Phys410 - Classical Mechanics
University of Maryland, College Park
Fall 2012, Professor: Ted Jacobson
Notes, Demos and Supplements

In these notes I'll try to just indicate the topics covered in class.
I'll also mention things I talk about in class that are not also in the textbook,
as well as supplementary material, if they are not in last years notes.
Please do not assume that these notes are even roughly complete.

Just announced discovery of a star with orbital period 11.5 years around the supermassive black hole at the galactic center:

Thursday, Oct. 18

+ Regarding the question from last class about how to explain Coriolis stabilizing effect on orbits near the Lagrange
points, from an inertial point of view, I suggested a simpler question first: Consider the potential of a single, central mass. In a frame
rotating with fixed angular velocity, the centrifugal potential combined with the -1/r potential produces a circular ridge of equilibrium
points, corresponding to the circular orbit with the given angular velocity. This looks unstable, but the Coriolis force must stabilize
it since we know that from an inertial point of view, a perturbed circular orbit is an elliptical (even closed), stable orbit. It would be
interesting to plot an elliptical orbit in the rotating frame...

+ Reviewed again the method of solving coupled oscillator problems.  Made the point that if det M is not zero, then the condition
det(K - w^2 M)=0 is equivalent to det(M-1K - w2 I)=0, so w2 is an eigenvalue of M-1K, and the vector a is an eigenvector of M-1K.

+ Started Special Relativity. See latex notes.

Tuesday, Oct. 16

EXAM: Thursday Oct. 25
 - Review session in class: Tuesday Oct. 23. Bring your questions about text & homework.
 - No homework this week.
 - Closed book, one 8.5 x 11" crib sheet (both sides) allowed.
 - Topics: everything up through non-inertial frames.

Earth ocean tides:

Depending on location on the Earth, the tides are semi-diurnal, diurnal, or mixed...

Jupiter's moon Io:

Discussed tidal friction taking energy but not angular momentum out of the orbit, circularizing the orbit.
Spin angular momentum counts too, but spin relative to a tidal bulge also dissipates energy. So moons
tend to get phase-locked in their rotation, so they rotate on their axis at the same rate as they orbit the planet.
Io maintains a small ellipticity ~ 0.004, due to perturbations of the other moons. This keeps tidal heating going,
which supplies the energy for the volcanism on Io.  A student suggested that maybe the solar perturbation could
play a role. In fact this is way too small: the tidal field due to a body of mass M at distance d is of order (GM/d^3)
times the separation of the points at which the field is evaluated. So the ratio of the tides on Io due to Jupiter compared
to those on the sun is (M_J/M_S)(d_S/d_J)^3 ~ (0.001)(1800)^3 ~ 6,000,000. So the sun's effect is really negligible.

Lagrange points:

Explained that stability is due to the Coriolis force. A student asked after class how to see this stability in the inertial
frame. I don't know. It seems hard to see.

Explained how to solve a coupled oscillator problem like this by writing the Lagrangian, finding the equations of motion,
writing them in matrix form, and assuming harmonic time dependence for a solution. The equation for harmonic motion
is then (K - w^2 M)a = 0, using matrices K and M. The existence of a vector a that solves this implies det(K - w^2 M)=0,
which is a quadratic equation for w^2, in the case of two degrees of freedom. There are two solutions for w^2; for each
solution there is an eigenvector a that specifies the ratios of the displacements of the two degrees of freedom.
In a normal mode the time dependence is harmonic and this amplitude ratio is constant.
You'll solve this on the next homework.

Several examples are worked out in the textbook.

- We worked out the Lagrangian for a double pendulum, with help from various students. Then I expanded it in small angles,
up to quadratic order. This is then a coupled oscillator and can be solved by the same method as above. This example
is also discussed in the textbook. I see there that Taylor obtained the kinetic energy of the second mass a little differently
than we did. In class we just worked out the Cartesian components of the velocity of the second mass and summed their squares.
In the book, he notes that the velocity of the second mass can be written as a vector sum of two vectors with a relative
angle of phi2-phi1, and then he just implicitly uses the vector formula (or law of cosines) (A+B)^2 = A^2 + B^2 + 2 AB cos theta.
That's definitely simpler than what we did.

