Phys410 - Classical Mechanics
University of Maryland, College Park
Fall 2011, Professor: Ted Jacobson
Problems from Taylor,
HW0 - 7due at
beginning of class, Tuesday 9/06/11
due at beginning of class, Tuesday 12/13/11
S12.1 (Fourier modes of
string in Lagrangian and Hamiltonian)
The Lagrangian for a string of length a, linear mass density µ, and tension
T, held fixed at the ends,
L = ∫ 1/2 [µ (∂y/∂t)2
- T (∂y/∂x)2]
where y(x,t) describes the transverse displacement of the
(a) Instead of using the function y(x) as the generalized
coordinate, rewrite the Lagrangian using the coefficients y_n, n
of the Fourier series y(x, t) = ∑ y_n(t) sin k_n x. (I'll let
you figure out how k_n depends on n and the string length.)
Show that the Lagrangian is a sum of harmonic oscillator
Lagrangians, one for each n. Find the frequency w_n of each
oscillator, and show that w_n = n w_1.
(b) Write out and solve Lagrange's equations for the system,
treating the y_n as the generalized coordinates.
(c) Find the canonical momentum conjugate to each y_n, and use
that find the Hamiltonian for the string using the phase space
these Fourier components and their conjugate momenta.
(d) You can "bend" a guitar note by pushing on the string to
increase the tension while the string is sounding. (The length
but I think the effect of this is smaller.) (The note is a
superposition of all the harmonics, dominated by the
(i) Explain why the variation of the tension is an adiabatic
change of the string Hamiltonian.
(ii) If the note is bent upwards by a whole step (2/12 of an
octave), by what percent does the energy of the string vibration
(Hint: The total energy
is the sum of the energies of all of the modes.)
(e) A quantum harmonic oscillator has a zero point energy hbar
w/2 in the ground state, and a string is a collection of
frequencies w_n = n w_1. Explain why the string doesn't have an
infinite zero point energy!
S12.2 (String with a
In class we considered a stretched string whose only restoring
force is due to the tension. Suppose the string is somewhat
has a potential energy associated with bending. The bending is
characterized by the curvature of y(x), i.e. ∂2y/∂x2.
The bending energy
has the form
∫ 1/2 ß (∂2y/∂x2)2
dx, where ß is
a bending modulus.
(a) Find the Lagrange equation for the stretched string
including this additional term, by imposing the condition that
the action is
stationary under all variations of y. It should be a modified
wave equation, with a fourth order spatial derivative term.
(b) The Fourier modes of the string are still harmonic
oscillators in the presence of the bending term.
Inspect your analysis for the Fourier series in S12.1 and read
off the frequencies w_n when the bending modulus is included.
You should find that the frequencies grow with n more quickly
than without the bending modulus. The string is no longer
"harmonic", in the sense that the frequencies w_n are not
integer multiples of the fundamental w_1.
string) A relativistic string has a rest
energy is equal to the string tension T times the proper length
of the string
(so its rest energy changes as it oscillates). The
action for a relativistic free particle is -mc^2 times the
proper time along the worldline.
If you replace mc^2 by the tension T times the proper length of
the string at each time, you get the action for the string: it
is -T times
the integral over proper length and proper time along the string
"worldsheet", i.e. -T times the proper spacetime area of the
Rather than ask you to work out the equations for a general
motion, I'll restrain myself and ask you only to consider
configurations of a closed loop of string whose center is at
rest in some inertial frame, so the string is
characterized at a given time by
its radius R(t) alone, where t is the time coordinate in that
frame. The action discussed above, when restricted to such
is -T ∫ dA = -2πT ∫ R [1 - (dR/dt)2/c2]1/2
dt. Let's choose units with c = 1 and
2πT = 1, so the action is just
S = - ∫ R [1 - (dR/dt)2]1/2
(a) Find the momentum p conjugate to R.
(b) Solve for dR/dt in terms of p and R.
(c) Find the Hamiltonian for the string, and compare it to the
Hamiltonian for a relativistic free particle.
