308 2008 Lecture summaries and lecture notes.
These are the lecture notes for Physics 308 - Mechaics of Discrete
and Contnuous Systems
taught in Fall 2008. This course was taught from Taylor - a wonderful
How to read these lecture notes:
Each page has 4 panels, representing a chunk of blackboard about 3 feet
wide. The sequence of panels on a page is TL, TR, BL, BR. A 55
minute lecture is usually ten
panels, 2.5 pages.
Note on errata: I'd
be grateful, if you find a mistake, if you let me know so I can
note it here.
Acknowledgements: Thanks to
Kay Warner for scanning these notes.
1 09/03/2008 Newtons laws and their domain of validity. Inertial
Frames. Conservation of Momentum. Reading: Newtons
laws: Inertial frames pp 15-17. Range of validity of Newton’s
ﬁrst and second laws pp 17. The Third law and conservation of momentum
pp 17-21. Validity of the third law pp 21-23.
2 09/05/2008 Rockets. Choice of coordinates: center of mass and
two-dimensional polar coordinates. Rockets
pp 85-87. Center of mass coordinates pp 87-89. Two dimensional polar
coordinates pp 26-33.
3 09/08/2008 Angular momentum. Reading: Angular
Momentum for a single particle pp 90-93. Angular momentum for several
particles pp 93-98.
4 09/10/2008 Kinetic energy, Work and potential energy. Path
independence and Conservative forces. Reading: 4.1
Kinetic energy and Work pp 105-109, 4.2 Potential energy and
conservative forces pp 109-116
5 09/12/2008 Force as the gradient of PE. Conditions for a force to
be conservative. Time dependent potentials. Reading: 4.4
The Second Condition that F be Conservative pp 118-121 4.5
Time-Dependent Potential Energy pp 121-123
6 09/54/2008 Energy in one dimensional systems and multiparticle
systems. Reading: Curvilinear
One-Dimensional Systems pp 129-133 Energy in Multiparticle systems pp
144 - 148
7 09/17/2008 Calculus of variations - shortest distance between two
points. Reading: 6.2 The Euler-Lagrange
Equation pp 218-221
8 09/19/2008 Frechet (or functional) derivatives. Applications:
optics from Fermat’s principle. Reading:
6.3 Applications of the Euler-Lagrange Equation pp 221-226
9 09/22/2008 More than one variable. First integrals. Reading: 6.4
More than Two Variables pp 226-229
10 09/24/2008 Newtonian mechanics from a variational principle:
Hamilton’s principle and the Lagrangian. From Lagrange to Newton
in Cartesian and Polar Coordinates. Example: projectile motion.
Reading: 7.1 Lagrange’s equations for
unconstrained motion. pp 237-245
11 09/26/2008 Degrees of freedom. Generalized coordinates. Holonomy
and holonomic systems. Lagranges equations with constraints. Reading: Degrees
of freedom pp 249 7.4 Proof of Lagrange’s equations with
constraints pp 250-254
12 09/29/2008 Examples. Reading: 7.5
Examples of Lagrange’s equations pp 254-260
13 10/01/2008 Ball in a surface of revolution. Physics from the
Lagrangian Reading: 7.5 Examples of
Lagrange’s equations pp 254-260 7.6 Generalized momenta and
ignorable coordinates pp 266-267 Conclusions p 267
14 10/03/2008 Noethers theorem. Symmetries and conservation laws.
Reading: 7.8 More about conservation laws.
15 10/06/2008 Constraints and the Lagrange multiplier approach.
Reading: 7.10 Lagrange multipliers and
Constraint forces pp275-280
16 10/08/2008 Action principles for other forces. The lagrangian
for electro-magnetic systems Reading: Section
7.9 pp 272-275
17 10/10/2008 More on the vector potential. Motion in a uniform
ﬁeld. Reading: Section 7.9 pp 272-275
18 10/13/2008 Center of mass frame. Reduced mass. Angular momentum
in COM frame. Section 8.1 pp 293-295 Section
8.2 CM and Relative Coordinates; Reduced Mass pp295-297. Section 8.3
The equations of motion pp 297-300
19 10/22/2008 The equivalent one-dimensional problem for central
forces. Centrifugal barrier. Small oscillations near a circular orbit.