- Introduced a nice coupled oscillator: a solid rod of length R hanging from a massless string of length l, moving in a plane.
There are two degrees of freedom. The rod kinetic energy can be evaluated from a center of mass part and a rotation about the
center of mass. You'll solve the small angle approximation to this on the next homework.

Thursday, Oct. 11

+ Non-inertial frames: rotational motion with constant angular velocity.

+ Rotating frame of reference: showed Rotating reference frame: movie, then wrote down the Lagrangian for free particle
motion in a plane as described in a uniformly rotating frame of reference. phi_in = phi_rot + Omega t, where Omega is the angular
velocity, so phido_in = phidot_rot + Omega. Insert this into the kinetic energy to find the Lagrangian

L = 1/2 m rdot2 + m r^2 Omega phidot + 1/2 m r^2 Omega^2

The second term is the velocity dependent Coriolis potential, and the third term is minus the centrifugal potential.
The Coriois potential is exactly what you'd get for a uniform magnetic field perpendicular to the plane, and the centrifugal
potential is an unpside down oscillator potential.

+ Wrote out the EL eqns. For the r-component found centrifugal and Coriolis terms, and also a -r phidot^2 term in the r-component
of the acceleration. I derived it by evaluating the second time derivative of the position vector using rhat and phihat unit vectors
which change with time. For some reason I neglected to point out that this term is nothing but the centripetal acceleration!

+ Compared size of Coriolis and centrifugal: F_cor/F_cent ~ phidot/Omega. This is for the radial component. I should have also
checked the ratio for the angular component of the Coriolis force, which gives F_cor/F_cent ~
rdot/r Omega. So the real relevant
comparison is the velocity of the ball compared to the tangential velocity of the platform. In fact, even for the largest
tangential velocity, at the edge of the platform, this is around 2.5, as you can see by playing the video in slow motion: the table turns
1/8 of a turn while the ball rolls all the way across: 2r/(2πr/8) = 8/π ~ 2.5. So, contrary to what I said in class, it seems the
Coriolis force is indeed dominant. [I also said, when I thought the opposite, that the centrifugal potential could produce the arc
of the ball, but that's clearly not true: look at the ball, it heads in  radially toward the center. The centrifugal potential vanishes
at the origin and in any case would repel radially rather than bending the path.]

+ Rotating water tank & parabolic surface: movie shows that the surface of water in a rotating tank assumes a parabolic form.
We can understand this as for the surface of the ocean tides: the surface must be an equipotential surface of the combined
gravitational and centrifugal potentials. (The only other force, water pressure, is normal to the surface.) Thus
mgh - 1/2 mr^2 Omega^2 = const, i.e.  h = (Omega^2/2g)r^2.

- 3d Coriolis and centrifugal forces: expressing the time derivative of a vector in the inertial frame in terms of the time derivative in the
rotating frame plus Omega x vector, I derived eqn (9.34). Magnetic interpretation of Coriolis force.

+ In class I got confused trying to give an intuitive explanation of why atmospheric wind flowing in toward a low pressure region along a
line of latitude from East to West would be deflected North. I think the point is that the air flowing radially towards the low pressure would not
flow along a line of latitude. Rather it would flow in, eg from slightly higher latitude. Since it's flowing into a lower latitude
where the earth surface has a higher tangential velocity, it falls behind, curving Northward. Similarly, for example, air flowing north
into the low pressure area comes from a part of the earth that is moving faster, so it shoots ahead, curving to the East.

+ I didn't say this but note: the Coriolis force 2m v x Omega generally has a component tangent to the earth but it also has a component
in the vertical down or up direction (perpendicular to the surface of the earth) , depending on the velocity vector. When you do problem 9.8,
be sure to take into account the vertical components...

+ Starting coupled oscillations, considered the Lagrangian for a mass on a spring hanging from a mass on a spring. Showed that the gravitational
potential term cancels out if one sets the generalized coordinates to zero at the equilibrium positions of the masses. A student pointed out
that you could also see this by completing the square in the Lagrangian.

Tuesday, Oct. 9

+ Non-inertial reference frames: case of translational motion with uniform acceleration.

+  Horizontally accelerating car example, as in book.

