Phys410 - Classical Mechanics
University of Maryland, College Park
Fall 2012, Professor: Ted Jacobson
Problems from Taylor, Classical Mechanics

HW0 - due at beginning of class, Tuesday 9/04/12

HW12 - pdf file - due at beginning of class, Tuesday 12/12/12

HW11 - pdf file - due at beginning of class, Tuesday 12/04/12

HW10 - due at beginning of class, Tuesday 11/20/12

13.12 (bead on spinning rod)

13.13 (particle on cylinder with restoring force)

S10.1 (
Hamiltonian for non-relativistic particle, and cyclotron motion)

(a) Find the Hamiltonian for a non-relativistic free particle of mass m.
(b) Now include an external electromagnetic field,  and show that for a charge e the Hamiltonian is H = (p - eA)2/2m+ eV, 
where V and A are the scalar and vector potentials. Note that in a magnetic field (with no scalar potential), the Hamiltonian is
given by H = 1/2 mv^2, the kinetic energy (but v must be regarded as a function of the momentum, the position, and the vector potential).
(c) Suppose there is just 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_0, 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 (see Problem S4.2). Where does the freedom of the y coordinate of the center of the orbit arise
in solving the equations?
Express the energy as a function of the radius of the orbit, w_0, and m.

S10.2 (Hamiltonian for relativistic particle, and cyclotron motion)
The action for a relativistic free particle is
-m∫ dt (1-v2)1/2, where v = dx/dt, i.e. the Lagrangian is L = -m (1-v2)1/2.
(a) Find the momentum p conjugate to x.
(b) Show that
p2 + m2 = γ2m2, where  γ = (1-v2)-1/2.
(c) Show that the Hamiltonian is H = (p2 + m2)1/2 (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 + m2]1/2+ eV, 
where V and A are the scalar and vector potentials. Note that in a magnetic field (with no scalar potential), 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 (as in the nonrelativistic case,  Problem S4.2).
(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.

(g) Express the energy
as a function of the radius of the orbit, w_0, and m. Show that in the non-relativistic limit you recover S10.1(d).
(Gravitational redshift)
Consider two light pulses sent radially outward from radius r_1 to radius r_2 in the Schwarzschild spacetime,

ds^2 = F (r) dt^2 − 1/F(r) dr^2/c^2 − (r^2/c^2)(dθ^2 + sin^2 θ dφ^2),  

where F(r) = 1 − r_g/r, and r_g
= 2GM/c^2 is the Schwarzschild radius. Suppose the emission events are separated 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 ∆s_1 between the emission of the two pulses at r_1? (c)
What is the proper time ∆s_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?
[Suggestions: (i) It may help to visualize this using a spacetime diagram, e.g. showing constant r lines as vertical and light pulses as
diagonal. (ii) To answer part (a) you should use the t-translation symmetry of this spacetime.]

S10.4 (Motion in a gravitational wave)

The line element describing a plane gravitational wave of frequency ω propagating in the x direction can be written (in c = 1 units) as

ds^2 = dt^2 − dx^2 − [1 + h sin[ω(t−x)] dy^2 − [1 − h sin[ω(t−x)] dz^2. 

(a) Show that if a test mass governed by the action −m ∫ ds is initially “at rest” in these coordinates (i.e. if dx/dt = dy/dt = dz/dt = 0 initially),
then it remains so for all times.
(b) Show that if a ring of independent test masses in the x = 0 plane is at rest with respect to these coordinates at t = 0, then the physical shape
of the ring, defined by the invariant distance between points on the ring at a constant value of the t coordinate, oscillates between two ellipses with
perpendicular major axes.

HW9 - due at beginning of class, Tuesday 11/13/12

15.46 (approaching and receding Doppler factors)

15.48 (transverse Doppler effect)

15.87 (pion decay to photons: pion velocity and photon angle)
(Suggestion: The velocity of the CM frame can be obtained from the total momentum divided by the total energy.)

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 and neutrino)
(Suggestion: You are only being asked for the speed, so try to get the gamma factor directly, without evaluating any energy or momentum
Express the neutrino 4-momentum in terms of the pion and muon 4-momenta, take its scalar product with itself,  and impose the mass shell conditions.)

S9.1 (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 relation
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.