(d) Find the equation of motion.
(e) (i) Explain how you know in advance that the Hamiltonian is
conserved, and (ii) use the equations of motion to show it
(f) Using (d) and (e), show that the string satisfies a harmonic
oscillator equation, and identify the frequency of the
(g) Show that when the string radius goes to zero, the string is
moving at the speed of light.
(h) A quantized string in its ground state has some zero point
motion. Estimate the size of the R fluctuations in the ground
by minimizing the energy subject to the uncertainty relation p R
≥ hbar. Use units with hbar = 1. Then find this zero point size
general units, i.e. restore the appropriate factors of hbar, c
- pdf file - due at beginning of class, Tuesday
HW10 - due at
beginning of class, Tuesday 11/29/11
13.12 (bead on spinning
13.13 (particle on cylinder
with restoring force)
13.14 (particle on cone)
(Note: It should say
"for a given energy [and angular momentum], this occurs at
exactly two values of z,
[except for the special case when z is constant]." Also, note
that when p_z = 0 this Hamiltonian is equal to the effective
for a fixed angular momentum.)
13.17 (nearly circular orbits
of particle on cone) (Suggestion:
Use the effective potential to do parts (a), (b), and (c).)
S10.1 (Hamiltonian for
relativistic particle and cyclotron motion) The action for a
relativistic particle is -m∫ dt (1-v2)1/2,
where v = dx/dt,
i.e. the Lagrangian is L = -m
(a) Find the momentum p
conjugate to x. (b)
Show that p2
+ m2 = γ2m2,
γ = (1-v2)-1/2.
(c) Show that the Hamiltonian is H = (p2
(in units with c = 1). (d) Now include an external
electromagnetic field, and show that for a charge e the
relativistic Hamiltonian is H = [(p - eA)2
where V and A are the
scalar and vector potentials. Note that in a magnetic field (with no
the Hamiltonian is given by H = γm.
(e) Suppose there is a uniform magnetic field in the z direction,
which can be
derived from the vector potential A
= Bx yhat. Show that the
orbits in the xy plane are circles with angular velocity w = w_0/γ,
where w_0 = eB/m is the nonrelativistic cyclotron
frequency. In your analysis you may for convenience set
the conserved quantity
p_y to zero (after finding Hamilton's equations), since it just
determines the x coordinate of the center of the circular orbit.
(f) Eliminate the velocity using v = Rw, and show that w =
w_0/(1+R^2 w_0^2)^1/2. Note that this ensures that the speed is
never greater than the speed of light, and that as the radius
grows the speed approaches the speed of light.
S10.2 (Proper time and orbits)
Consider three worldlines that all begin and end at the same
radius above the earth's surface: (i) the circular orbit at that
(ii) the static worldline held in place by a long pole or a
rocket, and (iii) the radial up and down free-fall path (with
chosen so that the path will return at the same time as the
circular orbit does). Paths (i) and (iii) are both stationary
paths of the
proper time since they are freely falling, while path (ii) is
not. Show that the proper times on these three paths satisfy
tau_i < tau_ii < tau_iii. You need not evaluate the proper
times; rather, only show that they satisfy these inequalities.
(i) and (ii) you can easily show it directly (again, without
actually evaluating the times). For (iii) the most convenient
is to argue that there must be a path of longer proper time than
(ii), and that path (iii) must be this path. (If you like a
try to actually demonstrate that tau_iii is longer by using the
conserved energy and effective potential. I didn't succeed in
a simple way --- I think it's rather tricky.)
S10.3 (Gravitational redshift)
Consider two light pulses sent radially outward from radius r_1
to radius r_2 in the Schwarzschild spacetime, with the emission
by a Schwarzschild coordinate time interval ∆t. (a) What is the
Schwarzschild coordinate time interval
between the arrival of the two pulses at r_2?
(b) What is the proper time ∆tau_1 between the emission of the
two pulses at r_1? (c) What is the
proper time ∆tau_2 between the reception of the
two pulses at r_2? (d) Suppose a photon is emitted radially from
r_1 with frequency w_1 measured by a static observer at r_1.