+  Application to tides, derivation of the shape of the ideal ocean surface.

+  Spring tides and neap tides.

Thursday, Oct. 4

+ Derivation of elliptical Kepler orbit: key is to define u = 1/r and write dr/dt = dr/du du/dphi dphi/dt...

+ Kepler's laws: 1) ellipse, 2) equal areas in equal times, 3) (period)^2 = (4π^2/G(m1+m2))(semi-major axis)^3.
The equal areas law is implied by conservation of angular momentum: dA/dt = l/µ, where l is the angular momentum.
The third law follows partly from dimensional analysis, but that does not explain why the coefficient is independent
of the dimensionless ellipticity and mass ratio m1/m2.
There must be a deep reason for this. Please let me know if you
figure this out. The textbook derives this relation by setting the area πab equal to the period times dA/dt = l/2µ.

+ Perturbations from solar oblateness (very small) and other planets (large, mostly Jupiter) cause Mercury's perihelion to precess
around 500 seconds of arc per century. You can get the planetary effect by averaging the potential over time, hence smearing
the mass of Jupiter along a ring coincident with Jupiter's orbit, and computing the corresponding contribution to the static gravitational
potential near the earth. There is a left over 43 seconds of arc per century that is explained by general relativity (GR).
(See last year's notes for more on this, and links.)

+ The GR correction can be described in Newtonian terms, as an addition to the effective potential with a -1/r^3. This changes
everything at small r (if the body is small enough). There is an innermost stable circular orbit (ISCO) that plays a key role
in black hole accretion astrophysics and mergers of black holes that spiral into coalescence due to gravitational wave emission.
See homework problem for more details. The ISCO of a non-spinning black hole is at 3 times the gravitational radius. For a solar
mass black hole this would be at 9 km. Remark: this radius is not a "distance to the center" (which is ambiguous in general relativity)
but rather a circumference/2π.

+ For spinning black holes the ISCO is pulled in towards the horizon, in fact all the way for a maximally spinning black hole.
This effect has been observed in the spectral line profile of an Iron K-alpha transition. The line shape arises from a combination of
Doppler shift and gravitational redshift, and has been fit to spinning black hole models.

+ The same GR correction also contributes to the precession of Mercury's orbit, and explains the 43 seconds. (I forgot to mention in
class that binary pulsar systems have been observed that have a GR precession of 4 degrees per year!

+ Dark matter: I mentioned the various lines of evidence pointing to the existence of dark matter, and discussed ways people are trying
to detect dark matter if it interacts non-gravitationally with ordinary matter. (See last year's notes for more on this, and links.)

Tuesday, Oct. 2

+ N bodies: let ∑ stand for the summation or integral over the N bodies.

Notable properties
Total mass M
M := ∑m_i
Center of mass (CM) position R MR := ∑ m_i r_i
Position s_i relative to CM r_i = R + s_i
∑ m_i s_i = 0
Velocity sdot_i relative to CM rdot_i = Rdot + sdot_i ∑ m_i sdot_i = 0

Decompositions relative to CM:

P_tot = P_cm = MRdot
T_tot  =T_cm + T_rel = 1/2 M Rdot.Rdot + ∑ 1/2 m_i sdot_i.sdot_i
L_tot = L_cm + L_rel = M R x Rdot + ∑ m_i s_i x sdot_i

Note: Momentum relative to the CM frame vanishes. So CM frame is also "zero momentum frame". This latter definition
generalizes to relativistic mechanics.

+ 2-body problem with R and r = r_1 - r_2 for generalized coordinates.
     - Reduction to effective one-body motion, reduced mass µ = (m_1 m_2)/M.
     - Angular momentum of effective one body = angular momentum relative to CM
     - conservation of angular momentum implies orbit planar in CM frame (actually just reflection symmetry implies this,
       but reflection in all planes implies rotational symmetry, which implies angular momentum conservation).
     - Reduction to one dimensional (radial) motion: use conserved angular momentum to eliminate angular velocity.
     - Caution: It would be wrong to substitute phidot = p_phi/mr^2 into the Lagrangian, because p_phi characterizes a particular
       class of solutions, so the Lagrangian would no longer generate the correct equation of motion. However it CAN be substituted
       into the conserved energy expression, for a particular motion. Then the angular part of the kinetic energy becomes a term in the
       "effective potential". Then imposing dE/dt = 0 yields the radial equation of motion. Alternatively, substitute it into the r-EL eqn,
       then re-express that in terms of the effective potential.
     - Circular orbits, noncircular orbits, unbound orbits, bound orbits, condition for closure of the orbits.
     - precession of Mercury perihelion, and success of general relativity in accounting for the extra 43 seconds of arc per century.