S9.2 (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, creating electron-positron
pairs. The reaction is gamma1 + gamma2 -> e+ +  e-. (a) If the two photons collide head-on, show that for a given
energy E2 of photon 2, the minimum
E1 of photon 1 required to 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) a mid-infrared photon of energy 25 meV.
(The actual energy cutoff of cosmic ray photons originating from more
than around 100 million light years comes from annihilation on this infrared background radiation.)

HW8 - due at beginning of class, Tuesday 11/06/12

15.7 (muon lifetime in cosmic ray showers)

15.12 (Lorentz contraction)
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?
Answer the question about the length two ways: i) Using the time for the pipe to pass, together with the speed of the pipe; ii) Using the length
contraction formula (15.15). This should explain how length contraction is the "flip side" of time dilation.

15.56 (rest energy in chemical reaction)

15.57 (rest energy in nuclear reaction)

15.58 (rest energy vs. kinetic energy)

15.71 (collision energy to create a particle)
See the sections in the textbook on CM frame and Threshold energies, p. 645. And/or, use Hint: The E^2 - p^2 c^2 is the same in all inertial frames,
where E is the total energy of all particles and p is the magnitude of the total momentum.

15.75 (parent particle reconstruction)

S8.1 (relativistic longitudinal Doppler effect) This problem refers to the situation and notation with observers O1 and O2 described in the latex notes.
(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 ∆x and ∆t 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.

HW7 - due at beginning of class, Tuesday 10/30/12

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 into account
that the ratio of carbon to oxygen mass is 3/4. Show that the observed frequency ratio is smaller
by about 12%.
(See 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 the difference 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 emitted in 
transitions between the vibrational levels. For an oscillator of frequency
ω the quantized energy levels have energies (n+1/2)ℏω.
The energy difference between two adjacent levels is
therefore ℏω, so the emitted photon frequencies are ω. The inverse wavelength
is thus proportional to
ω, the frequency of the oscillator.

S7.1 (masses suspended by springs)
Consider a mass m suspended by a spring with spring constant k from another mass m which is suspended from a fixed support by another
spring with spring constant k. Let y_1 and y_2 denote the displacements from their equilibrium positions of the top and bottom masses
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 frequencies are
(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 higher one?
Try to explain "why", without reference to equations.

S7.2 (physical pendulum hanging from a string)
Consider a 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 M=R=g=1 for part (b). (iii) Answers for part (c): case 1: 3.1√g/R, 0.8
√g/R; case 2: √6g/R, √g/l; case 3: √3g/2R, 2√g/l.)

OPTIONAL EXTRA CREDIT PROBLEM: not difficult, but nice. If you choose to do this problem it will be graded and added to your worst previous
homework score, normalized as 25/100. E.g., if you previously got 60/100, that will be adjusted to 85/100 (totals greater than 100/100 are allowed).

S7.3 (charged particle on a cylinder in a transverse magnetic field)

A particle of charge q and mass m is confined to a cylinder of radius R, centered on the z axis, in the presence of
a uniform magnetic field in the x direction, B = B xhat. Use cylindrical coordinates (rho, phi, z) for the position of the charge. 
(a) Show that A = By zhat is a vector potential for this magnetic field.
(b) Write the Lagrangian for the charged particle.
(c) Now rewrite the Lagrangian in units with $m=R=qB=1$ (check that this is allowed), and find an expression for the energy of the particle.
(d) Show that the canonical momentum p_z conjugate to z is conserved. (What symmetry is responsible for this?)
(e) Using the Lagrangian method, find the force of constraint that holds the particle on the cylinder as a function of its position and velocity.
(f) Find the effective potential for a given value of p_z, and carefully sketch or plot it (from -π to π) for the values p_z = 0, 1/2, 1.
(g) Describe the qualitatively different possible motions of the charge for each of the three values of, p_z in part (f).

HW6 - due at beginning of class, Tuesday 10/16/12

9.2 (artifical gravity in rotating space station)

9.7 (derivative of vector in rotating frame)

9.8 (direction of centrifugal and Coriolis forces)
[Be sure to describe the components both tangent and normal to the Earth's surface.]

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, with constant angular velocity, it's not hard to use coordinates and it eliminates the
need for fancy footwork with vector identities. So, instead of using vectors, I suggest you adopt Cartesian coordinates, with
the z axis along the angular velocity of the rotating frame. The coordinates in the two frames are related by

equation1,   so      equation.