When the photon arrives
at r_2, what will be its frequency as measured by a static
observer at r_2? (e) As r_1 approaches the Schwarzschild
radius, what happens to the frequency
w_2, if w_1 is fixed?
HW9 - due at
beginning of class, Tuesday 11/15/11
relativistic relation between energy and momentum)
Consider a particle of rest mass m, energy E and momentum p, and
first set c=1.
(a) Show that when p << m, we have E ≈
m + p2/2m.
(b) Show that when p >> m, we
have E ≈ p
+ m2/2p, and p ≈
E - m2/2E.
(c) Rewrite your answers for (a) and (b) including the
appropriate factors of c.
S9.2 (Inverse Compton
If a high energy electron collides with a low energy photon,
much of the electron energy can be transferred to the outgoing
This is called "inverse Compton scattering", even though it's
the same as Compton scattering when viewed in a frame in which
is initially at rest. Let's consider the case in which the
incoming electron is highly relativistic (E0 >> m) and
moving in the +x direction,
and collides head on with a photon of frequency w0 moving along
the -x direction. Suppose that after the collision the outgoing
is moving along the +x direction.
(a) Let p0, k0, p, and k, be the initial and final 4-momenta of
the electron and photons. Show that k0.p0 = k.p
(b) Using (a) show that w = w0(E0 +
p0)/(E - p) ≈ 2w0
E0/(E - p) (the last approximation follows from E0 >> m).
(c) Use conservation of energy and momentum to show that
E - p = E0 - p0 + 2w0.
(d) Combine the previous to results and S9.1 to show that w/E0 =
1/[1 + m^2/(4E0 w0)]. When the incoming electron energy is
sufficiently large, the outgoing photon therefore has almost all
of the energy.
(e) If w0 = 1 eV, what must be the incoming electron energy E0
if the final photon energy is to be approximately equal to E0/2?
S9.3 (Motion in the Schwarzschild spacetime)
S9.4 (Motion in a gravitational wave)
HW8 - due at
beginning of class, Tuesday 11/08/11
Answer only (b) How long is the pipe as "seen" (better word
would be "reckoned") by the pions, and how long does it take to
pass the pions?
15.56 (rest energy in chemical
15.57 (rest energy in nuclear
15.58 (rest energy vs. kinetic
15.71 (collision energy to
create a particle)
15.75 (parent particle
15.87 (pion decay to photons:
pion velocity and photon angle)
15.90 (pair creation off
nucleus) Do part (a) by showing that the sum of two
future timelike 4-vectors cannot be lightlike.
15.92 (pion decay to lepton
S8.1 (no vacuum Cerenkov
radiation) Show that the reaction e -> e + gamma
cannot satisfy both energy and momentum conservation.
You could do this at least three different ways: by (i) writing
out energy and momentum conservation and applying the mass
in a general frame, (ii) doing the analysis in the rest frame of
the initial electron, or (iii) using 4-vectors and proving the
result for problem 15.53.
S8.2 (cosmic gamma ray cutoff)
Although gamma rays of energies up to 1020 eV are
created by ultra high energy cosmic rays throughout the
universe, they cannot travel all the way to the earth, because
on the way they collide with other photons and disappear,
pairs. The reaction is gamma1 + gamma2
-> e+ + e-. (a) If the two
photons collide head-on, show that for a given E2
the minimum energy E1
create the pair is E1 = me2/E2.
(b) Evaluate this energy (in eV) assuming E2
is the energy of (i) a typical cosmic microwave background
photon, 3K = 0.0003 eV,
or (ii) an infrared photon of energy 0.5 eV. (The
actual cutoff of cosmic ray photons originating from more than 100
million light years (?) comes from
annihilation on the infrared background radiation. I'm not sure
about the numbers here...)
HW7 - due at
beginning of class, Tuesday 11/01/11
11.32 (CO2 vibrations) Do the eigenvalue and eigenvector
calculations by hand. (Suggestion:
Once you have worked out the equations
of motion, adopt units with k = m = 1.) Add to part (b):
Which mode has the higher frequency?
