LECTURE NOTES FOR PHYSICS 307: CLASSICAL MECHANICS
TEXT: THORNTON AND MARION'S CLASSICAL MECHANICS OF PARTICLES AND SYSTEMS
SOME USEFUL REFERENCE STUFF: Greek alphabet, metric prefixes, conversion factors
ASIGNMENTS: (Subject to change: check back often.)
HW #10: Due Friday, Dec. 5:


m1r1  m2 r2
m2 
m1 
  

 
 
1A. Given rcm 
and r  r1  r2 , show that r1  rcm 
r and r2  rcm 
r.
m1  m2
m1  m2
m1  m2
m2 
m1 


 
 
1B. Given r1  rcm 
r and r2  rcm 
r , show that T  12 m1  m2 rcm2  12 r 2 .
m1  m2
m1  m2
 

2. Given F  rˆF ( r ) , show that   F  0 . (Use spherical coördinates.)
3. Problem 8.2. (HINT: Start with Eq. 8.39, and use the "arccosine integral".)
"Dailies":
Thursday, Dec. 4: Verify that dH/dp for a pendulum in an upwardly accelerating elevator is consistent with the
definition of p:
 at 2

p2
at
2 2
2
1
H
 2 ma t sin   p sin   mg
 l cos  
2
l
2ml
 2

WEEK 14: GO TO LECTURE 23, 24
LECTURE 23: SECTION 8.1-4 (Return to top.)
We will review the basics of rotational motion, with some mention of how the more general case, in which roation
is not along any of the principal axes of the object, is more complicated. The text assumes that there is no need of
a review before looking at this more general case, so I would advise you look at an introductory text for a quick
review. A few websites that I found that might find useful are Rotational kinetic energy and angular momentum
and Angular Quantities.
CHAPTER 8: CENTRAL FORCES
SECTION 2: CENTER OF MASS
The important message here is that we can take problems involving two objects and simplify them into two
independent motions. One of these motions is the motion of the center of mass, and the other is the relative
motion -- motion about the center of mass. We can consider the relative motion as being due entirely to internal
forces between only the two objects. The motion of the center of mass is due only to forces acting on the system
from outside. (This is like separating projectile motion into the x motion and the y motion. Remember that the
component of acceleration in the x direction is due only to components of forces pointing in the x direction.
Likewise for the component of acceleration in the y direction. )
What is important to take away from this treatment of the center of mass? You should be able to calculate the
center of mass coordinates of a system of two or more masses,


m1 r1  m2 r2

rcm 
m1  m2
and you should be able to calculate the reduced mass of that system. Because the expression for the reduced
mass is

m1 m2
m1  m2
,
whenever these two masses are quite different, the reduced mass is roughly equal to the smaller mass. In this
case we can consider this small object as orbiting the larger object, as the Earth circles the Sun, or an electron
circles the nucleus. (In reality, both masses move around the center of mass.) However, when these two masses
are roughly equal, the reduced mass is equal to half of either object mass. Then it is more obvious that neither
mass is stationary.
SECTION 3: CENTRAL FORCES
What are central forces? If an object is in a potential energy field that depends only on its distance from another
object,
U(
) = U(r,θ, φ) = U(r),
then the force between them depends only on distance, and points along the line connecting the two objects. This
is a "central force”. From what we learned about symmetries in Chapter 2, this means that angular momentum
must be conserved.
  
L  r  p = constant
We begin our consideration of central forces by using the Lagrangian formalism to analyze the motion. Since the
potential depends only on radius, we would use spherical coordinates, but the motion will be in a plane. Thus we
use plane polar coordinates. Follow the math on pages 293-295. Applying Lagrange’s equation to both the radial
and angular coordinates, r and θ, we find an expression for the energy in terms of radius
E  12 r 2 
L2
 U (r )
2r 2
At this point in class, I will make two digressions. First of all, I will look at the conservation of angular momentum.
We can ascribe this conservation to the fact that there is no torque, or we can see angular momentum as being a
“first integral” of the motion (“popping out” of the Lagrangian method). We will also show that this conservation
explains Kepler’s (1571-1630) Second Law. (more on those laws later)
The second digression will be to consider the similarities between our equation for energy (above) and
Schrödinger’s equation for three-dimensional systems in radial coordinates, with its centrifugal term:

