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#9: Due Friday, Nov. 21: Calculate the Lagrangians for Problems 7.7, 9, 12, 15.
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 rcm2  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":
Tuesday, Nov. 18: Calculate the Lagrangian for Problem 7.7.
Tuesday, Dec. 2: Solve the Hamiltonian for the simple Atwood machine: H = p2/2(m+M) + (m-M)gy.
Thursday, Dec. 4: Verify that dH/dp for a pendulum in an upwardly accelerating elevator is consistent with the
definition of p:
H
 at 2

p2
at
2 2
2
1


ma
t
sin


p
sin


mg
 l cos  
2
2
l
2ml
 2

WEEK 13: GO TO LECTURE 21, 22
LECTURE 21: SECTION 7.1-6 (Return to top.)
SECTION 7.10: HAMILTONIAN DYNAMICS
There is just one more item for us to look at which is a result of Hamilton's Principle. It is yet another procedure for
solving dynamics problems, known as Hamilton's equations. Its main charms relative to the Lagrangian formalism
are the following:
1) It produces first order differential equations instead of second order.
2) It introduces a class of generalized momenta corresponding to the various generalized coordinates.
3) It is related to the left-hand side of Schrödinger's equation, and thus provides a bridge between
classical mechanics and quantum mechanics.
4) It replaces the philosophically troubling quantity "action" with a quantity, the "Hamiltonian", equal to the
mechanical energy for some systems.
The first step in developing this new formalism is to define the generalized momenta. For each of the generalized
coordinates, qj, the corresponding generalized momentum, pj, is given by
pj 
L
q j
we can now define the Hamiltonian as
H   q j p j  L
(If the coordinates are independent of time, and if the potential energy is independent of velocity, then this
Hamiltonian, H, is equal to the total mechanical energy: H = T + U.) Lagrange's equations of motion give rise to
the following "canonical equations of motion", also known as Hamilton's equations of motion:
q j 
H
p j
 p j 
H
q j
LECTURE 22: SECTION 7.7-10 (Return to top.)
We will do a handful of problems that we have already done as Newtonian dynamics problems, or even as
Lagrangian dynamics problems, using this new technique --- the projectile, a block on an incline, the spherical
pendulum, and/or the the double pendulum. Again, as with the Lagrangian problems, we will only set up the
differential equations, rather than solve them. Generally, problems that invite the use of Hamilton's equations are
difficult problems which give rise to ugly differential equations. Our primary goal in this chapter is to show you how
to set up these problems and get to those ugly differential equations, but not to sweat the math. (That's for your
Math Methods prof to make you do.)
YSBATs
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