Try to explain "why", without reference to equations. Add parts:
(d) Show that the ratio of the two frequencies is √11/3, taking
that the ratio of carbon to oxygen mass is 3/4. Show that the
observed frequency ratio is smaller by
for information on carbon dioxide vibrations.*)
(e) Try to give a physical reason (or reasons) why the observed
ratio is smaller than that in our simple model. I don’t think
between quantum and classical mechanics is the most important
issue in this case.
(Hint: Think about the
physical nature of the “springs” in the molecule and how they
differ from those in our model.)
*The frequencies are given on this web page in units of cm−1.
This refers to the inverse wavelength of the photons that are
transitions between the vibrational levels. The inverse
wavelength is proportional to the frequency of the photons,
which is proportional to the
frequency of the vibrations.
15.7 (muon lifetime in cosmic
S7.1 (masses suspended by
Consider a mass m suspended by a spring with spring constant k
from another mass m which is suspended from a fixed support by
spring with spring constant k. Let y_1 and y_2 denote the
displacements from their equilibrium positions of the top and
respectively, with the downward direction taken as positive.
Consider only vertical motion. (a) Write the Lagrangian for the
system, and find
the equations of motion. (Note the
equilibrium spring forces balance the gravitational forces,
which therefore drop out of the problem.)
(b) Determine (by hand) the normal mode frequencies and
displacement ratios y_2/y_1. (To check your result: The squared
(3± √5)/2 and the displacement ratios are y_2/y_1 = (1
-/+ √5)/2.) (c) Describe and indicate with arrows the nature of
the two normal mode motions,
showing both direction and approximate relative displacement of
each mass. Label with the frequency of each mode. Which is the
Try to explain "why", without reference
S7.2 (physical pendulum)
In class we saw the demonstration D2-13
RACING PENDULA. Show that the motion of a physical
pendulum consisting of a uniform rod of mass
M and length R is equivalent to that of a simple pendulum of
mass M and length 2/3 R.
S7.3 (physical penulum hanging
from a string)
uniform rod of mass M and length R hanging by its end from a
massless string of length l. (a) Write the Lagrangian using the
angles of the string
and rod as generalized coordinates, and expand it to quadratic
order in the angles. (b) Find the frequencies of the normal mode
oscillations for motion in a plane.
(c) Evaluate the frequencies in three cases: l=R, l >> R,
and l << R.
(Notes: (i) When you
write the kinetic energy, decompose it into the center of mass
motion and the motion relative to the
center of mass, and for the latter make use of the moment of
inertia relative to the center of mass (not relative to the end
of the rod).
(ii) I suggest you adopt units with R=g=1 for part (b). (iii) Answers for part (c): case 1:
case 2: √6g/R, √g/l; case 3: √3g/2R,
longitudinal Doppler effect) This problem refers to the
situation and notation with observers O1 and O2 described in the
notes of 10/20.
(a) Explain why the ratio t1/t0 is the Doppler factor for light,
i.e. the ratio of received frequency to transmitted frequency
for light sent from O1 to O2.
(b) Show that t1/t0 = √t1/t2. (c) The the
expression of t1 and t2 in terms of Dx and Dt to show that the
Doppler factor is √(1-v/c)/(1+v/c). (d) For v << c
it is useful to use an approximation. Expand the Doppler factor
to linear order in v/c to find this approximation.
HW6 - due at
beginning of class, Tuesday 10/18/11
9.2 (artifical gravity in
rotating space station)
9.7 (derivative of vector in
9.8 (direction of centrifugal
and Coriolis forces)
9.11 (Lagrangian in rotating
frame and equations of motion) The textbook suggests to
do this problem using vector notation,
without coordinates. However in this case a judicious choice of
coordinates simplifies matters, or at least eliminates the need
fancy footwork with vector identities). So, instead of using
vectors, I suggest you adopt cylindrical coordinates, with the z
along the angular velocity of the rotating frame. Then phi_0 =
phi + Omega t. Once you get the three Lagrange equations, you
can then show they are equivalent to (9.34).