Once you get the three Lagrange equations, you can then show they are equivalent to (9.34).
[Tip: The kinetic energy term is invariant under a rotation of the velocity vector!] 
[You could also do it with cylindrical coordinates, in with phi_0 = phi + Omega t.]

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 that the ocean tides can be traced to the fact that the moon's
gravitational force gets weaker with distance. But in fact we would have tides even if the moon's force on the earth were
independent of distance, because of the direction dependence.  To see how this works explicitly, let's assume that the force the moon
exerts is central and everywhere has the same magnitude as the true gravitational force of the moon has at the center of the earth.
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 (9.18)?

HW5 - due at beginning of class, Tuesday 10/09/12

8.2 (two bodies in an external field) Add parts: (c) If the external field is not uniform 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 presence of the sun. Write the Lagrangian of the earth-moon system using the earth-moon center of mass and
relative position vector as your coordinates. Explain which terms spoil the separation of the center of mass and relative motions.
(d) Look at en.wikipedia.ord/Lunar_theory#Newton to see how Newton analyzed this using the vector sum of forces.

8.3 (two masses, spring and gravity) Note: 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.

S5.0 (spinning rod with finite inertia) Heretofore we considered a rod, forced to spin at a fixed angular velocity, and with a
bead constrained to slide along the rod. In that problem, neither energy nor angular momentum are conserved. Now let's
change the problem and let the rod have finite moment of inertia I, and let it spin freely on a frictionless axis, so that energy
and angular momentum are conserved. (a) Write the Lagrangian of the rod + bead system, using for generalized coordinates
the angular position of the rod and the radial position of the bead. Let the moment of inertia of the rod be I. (b) Write the
total energy for a fixed angular momentum in terms of the radial kinetic energy of he bead plus an effective potential, and
sketch the effective potential. (c) Suppose the rod is spinning and the bead begins with no radial velocity at some nonzero
radius. Describe qualitatively the subsequent motion of the system, including what happens in the limit of infinitely late time.
(d) Suppose the rod is spinning and the bead is moving towards the center. Describe the three possible subsequent motions.
(Which of these occur is determined by the energy for a fixed angular momentum.)

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 fact that
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 scale
with the radius of the orbit? (b) Suppose the mass of a galaxy is dominated by a spherical distribution of some kind of dark matter
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? (c) For this rho(r), how does the mass M(r) inside r, the force F(r) on a test mass, and the potential U(r) of a
test mass depend upon 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
radius" or "Schwarzschild radius" (for M = M_sun the gravitational radius is 3 km), and c is the speed of light.
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) Show that no circular orbits exist for l < √12, and that for each l > √12 there is one stable and one unstable circular orbit.
Sketch the form of the potential for these two cases, as well as for the critical case l = √12.

(b) Find the lower limit of r for which a stable circular orbit exists, and find the corresponding orbital energy and speed.
(The limiting orbit is marginally stable, and is called the ISCO, "innermost stable circular orbit".) Use the above units in
your algebra, but once you have the result re-express the results in terms of the dimensionful quantities GM, c, and m.
(The speed will not be small compared to c, so the Newtonian treatment is not justified. Nevertheless, when properly
understood this result for r_ISCO agrees with the one in general relativity.)

(c) Show that the unstable circular orbits exist between r = 6 and r = 3, and find the energy of the orbit at r = 3.

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 oscillation frequency
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 number r_g/r
(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 elliptical
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 = 3r^(-5/2),  42"/century.)

HW4 - due at beginning of class, Tuesday 10/02/12

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.)
For orbits of constant rho in the xy plane, show that the energy is proportional to the z-component of angular momentum, and find
the coefficient of proportionality. (Be careful about the sign of the angular momentum.)
(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 is
quantized in integer multiples of hbar, and using your classical orbit results, find the quantized energies of a charge in a planar orbit
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)
We know that adding a total time derivative to the Lagrangian only adds a constant to the action (for fixed endpoints), so does not change the EL-equations.
This problem checks this explicitly, and is an exercise in using partial derivatives and index notation.
(a) Show explicitly that if you add
a total time derivative (d/dt)f(x,t) to the Lagrangian for a particle in one dimension, all the added terms in Lagrange's equations cancel.
(b) Now allow for an arbitrary number of generalized coordinates q^i, and do the same analysis. 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 zero 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 (!).

