Phys410 - Classical Mechanics
University of Maryland, College Park
Fall 2011, Professor: Ted Jacobson
Homework
Problems from Taylor, Classical Mechanics

HW0 - 7due at beginning of class, Tuesday 9/06/11

HW12 - due at beginning of class, Tuesday 12/13/11

16.31 (Earthquake distance)

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) - T (∂y/∂x)2] dx

where y(x,t) describes the transverse displacement of the string.

(a) Instead of using the function y(x) as the generalized coordinate, rewrite the Lagrangian using the coefficients y_n, n = 1,2,3,...
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 of
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 also changes,
but I think the effect of this is smaller.) (The note is a superposition of all the harmonics, dominated by the fundamental.)
(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 increase?
(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 oscillators of
frequencies w_n = n w_1. Explain why the string doesn't have an infinite zero point energy!

S12.2 (String with a bending modulus)

In class we considered a stretched string whose only restoring force is due to the tension. Suppose the string is somewhat stiff and
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/∂x
2)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.

S12.3 (Relativistic 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 "worldsheet".

Rather than ask you to work out the equations for a general motion, I'll restrain myself and ask you only to consider circular
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 configurations,
is  -T ∫ dA = -2πT ∫  R [1 - (dR/dt)2/c
2]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 dt.

(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 explicitly.
(f) Using (d) and (e), show that the string satisfies a harmonic oscillator equation, and identify the frequency of the oscillator.
(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 state
by minimizing the energy subject to the uncertainty relation p R ≥ hbar. Use units with hbar = 1. Then find this zero point size using
general units, i.e. restore the appropriate factors of hbar, c and, 2πT

HW11 - pdf file - due at beginning of class, Tuesday 12/06/11

HW10 - due at beginning of class, Tuesday 11/29/11

13.12 (bead on spinning rod)

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 potential
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 (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.
(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 radius,
(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 initial velocity
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. For
(i) and (ii) you can easily show it directly (again, without actually evaluating the times). For (iii) the most convenient thing
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 challenge,
try to actually demonstrate that tau_iii is longer by using the conserved energy and effective potential. I didn't succeed in finding
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 events 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 ∆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

S9.1 (Approximate 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 scattering)
If a high energy electron collides with a low energy photon, much of the electron energy can be transferred to the outgoing photon.
This is called "inverse Compton scattering", even though it's the same as Compton scattering when viewed in a frame in which the electron
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 photon
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

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?

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)

15.75 (parent particle reconstruction)

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 and neutrino)

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 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.

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, creating electron-positron
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 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) 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 into account
that the ratio of carbon to oxygen mass is 3/4. Show that the observed frequency ratio is smaller
(See
www.phy.davidson.edu/StuHome/jimn/CO2/Pages/CO2Theory.htm 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. 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 ray showers)

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)
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)
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 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.)

S7.4 (relativistic 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 rotating frame)

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 for
fancy footwork with vector identities). So, instead of using vectors, I suggest you adopt cylindrical coordinates, with the z axis
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 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 10/11/11

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 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.

S5.1 (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?

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 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) 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 ISCO
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 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),  43"/century.)

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, and find
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 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)
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 sphere), and
(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, 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-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 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.

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 zero.)

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 dimensional analysis.
(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 t-dependence.
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 settings)

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 eliminate
phidot from the equation for r. (d) Find the circular orbit (motion with rdot=0) for a given J. (e) Describe the radial motion
for orbits with nonzero rdot and nonzero J.

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, 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 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 9/13/11

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 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.