9.22 (Larmor precession via
rotating frame trick)
9.26 (approximate trajectory
of falling particle on spinning earth)
S6.1 (tides for a constant
central force) It is often said - and I probably said
it - that the ocean tides can be traced to the fact that the
moon's gravitational force gets weaker with distance. But
actually we would have tides even if the moon's force on the
independent of distance, because of the direction dependence.
To see how this works explicitly, let's assume that the force
exerts is central and everywhere has the same magnitude as the
true gravitational force of the moon has at the center of the
That is, F =
-(GMm/d_0^2) dhat, where
M is the mass of the moon, m is the mass of the particle it is
acting on, d_0 is the distance from
he center of the moon to the center of the earth, and dhat is a unit vector
pointing from the moon to the particle. For this problem, you
will go through the same steps as in the textbook, but for the
modified force field. (a) Write down the tidal force field (as
in Eq. 9.12).
(b) Sketch the tidal force field at the surface of the earth (as
in Fig. 9.4). (c) Find the tidal potential (as in Eq. 9.13). (d)
Find the height
difference between high and low tides in the idealized model (as
in Eq. 9.18). How does your result compare with the true result
HW5 - due at
beginning of class, Tuesday 10/11/11
8.2 (two bodies in an
external field) Add parts: (c) If the external field is
not uniform but is, say, a -1/r potential, then the
center of mass motion no longer separates completely from the
relative motion. For instance, this is the case for the
earth-moon system in the field of the sun. Write the Lagrangian
of the earth-moon system using the center of mass
and the earth-moon relative vector as your coordinates. (d) Make
a Taylor expansion in the small ratio of the earth-moon
distance divided by the distance from the center of mass to the
sun, to extract the leading order term in this ratio that
appears in the Lagrangian. (If you're ambitious (not required),
work out the next to leading order term as well.)
(e) Try to guess (after a bit of thought) what the effect of
this extra term is on the earth-moon system.
8.3 (two masses, spring and
The solution in the book is only valid if the table is removed
at t=0. You can assume
this is the case, to simplify the problem.
S5.1 (dark matter)
One of the first pieces of evidence for the existence of dark
matter was the "flat rotation curves" of galaxies, meaning the
at large radii the orbital velocity of stars (as measured by the
Doppler effect) approaches a constant, rather than falling off
with distance. (a) How does the velocity of a test body in a
circular Newtonian gravitational orbit around a central mass
with the radius of the orbit? (b) Suppose the mass of a galaxy
is dominated by a spherical distribution of some kind of dark
with a radial mass density profile rho(r). What form must the
function rho(r) take in order for the orbital velocities of
stars to be
independent of r?
S5.2 (innermost stable
circular orbit in GR)
In general relativity there is a correction to the Newtonian
orbit around a central gravitating body that can be approximated
for weak fields and low velocities by an additional, attractive
1/r^3 term in the potential energy. The total effective
potential for a
body of mass m then has the form U =
-a/r + b/r^2 - d/r^3. Here a = GMm, where M is the mass of
the central body, assumed much
larger than m, b = l^2/(2m), where l is the angular momentum of
the orbiting mass, and d = b r_g, where r_g = 2GM/c^2 is the
"gravitational radius" or "Schwarzschild radius"
(for M = M_sun the gravitational radius is 3 km), and c is the speed
We can simplify the algebra by choosing units with GM = c = m = 1,
in which case a = 1, b = l^2/2, and d = l^2.
Assume you can use Newtonian dynamics.
(a) Are there any unstable circular orbits? If so, over what range
of radii do they exist?
(b) Find the minimum value of r for which a stable circular orbit
exists, and find its orbital speed. (That orbit is called the ISCO,
"innermost stable circular orbit".) Use the above units in your
algebra, but once you have the result re-express the radius of the
in terms of the gravitational radius and the orbital speed in terms
of the speed of light.
