connected
to a spring with spring constant k and equilibrium length ` which is attached to the ceiling a height L
above the wire. Describe the position of the bead with coordinate x.
x
m
L
k,ℓ
(a) Write down the Lagrangian for the motion of the bead. [4 Points]
(b) Are there any conserved quantities? What symmetries are they associated with? [3 Points]
(c) Write down the Euler-Lagrange equation for x. [3 Points]
(d) For ` L, find the stable equilibrium position of the bead. What is the frequency of oscillations
around that equilibrium, to lowest order in small displacement from that equlibrium? [5 Points]
Page 1 of 6
(e) for ` L, find the stable equilibrium position of the bead. What is the frequency of oscillations
around that equilibrium, to lowest order in small displacement from that equlibrium? [5 Points]
2. [ 30 Points ] You spin me right ’round. Four equal, uniform and infinitely thin rods of mass m and
length 2a are hinged together at points A, B, C, D to form the rhombus shown below. The rhombus
is mounted on a vertical rod, shown as the gray line. Hinge A is fixed on the rod, while hinge D can
slide up or down. The whole system is free to rotate around the vertical.
θ
ϕ
A
B C
D
(a) Write down the Lagrangian of this system. (Hint: remember it’s the rods that have mass, not the
black dots in the above diagram.) [7 Points]
(b) Show that there are two conserved quantities. [3 Points]
(c) Write down the Euler-Lagrange Equations. [5 Points]
(d) Imagine rotating this system around the vertical axis with fixed angular speed ω. What is the
equilibrium configuration of the rhombus? What is the minimum angular speed for a non-trivial
equilibrium? [5 Points]
(e) Find the Hamiltonian of this system. [4 Points]
(f) Express the conserved quantities in terms of the canonical variables (q’s and p’s). [3 Points]
(g) What is the poisson bracket of the integrals of motion. [3 Points]
3. [ 20 Points ] Let’s go to spaaaace. The movie 2001: A Space Odyssey, featured a ring-shaped space
station that spins around in order to create “artificial gravity” for the inhabitants living in the ring.
Page 2 of 6
Let’s work through some of the details and see in what ways this does or does not mimic natural
gravity on the Earth surface. Imagine the space station as a ring-shaped tube with outer radius R
that rotates with angular velocity ω. If you live on that space station, the coordinate system that is
relevant to your daily life is the body-fixed coordinate system (x′′, y′′, z′′), which moves and rotates
with your surroundings as the space station rotates (2nd figure).
x′ ′
y′ ′
z′ ′
R
ω
x′ ′
y′ ′ z′ ′

(a) Write down the Lagrangian that applies to a particle of mass m in your body-fixed (x′′, y′′, z′′)
coordinate system. [5 Points]
(b) What are the Euler-Lagrange equations for this free particle? [4 Points]
(c) Assume that the space station rotation mimics normal earth gravity. For a given radius R, what
is the required period of rotation T? [3 Points]
(d) Say your height is h and the “gravity” at your feet is earth standard. What is the apparent
strength of gravity at your head? [3 Points]
(e) Would a game of billiards on the space station proceed the same way it would on earth? How
about a game of basketball? Explain all the ways in which the game is/is not affected in qualitative
detail. [5 Points]
4. [ 15 Points ] Weird vibes. Consider the Hamiltonian of a general oscillator:
H =
p2
2m
+Bqn
where n ≥ 2 is an even integer.
(a) Let’s simplify this Hamiltonian. Find constants C,D such that the coordinate transformation
Q = Cq, P = Dp is canonical and H = α(P 2 +Qn). What is α? [5 Points]
(b) Is energy conserved in this system? Why? [1 Points]
(c) Sketch a typical phase space trajectory for n = 2 in the (P,Q) plane. [4 Points]
(d) Define H˜ = H/α, and imagine different cases for n while fixing H˜ = 1. In the same plot, sketch
the phase space trajectories for n = 2, n = 100. [5 Points]
Page 3 of 6
5. [ 30 Points ] All Hail Jupiter, King of the Planets. Jupiter is huge, about 300× more massive than
Earth and orbiting around 5× further away from the sun. As humungous as that is, it’s still dwarfed
by the sun, which is 1000× more massive than Jupiter again. No wonder that the motion of the Earth
around the sun is well-described as a 2-body problem in a central gravitational potential. However,
Jupiter’s presence is by far the next-most significant effect after that of the sun. So let’s see if we can
start figuring it out.