Thursday, Sept. 27

+ Lagrange multipliers and constraints

+ the action in quantum mechanics, Feynman's path integral formulation, stationary phase and stationary action as classical limit

+ coupling to a vector potential: phase = q/hbar times line integral of A along path. Phase difference between two paths = q/hbar times magnetic flux enclosed.
Note that the effect is periodic in units of the flux quantum h/q (where h = 2π hbar).

+ Aharonov-Bohm effect: a quantum particle can sense a magnetic field even when B = 0 everywhere the particle goes, if two particle paths can enclose flux.
As ramp a magnetic field strength, the interference pattern oscillates as a function of the flux, with a period of one flux quantum h/q.

Tuesday, Sept. 25

+ Re-derivation of Lorentz force from Lagrangian, using vector notation. Uses identity grad(v.A) - (v.grad) A = v x curl A, if v is independent of position.

+ Charge in uniform magnetic field

+ Hamiltonian = energy for charge in em field

+ start Lagrange multipliers

Thursday, Sept. 20

+ When is action least?

+ When is Hamiltonian = energy?

+ Lorentz force, Maxwell's equations in differential form, vector potential, gauge transformations

+ action for charged particle in electromagnetic field:determined using gauge invariance of the Lagrangian up to a total time derivative

+ recover Lorentz force law from EL equations

+ example of charge in a uniform electric field, described with time-independent scalar potential and by space-independent vector potential

+ canonical momentum versus mechanical momentum

Tuesday, Sept. 18

+ symmetries and conservation laws

+ energy and Hamiltonian

+ index notation

+ ambiguity of Lagrangian: can add total time derivative, which changes the action by a constant for fixed endpoints.

+ Galilean symmetry

Thursday, Sept. 13

+ The history of variational calculus and its application to mechanics is not so simple. To put some perspective into the
names "Lagrangian" and "Hamilton's principle", you might like to read this five page historical introduction from
the treatise Calculus of Variations by Giaquinta and Hildebrandt.

+ Action principle applies to fields as well, including the electromagnetic and gravitational fields.
I mentioned the story of Einstein and Hilbert, since Hilbert used the action principle to find the find equations of gravity:
See for instance, Einstein and Hilbert: The Creation of General Relativity, by Ivan T. Todorov. 

- Some amazing properties of the action principle:

1. include any number of particles, just add terms to L
2. use any coordinates to label the paths (Cartesian, polar, center of mass and relative, etc.)
3. impose constraints by restricting configurations, even if time dependent (no need to compute constraint forces).

Taylor offers a proof that the constraints can be imposed in the Lagrangian, but I don't understand why the proof is needed.
The restricted variations fully determine the equation of motion.

+ Example of simple pendulum. The potential U(r) can be omitted, but if you include it and impose the r-EL eqn,
you find the tension dU/dr in the string! Showed this and compared with Newtonian force analysis.

+ Example of spherical pendulum: phi is an ignorable coordinate, whose conjugate momentum is the angular momentum about the vertical axis.
Considered the special case of motion with theta = constant: the value of theta determines the angular velocity. Compared this with Newtonian analysis.
In limit theta = 0 this circular motion is a superposition of two linear motions, π/2 out of phase, with same frequency (like circular polarization
by superposing orthogonal linear polarizations). In limit theta = π/2, angular frequency is infinite (the tension must be infinite, otherwise the
vanishing vertical component of the tension force cannot balance the weight).

+ Example of the double pendulum: position of the lower bob depends not only on the angle theta_2 but on the angle theta 1. To figure out the
Lagrangian write in terms of Cartesian coordinates of m2 position and express these in terms of the two angles.