S5.3 (precession of perihelion of
Mercury in GR)
Using your results from S5.2, consider an elliptical
perturbation of any stable circular orbit, and compute the radial
w_r in terms of the radius r. (Eliminate the explicit l
dependence by solving for l in terms of r). The precession rate is
w_p = w_phi - w_r,
where w_phi is the angular velocity. Assuming that r >> r_g,
find the leading order precession rate by expanding in the small
(which is 2/r in our nice units). Restore the dimensionful
quantities using dimensional analysis, and evaluate the precession
rate for the
perihelion of Mercury, expressing the answer in seconds of arc per
century. For the radius use the semimajor axis of Mercury's
orbit. [Tip: When
restoring the dimensionful quantities, you need only multiply by
appropriate powers of r_g/2 and c (which are both equal to 1 in
these units) so that you wind up with something that has the
dimensions of inverse time.]
(Answer: w_p =
HW4 - due at
beginning of class, Tuesday 10/04/11
7.49 (charged particle in
uniform magnetic field - polar gauge)
Add parts: (d) Show that the Hamiltonian is the kinetic energy.
(e) Find the conserved quantity conjugate to the angle
about the z-axis.
This is the z-component of angular
momentum, which is not the same as "mvr" (much as the linear
momentum is not mv.)
(f) For orbits of constant rho in the
xy plane, show that the energy is proportional to the
z-component of angular momentum,
the coefficient of proportionality. (g) Show that dimensional analysis gives
the same result for the energy, up to a dimensionless constant
(which happens to be -1, and you could infer the sign from the
sign of the angular momentum). (h) Assuming the angular momentum
quantized in integer multiples of hbar, and using your classical
orbit results, find the quantized energies of a charge in a
in a uniform magnetic field. These
are the "Landau levels", except for a missing zero point energy.
(The quantum ground state has
zero angular momentum, but has a zero point energy hbar omega/2,
where omega is the orbital angular frequency.)
7.51 (pendulum with
constrained Cartesian coordinates)
S4.1 (ambiguity of Lagrangian)
Show explicitly that if you add a
total time derivative (d/dt)f(q^i,t) to the Lagrangian
all the added terms in Lagrange's equations cancel.
Allow for an arbitrary number of generalized coordinates q^i, and
make use of the Einstein summation convention when appropriate.
S4.2 (charged particle in uniform magnetic field - Cartesian gauge)
A uniform magnetic field of strength B in the zhat direction is
described by a vector potential A = Bx yhat.
Unlike the potential in problem
7.49 for the same magnetic field, this one is not obviously
rotation invariant, but it is y-translation invariant. (a) Write
the Lagrangian for a particle of
mass m and charge q using this gauge, and find the equations of
motion. (b) From now on, choose units with m = qB = 1, and show
that the solutions
that have no velocity in the z-direction are uniform circular
orbits with any center, with the same angular velocity you found
in problem 7.49.
(c) Show that the conserved quantity associated with the
y-translation symmetry is qB times the x-component of the center
of the circular orbit (!).
(d) Show that the Hamiltonian is the kinetic energy. (e) Show
directly that the equations of motion imply that the kinetic
energy is conserved.
S4.3 (constraint for bead on
spinning rod) Consider problem 7.21 again. (a)
This time include the angular coordinate as a degree of freedom,
and write the Lagrangian with a constraint with Lagrange
multiplier that imposes the condition that the rod spins with
constant angular velocity
omega. (b) Write the radial and angular equations of motion. (c)
Show that the Lagrange multiplier term in the angular equation
of motion yields
the torque on the bead.
S4.4 (bead on a
circular ring) Consider a bead of mass m sliding on a
frictionless ring of radius R in a fixed vertical plane.
(a) Write the Lagrangian and equations of motion using polar
coordinates based at the center of the ring, with angle theta =
0 at the top, including
the constraint r = R with a Lagrange multiplier. (b)
Suppose the bead starts from rest nearly at the top. Use the
Lagrange multiplier to determine
(i) at what angle the force of constraint goes to zero (this is
where a bead sliding on the surface of a sphere would leave the
(ii) the magnitude and direction of the force of constraint when
the bead reaches the bottom of the ring.