Since we can’t solve the full three-body problem analytically, let’s work towards finding an approximate
solution, making use of the fact that MS MJ ME (masses of the Sun, Jupiter and Earth) and
that the distance of Jupiter to the Sun is significantly larger than the distance of the Earth to the sun.
Kepler’s law tells us that the inner planet will have a much shorter orbital period than the outer one.
Therefore, as a first pass at this problem, let’s assume that Jupiter is stationary a distance rJ from
the Sun, while the Earth orbits around the sun.
x
y
z
⃗r
rJ
E J
S
(a) Working in the indicated coordinate system with origin at the position of the sun, write down
the Lagrangian for the motion of the earth at position ~r under the influence of the external
gravitational potentials of the sun and Jupiter. Assume that all bodies move in the x − y plane
only, and use spherical polar coordinates. [4 Points]
(b) Does this Lagrangian have any conserved quantities? What are they? [3 Points]
(c) Assuming that = MJ/MS 1 and that r rJ , expand your Lagrangian to lowest non-trivial
order in and r/rJ . [5 Points]
(d) What is the Hamiltonian for this approximate Lagrangian? [3 Points]
(e) What are Hamilton’s equations for this approximate Hamiltonian? [4 Points]
(f) We will solve for the effect of Jupiter as a perturbation on Earth in a circular orbit without
Jupiter. Write the trajectory of the Earth in terms of distance from the sun and polar angle as
well as conjugate momenta
r(t) = r0(t) + r1(t) , ϕ(t) = ϕ0(t) + ϕ1(t)
pr(t) = pr,0(t) + pr,1(t) , pϕ(t) = pϕ,0(t) + pϕ,1(t)
where r0, ϕ0, pr,0, pϕ,0 is the solution without Jupiter, and r1, ϕ1, pr,1, pϕ,1 is the first correction
due to Jupiter’s presence to lowest non-trivial order in and 1/rJ . What is the unperturbed
0th-order solution, assuming the Earth is at the position indicated in the above diagram when
t = 0? [3 Points]
(g) What are the Hamilton’s Equations for r1, ϕ1, pr,1, pϕ,1. [5 Points]
(h) What is pϕ,1(t)? [3 Points]
6. [ 25 Points ] Chilled Dreidel. In class, we carefully worked through derivations to understand the Heavy
Symmetric Top, which was a symmetric top (I1 = I2 6= I3) under the influence of gravity, free to rotate
Page 4 of 6
around its bottom tip, but the position of the bottom tip was fixed. Therefore, it wasn’t really like a
spinning top but more like a gyroscope apparatus.
Well, here we are going to slightly extend that derivation from lecture notes to describe a spinning top
on a frictionless surface. Like a dreidel on ice, illustrated below.
x
y
z
ℓ
θ
·ψ
In the diagram on the right, the bottom tip of the spinning top is always in contact with the surface,
which is in the x− y plane. The center-of-mass of the top is indicated with the black dot.
Let the body-fixed coordinate system for the heavy top be centered on the center-of-mass (rather
than the bottom tip as we did in lectures), which is distance ` from the bottom tip. Let ~r = (x, y, z)
be the position of the center of mass in the lab frame, and use the same Euler angles φ, θ, ψ for the
orientation of the top as we did in lectures. The spinning top has mass m and principial moments of
inertia I1 = I2 6= I3 around its center of mass.
(a) Carefully write down the Lagrangian for this spinning top. You can closely follow the derivation
from lecture notes, with some necessary modifications. [8 Points]
(b) What are the symmetries of this Lagrangian? What are the associated conserved quantities?
[4 Points]
(c) Using the conserved quantities and the same methods and notation as we did in lecture notes,
show that the angle θ obeys the equation and has the form
u˙2 =
f(u)
g(u)
where u = cos θ, f(u) is the same as in the lecture derivation (except the I1 moment of inertia is
around the center of mass and not the bottom tip), and
g(u) = 1 +
m`2
I1
(1− u2) .
[7 Points]
(d) Show that the qualitative behaviour of this spinning top is the same as for the heavy symmetric
top from lectures with the bottom fixed in place, i.e. there is nutation between θmin, θmax,
and the precession can always be in one direction or change sign. Furthermore, without solving
for θmin, θmax explicitly, show that these min/max values are entirely dictated by f(u) and not
modified by g(u).
Page 5 of 6
This shows that the lecture derivation where the bottom tip of the spinning top was fixed in place
was still representative of the behaviour of a real spinning top on a frictionless surface. :)
[6 Points]
Page 6 of 6

欢迎咨询51作业君