+ Example of bead on hoop rotating with fixed angular frequency w: this is a time-dependent constraint.
Digression on dimensional analysis: can set m=g=R=1 by choice of units. (I explained why.)
Then the frequency w is measured in units of √g/R, the oscillation frequency at the bottom of the non-spinning hoop.
Then the Lagrangian is simpler to write and work with. The azimuthal part of the kinetic energy is determined by w and
theta, with no time derivatives, so acts like a potential term. Define the effective potential including this and look at the
equilibrium conditions. U_eff = -cos theta - w/2 sin^2 theta = -1 + 1/2(1 - w)theta^2 + O(theta^4). This shows that the equilibrium
point at theta=0 becomes unstable when w > 1. We found the equilibrium points for any w. On the homework you work this out...

+ Example of rigid body: think of as a huge number of mass points rigidly connected. Write the Lagrangian for rotation about a fixed
axis and show that L = 1/2 I thetadot^2 + MgR_cm cos theta, where I is the moment of inertia and R_cm is the distance from the axis to the center of mass.

Tuesday, Sept. 11

- Variational calculus

+ general idea of functional S[y(x)] = ∫ dx f(y, y', x)  (this is a restricted form, general enough for many purposes)
+ variational derivative
+ examples: brachistochrone, soap film on two rings, shortest path (geodesic)
+ freedom to choose y(x) or x(y) as function
+ Euler-Lagrange equation derived from variational principle using path variation δy
+ E-L is a 2nd order ODE (if the integrand of the functional depends on no higher than 1st derivative of the function)
+ E-L reduces to a 1st order ODE in two cases:
       1. If f(y,y',x) = f(y',x) is independent of y, then ∂f/∂y' = const.
       2. If f(y,y',x) = f(y,y') is independent of x, then f - y' ∂f/∂y' = const.
+ example of soap film: using y(x), x does not appear explicitly, as in case 2; using x(y), x(y) does not appear explicitly, as in case 1 (simpler).
+ example of geodesics in plane, using x(y) or y(x)
+ example of geodesics in plane using x(s), y(s): get TWO E-L eqns.
+ example of geodesics in plane using polar coordinates: r or θ as independent variable. Using r, thε E-L eqn is r2θ'/√1 +
r2θ'2 = const. 
  This simplifies if we choose the path to go through the origin r=0, in which case the constant is 0, so θ' = 0, i.e. the path is a straight line through the origin.

- Mechanics

+ can solve 1d problems using energy conservation, but not 3d problems: energy conservation is one equation, while Newton's 2nd law in 3d is three eqns.
+ pulled out of a hat: Lagrangian L = T - U and action S = ∫ L dt.  Then
δS = 0 implies Newton's 2nd law for every particle. This is amazing. The one scalar
function L captures all the dynamics of any number of particles. Even works for a continuum, like a string or a solid, and for fields like the electromagnetic field.

Tuesday, Sept. 4 and Thursday, Sept. 6

Prof. Peter Shawhan taught the class this week. His lecture notes are available here:


He addressed:

+ energy conservation and ways to use it
+ generalized forces
+ conservation or non-conservation of energy, linear momentum, and angular momentum in collisions
+ action integral, variational calculus, Euler-Lagrange equation
+ the brachistochrone

Thursday, Aug. 30

- Intro to the class, syllabus, website, course plan, homework 0, piazza, email, etc.

- We're starting out by reviewing material from Ch. 1 and Ch. 4, after which we'll move on to Ch. 6.
   Here's what i covered or mentioned, in varying depth:

+ structure of Newtonian space and time (Euclidean distance, translation invariance, absolute time)
+ contrast with spacetime of special and general relativity
+ vector description of position, velocity, acceleration, relative position
+ Newton's 2nd law, inertial frames, vector addition of forces
+ Galilean invariance of Newton's 2nd law
+ work, kinetic energy, work-kinetic energy theorem, power
+ conservative forces (gradient of a function of space),
+ potential energy for one particle, total mechanical energy
+ relation between curl F = 0 and conservative condition on F
+ addition of potential energies corresponding to more than one force
+ potential energy for a central force, example of gravitation
+ potential energy for two particles using example of gravitation [very briefly]