S5.5 (catenary) A
flexible rope or chain hanging from two points forms a shape
called a "catenary", which is actually a hyperbolic cosine!
You can show this directly using the condition that the net
force on an infinitesimal segment of rope vanishes, but you can
also show it using
variational calculus. The calculation is nearly identical to the
problem of the soap film bounded by two rings that you already
solved on HW1,
but you need to impose the constraint that the rope has a given
fixed length b. Suppose the rope hangs from two points (x1,
y1) and (x2,
express the gravitational potential energy of of rope as an
integral involving the function x(y). Using the method of
find the equation on x(y) implied by the fact that the potential
is a minimum for all variations of x(y) that keep the rope
Then re-expressed this as an equation on y(x), and show that the
solutions are given by y = c + y0 cosh[(x-x0)/y0] (this should
Compared to the soap film problem, there is an extra constant
here, c. This is necessary since there is one more condition to
besides the positions of the endpoints, the length of the rope
must be matched.
HW3 - due at
beginning of class, Tuesday 9/27/11
7.20 (particle on a helix)
(Note: this could equally well be solved by writing down the
total mechanical energy and setting its time derivative to
7.21 (bead on a spinning rod)
Let's add two interesting parts (b): Suppose the rod has length
L from the pivot to one end, and the bead is released with zero
radial velocity at some initial radius r_0. In the limit that
r_0 goes to zero, what velocity does the bead have when it
reaches the end of the rod? Give the
components of the velocity along the rod and perpendicular to
the rod. Compare your result to what you would expect from
(c) (i) Explain why the Hamiltonian is a conserved quantity, and
show that it is equal to E - Ω J,
where E and J are the kinetic energy and angular momentum
of the bead, and Ω is the angular velocity of the
rod (called lower case omega in the book). (ii) Use this
conservation law to evaluate the kinetic energy when
the bead leaves the rod, and show that it agrees with what you get
using your results from part (b).
7.22 (pendulum in an
accelerating elevator) Change the second part: instead
of finding the equation of motion, show that the Lagrangian
itself is equal,
up to addition of a total time derivative, to the Lagrangian for
a non-accelerating pendulum with g replaced by g+a. (The total
time derivative does not
affect the equations of motion.) (Hint: You'll need to do an integration by
parts on t d(cos theta)/dt.)
7.37 (two masses connected by
a string through a hole in a table) Add part (e) What
does dimensional analysis (and a bit of physics logic) tell you
about the oscillation frequency in part (d)? (The problem
involves m, L (length of the string), g, and r_0.)
S3.1 (double pendulum)
(a) Write the Lagrangian for the double pendulum illustrated in
Fig. 7.3, assuming the masses are equal. (b) Expand the
Lagrangian to quadratic order in the amplitude of the angles,
dropping the higher order terms. (This describes small
oscillations about the equilibrium.
Later we'll find the normal modes.)
S3.2 (conserved quantity
for bead on a spinning hoop) Consider Example
7.6, but call the angular velocity of the hoop Ω. (a)
Explain why neither the
total mechanical energy E nor the angular momentum J about the
vertical axis are conserved. (Hint:
What work and torque are exerted on the bead?)
(b) Show that the Hamiltonian is equal to E - Ω J. (c)
Explain why the system (bead on a spinning hoop)
is neither symmetric under time translation nor under rotation.
[N.B. I refer here to the system,
not to the Lagrangian
(d) On general grounds, the
Hamiltonian is conserved, since the Lagrangian has no explicit
What is the symmetry whose corresponding conserved
quantity is the Hamiltonian, E - Ω J?
S3.3 (least action for
free-fall) Consider a particle of mass m in a uniform
gravitational potential U(y) = mgy. Let y represent the height
of the particle,
and consider the action for vertical paths y(t) over the time
interval [-T,T], with y(-T) = 0 = y(T). The action for the path
y(t)=0 is zero, of course.