Reading: Section 8.4 The equivalent
one-dimensional problem pp 300-305.
20 10/24/2008 Finding orbits. Conic Sections. The Kepler orbits.
Parabolic orbits and galaxy collisions. Reading: Section
8.5 The Equation of the Orbit pp 305-308 Section 8.6 The Kepler Orbits
21 10/28/2008 Unbounded orbits. Changes of orbit. Mission planning.
Reading: Section 8.7 The unbounded Kepler
orbits pp 313-315. Section 8.8 Changes of orbit pp 315-319
- Lecture 22
10/29/2008 Class visit by Robin Selinger, Kent State University
- Lecture 23 10/31/2008 Class Cancelled
24 11/03/2008 Changes of orbit - mission planning. Reading: Section
8.8 pp 315-319. Also review Taylor Section
8.6 The Kepler Orbits pp 308-313
25 11/05/2008 Geometry of bound orbits. Transfer orbits.
26 11/07/2008 Hamiltonian mechanics. Canonically Conjugate
variables. Liouville’s Theorem. Symplectic structure of Hamiltons
equations Reading: Taylor 13.1 The Basic
Variables pp 522-523. Taylor 13.2
Hamilton’s equations for one-dimensional systems. pp 524-528
27 11/10/2008 The time evolution operator. Geometric interpretation
of symplectic structure. The Poisson bracket. Reading: Hamilton’s
equations in several dimensions pp 528-535, Taylor Section 13.6
Phase-space orbits pp 538-543 Taylor section 13.7 Liouville’s
theorem pp 543-549
28 11/12/2008 Hamiltonian Mechanics - more than one degree of
freedom. Geometric meaning of symplecticness. The Poisson Bracket.
Canonical Brackets. Canonical transformations are bracket preserving.
Reading: Numerical Hamiltonian problems, J.
M. Sanz -Serna and M. P. Calvopp, Chapter 1 pp 1-3 Chapter 2 pp15-23.
Elements of Hamiltonian Mechanics, Ter Haar, Chapter 5 pp 95-114
29 11/14/2008 Observables on phase space are a Lie algebra. Lie
operators. The Hamiltonian generates time evolution. From Poisson
Brackets to commutators. Flows - basic concepts. The material
derivative. Reading: Numerical Hamiltonian
problems, J. M. Sanz -Serna and M. P. Calvopp, Chapter 2 pp15-23. D. J.
Tritton, Physical Fluid Dynamics pp 48-55 Circulation
and vorticity. Divergence and incompressible ﬂows. Euler’s
30 11/17/2008 Point vortices. Equation for vorticity in an ideal
ﬂuid. How do point vortices move in the ﬂow they deﬁne? From vorticity
to velocity in 2 − D. Reading: D.
J. Tritton, Physical Fluid Dynamics pp 81-88
31 11/19/2007 Point vortex motion. Equations of Motion. Hamiltonian
Structure. Flow due to one vortex. Motion of two vortices. Reading: Review
reading for Lectures 28 and 29.
32 11/21/2007 Viscous forces and the Navier Stokes equation.
Newtonian Fluids. Reading: Tritton, Section
5.6, The Navier Stokes equations pp 55-61
33 11/24/2007 Flows with vorticity everywhere: channel ﬂow.
Reading: Tritton, Chapter 2, Sections 2.1,
2.2 Channel ﬂow
34 12/01/2008 Flow rate in channel ﬂow. Transition to turbulence in
channel ﬂow. Reynolds number. Reading: D.
J. Tritton, Physical Fluid Dynamics, pp 7-20
35 12/03/2008 Stream function, streamlines, streaklines etc.
Reading: D. J. Tritton, Physical Fluid
Dynamics, Chapter 6, pp 73-80
36 12/05/2008 Exact solutions. Reading: D.
J. Tritton, Physical Fluid Dynamics, Chapter 9, pp 106-112
37 12/08/2008 Stokes ﬂow past a sphere. Reading: D.
J. Tritton, Physical Fluid Dynamics, pp 109-111
38 11/26/2008 Dynamical Similarity. Reading:
D. J. Tritton, Physical Fluid Dynamics, pp 89-95
39 11/28/2008 Turbulence Reading: D. J.
Tritton, Physical Fluid Dynamics, pp 295-311