2  l l  1 2
g   2 
 U (r )  E  g (r )  0
2
  2r

The thing to note here, for both Schrödinger's equation and for this classical central force problem, is that the
plane polar problem reduces to a one-dimensional problem for the radius that looks like one-dimensional motion
except for the extra centrifugal term. This is the same pseudoforce that you feel when you go around a merry-goround, but in this case, it seems to have a real physical function.
LECTURE 24: SECTION 8.5-8 (Return to top.)
SECTION 4: EQUATIONS OF MOTION
We will next take two independent approaches to developing an equation for the orbit of the central force
problem. In approach 1, we start with the energy equation from above to derive an expression for θas a function
of radius. This gives us an equation for how the angle evolves as a function of radius. Notice that this tells us
nothing about the time dependence, only about the shape of the orbit. The ugly equation that we come up with is
 (r )  
 Ldu
2 ( E  U )  L2 u 2
Now, do I want you to memorize this equation? No. But you should be able to do simple problems involving this
integral, and you should also keep this expression somewhere handy for future reference.
Approach 2 starts with the Lagrangian, and we do all the usual stuff with it. Finally, to make the math more
elegant, we define the quantity u=1/r. Then we get the following second-order differential equation:
u   u  
 f (1 / u )
l 2u 2
Notice again that this is an equation for radius as a function of angle. μ is the reduced mass of the object. Also
note that u is a function of θ, and so the derivative is a derivative with respect to θ, not time. In other words, both
of these equations that we've just derived -- the θ(r) integral above and this equation -- describe the orbits, not the
time evolution of the motion. Now that is not as bad as it seems, because, since angular momentum is conserved,
we can calculate how angle changes with time. From this we can deduce how radius and angle evolve with time,
if we want to. If you grow up to be a rocket scientist, this would obviously be a practical thing to be able to
calculate.
SECTION 5: ORBITS
Time to define a few new terms. "Apsides" are turning points. They are the points in an orbit at which an object
reaches its maximum or minimum radius. We can consider three cases: the general case, the case of the earth's
motion around the Sun, and the case of the moon's motion (or the International Space Station's motion) around
the earth. For each of these cases, we have special words for the maximum and minimum radius.
Rmaximum = pericenter, perihelion, perigee for general case, Sun-centered, Earth-centered cases,
respectively
Rminimum = apocenter, aphelion, apogee
The other important thing about orbits is to distinguish between "closed" and "open" orbits. "Closed" orbits are
those that return to the same point in space eventually. (after one, two, or more orbits) "Open" orbits are those
that don't. An ellipse (incl. circle) is the most obvious class of "closed" orbit.
SECTION 7: KEPLER'S PROBLEM: PLANETARY MOTION
Up till now, we have been considering the general case of central forces. We have said nothing specific about the
force between the two objects that we are considering. Now, the most common problems that we would want to
consider (well, actually the simplest problems) are those that involve either gravity or electromagnetic forces.
What we are going to do now is consider gravitational and electromagnetic forces. Luckily, in both cases, the
force is inverse square law, so we lose no generality by expressing the force as
F  k / r 2 ,
or equivalently,
U  k / r
if we plug these into our expression for θ(r), we can show that the angle is a simple function of radius:
cos  
 / r 1

where
  l 2 / k
and
  1
2 El 2
,
k2
where ε is a dimensionless quantity known as the eccentricity of the orbit. It is not obvious, but the first of these
equations can produce any of the conic sections --- hyperbola, parabola, ellipse, and circle. Imagine ε on a
number line going from the zero to positive infinity. The value of the eccentricity tells you which kind of conic
section you have:
Eccentricity, ε
Orbit
0
circle
0-1
ellipse
1
parabola
1+
hyperbola
The most important of these conic sections are the ellipses and the very special case of ellipse, the circle. We will
focus our attention for the second time in this section (The first time it was going from a general central force
problems to inverse square force problems.) as we go from the conic section in general to the ellipses in
particular.
The "pericenter" and the "apocenter" of the ellipse are given by
Rmin = α / (1 + ε )
and
Rmax = α / (1 - ε )
These are related to the semimajor axis, a, and semiminor axis, b, of the ellipse by
a  k/2E
and
b  1 / 2 E
Notice in all of this that the energy is negative -- the particle can only attain zero or positive energy if it is no longer
bound to whatever it is orbiting.
KEPLER'S LAWS:
From our results so far -- derived using Newton's (1642-1727) Laws -- we can derive Kepler's (1571-1630) Laws.
(What does this tell us about the historical authenticity of our approach?)
Kepler 1: The planets move in ellipses which have the sun at one of the foci of the ellipse.
Kepler 2: The rate at which a planet's radius sweeps out area is a constant.
Kepler 3: The square of a planet's period is proportional to the semimajor axis cubed:
2 
4 2  3
a
k
Clearly, none of these is obvious, nor is any of them easily generalizable from the circular case. Kepler did an
amazing job of formulating these laws long before Newton came along with a theory that allows one to deduce
these from first principles. Kepler based his work on the amazing compilation of data compiled by Tycho Brahe
(1546-1601), "the man with the golden (silver?) nose". To recap: Brahe's observations ---> Kepler's theories --->
Newton's laws.
SECTION 8: ORBITAL DYNAMICS
We won't do a lot with this material. We will talk about Hohmann transfers, a way of firing a rocket from one orbit
to another. What is important here is to see how one goes about changing orbits. One doesn't simply the "aim at"
Mars in order to get there. In order to go from one nearly circular orbit to another nearly circular orbit, you need to
put yourself in an elliptical orbit which intersects the two circular orbits, ideally one tangent to both circles. (Plus,
of course, you need to time things, so that you reach Mars' orbit at the time and place where Mars is there as
well.)
We will skip SECTIONS 9 and 10
YSBATs
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