(a) Show that the paths with uniform velocity on the way up and
the way down have minimum action if they reach the height h =
1/2 g T^2, and
show that the value of that minimum action is -1/4 m g^2 T^3.
(b) Show that the classical path reaches same same height as you
found in part (a),
and that the classical action on that path is -1/3 m g^2 T^3.
(c) Show that the m g^2 T^3 factor
follows from dimensional analysis.
HW2 - due at
beginning of class, Tuesday 9/20/11
6.6 (ds in various
6.16 (geodesics on a sphere)
6.19 (soap bubble)
Note: Assume y1 and y2
are both positive.
7.3 (2d oscillator using
Lagrangian) Modify this problem as follows: (a) Write
the Lagrangian using polar coordinates.
(b) Find the equations of motion for phi and r. (c) define the
angular momentum by J = mr2phidot, and use this to
phidot from the equation for r. (d) Find the circular orbit
(motion with rdot=0) for a given J. (e) Describe the radial
for orbits with nonzero rdot and nonzero J.
and relative motion of two bodies in 1d) Modify
this problem as follows: Instead of a spring, suppose the
U is an arbitrary function of the separation of the two
particles, U(x), with x = x1 - x2. Replace
part (c) by "describe the resulting
dynamics". Add part (d) Now let the two masses be different.
Find the Lagrangian in terms of the center of mass position
X = (m1x1
is the total mass M = m1 + m2, and x, and
again describe the resulting dynamics.
(The reduced mass µ = (m1m2)/M will
arise. Express your result using this quantity.)
S2 (motion about equilibria
for bead on spinning hoop)
Example 7.7 discusses motion near equilibria of a bead on a
spinning hoop. It does so by expanding the equation of motion
about the equilibrium. Instead, let's expand the Lagrangian
about the equilibrium. Hence: the Lagrangian (7.68) can be
in the form L = 1/2 m R^2 qdot^2 - U(q), where I'm using q to
stand for theta. At an equilibrium point q_0 we have U'(q_0)=0.
(a) Make a Taylor expansion of U(q) about an equilibrium point
q_0. Keep terms out to quadratic order in the displacement from
equilbrium. (b) Stability is determined by the sign of the
quadratic term in U(q): if it's positive the motion is a stable
it's negative the motion runs away exponentially. (i) Show that
below the critical angular velocity, the equilibrium at theta=0
stable and that at theta=π is unstable. (ii) Show that above the
critical angular velocity the equilibria at theta = 0 and π are
while the two new equilibrium points are stable.
HW1 - due at
beginning of class, Tuesday 9/13/11
Read the Chapter summaries of Chapters 1-5, and browse the
chapters. If you see anything you're not comfortable
with read that part of the chapter.
1.32 - (Newton's 3rd law
failure for magnetic forces between point charges)
read, but do not do, this problem
1.33 (Newton's 3rd law for
magnetic forces between closed current loops) This problem
A couple of hints: (i) Evaluate grad1(1/s),
a part of the integral vanishes
because it's the line integral of a gradient around a closed loop. (ii) To establish the 3rd law,
the force of loop 2 on loop 1 is proportional to ∫∫ (dr1. dr2) s/s3, which
obviously changes sign when
considering the force of loop 1 on loop 2.
4.4 (Particle on table
attached to string through hole)
4.24 (Gravitational potential
of an infinite rod) Do only parts (a) and (d). [Once
you find a potential that
gives the force, you don't need to check that the curl of the
force vanishes, since curl grad U = 0 for any U.]
4.36 (ball-pulley-mass gizmo)
4.38 (pendulum period with
large amplitude) For part (b), do not use the "complete
elliptic integral" function.
Instead, just use whatever software you like to evaluate
numerically, by brute force, to sufficient accuracy, the
integral, and make the plot in question.
4.39 (pendulum period,
amplitude expansion) Add part (d): Plot the analytic
result you get for the approximate
period on the same graph as the numerical result from 4.38.
4.41 (virial theorem)
4.43 (central, spherically
symmetric forces) Add part (c) Better yet, show
directly by construction that one can
find a U such that F = -