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 bead slides on a rod that 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 (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, y2), and
express the gravitational potential energy of of rope as an integral involving the function x(y). Using the method of Lagrange multipliers,
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 length fixed.
Then re-express this as an equation on y(x), and show that the solutions are given by y = c + a cosh[(x-x0)/a] (this should be easy).
Compared to the soap film problem, there is an extra constant here, c. This is necessary since there is one more condition to be met:
besides the positions of the endpoints, the length of the rope must be matched. [For this problem you need not find the relation between a and b.
However, if you are curious, assume the two ends are at the same height, choose the origin of coordinates so that x0=0 and c=0,  and use the length
constraint to show that b/2a = sinh(d/2a), where d is the distance between the ends of the chain. This transcendental equation determines a uniquely.] 

HW3 - due at beginning of class, Tuesday 9/25/12

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 zero.)

7.21 (bead on a spinning horizontal 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 dimensional analysis.
(c) (i) 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?)
(ii) 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. (iii) 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 a time derivative.)

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, and show that it takes the form of two oscillators
coupled by a term quadratic in the velocities. (This describes small oscillations about the equilibrium. Later we'll find the normal modes.)
S3.2 (bead on a spinning tilted rod)  Reconsider Problem 7.21, where now the rod makes an angle α with the vertical and spins about a vertical
axis that goes through the bottom end of the rod. (a) Find the Lagrangian for the bead on the rod in a uniform downward gravitational field g, using
the distance s from bottom end of the rod as the generalized coordinate. (b) Identify the effective potential, and sketch it for no rotation and for a
nonzero angular velocity ω
of the rod. (c) For a generic value of ω, show that there is one equilibrium point (other than s = 0), and determine
whether it is stable or unstable.

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 among the paths with uniform velocity on the way up and the way down, the action is minimized for the one that reaches 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 Newtonian path reaches same same height as
you found in part (a) (a fluke?), and that the action for that path is -1/3 m g^2 T^3. (c) Show that the
m g^2 T^3 factor follows from dimensional
analysis, and show that you can choose units with m=g=T=1 (if you want to, you can make this choice at the beginning of the problem).

HW2 - due at beginning of class, Tuesday 9/18/12

6.6 (ds in various settings)

6.16 (geodesics on a sphere)

6.19 (soap bubble) Note: Assume y1 and
y2 are both positive. [Hint: I found it simplest to start with x(y) in the area functional,
but then to switch to y(x) after deriving the E-L equation.] [Note: This problem contains considerable subtleties which you need
not enter into unless you feel like it. Namely,  (i) there is no solution of this form when the two rings are sufficiently far apart
given their sizes, and (ii) not every solution to the E-L eqn is a minimum area solution. In fact, for each ring configuration
admitting a minimum area surface connecting the rings, there is a second surface at which the area functional is stationary
but not at a local minimum.]

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 show that the phi
E-L eqn implies that it is conserved. (d) Express the angular part of the kinetic energy in terms of J and r (with no phi
dependence), and combine this with the oscillator potential to obtain an "effective potential". Sketch the effective potential.
(e) Use energy conservation to express the radial acceleration in terms of the effective potential. Find the circular orbit
(motion with rdot=0) for a given J, and describe the radial motion for orbits with nonzero rdot and nonzero J.
(f) It is not obvious using these variables that the orbits are closed. Start over and do the problem using Cartesian coordinates,
and using that prove that the orbits form closed ellipses.

7.8  (
CM and relative motion of two bodies in 1d) Modify this problem as follows: Instead of a spring, suppose the potential
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 + m2x2)/M,  where M is the total mass M = m1 + m2, and x. The so-called "reduced mass" μ = (m1m2)/M will arise.
Express your result using this quantity.
Show that the center of mass moves at constant velocity, and that the relative position
satisfies Newton's 2nd law with a mass

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 written
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 oscillation, if
it's negative the motion runs away exponentially. (i) Show that below the critical angular velocity, the equilibrium at theta=0 is
stable and that at theta=π is unstable. (ii) Show that above the critical angular velocity the equilibria at theta = 0 and π are both unstable,
while the two new equilibrium points are stable.

HW1 - due at beginning of class, Tuesday 9/11/12

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 is intriguing.
A couple of hints: (i) Evaluate grad1(1/s), and use that to show that 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, show that
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 = - grad U.