Physics 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 text

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.

- Lecture 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.
- Lecture 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.
- Lecture 3 09/08/2008 Angular momentum. Reading: Angular Momentum for a single particle pp 90-93. Angular momentum for several particles pp 93-98.
- Lecture 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
- Lecture 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
- Lecture 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
- Lecture 7 09/17/2008 Calculus of variations - shortest distance between two points. Reading: 6.2 The Euler-Lagrange Equation pp 218-221
- Lecture 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
- Lecture 9 09/22/2008 More than one variable. First integrals. Reading: 6.4 More than Two Variables pp 226-229
- Lecture 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
- Lecture 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
- Lecture 12 09/29/2008 Examples. Reading: 7.5 Examples of Lagrange’s equations pp 254-260
- Lecture 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
- Lecture 14 10/03/2008 Noethers theorem. Symmetries and conservation laws. Reading: 7.8 More about conservation laws. pp 268-272
- Lecture 15 10/06/2008 Constraints and the Lagrange multiplier approach. Reading: 7.10 Lagrange multipliers and Constraint forces pp275-280
- Lecture 16 10/08/2008 Action principles for other forces. The lagrangian for electro-magnetic systems Reading: Section 7.9 pp 272-275
- Lecture 17 10/10/2008 More on the vector potential. Motion in a uniform ﬁeld. Reading: Section 7.9 pp 272-275
- Lecture 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
- Lecture 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.
- Lecture 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 pp 308-313
- Lecture 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
- Lecture 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
- Lecture 25 11/05/2008 Geometry of bound orbits. Transfer orbits.
- Lecture 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
- Lecture 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
- Lecture 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
- Lecture 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 equation.
- Lecture 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
- Lecture 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.
- Lecture 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
- Lecture 33 11/24/2007 Flows with vorticity everywhere: channel ﬂow. Reading: Tritton, Chapter 2, Sections 2.1, 2.2 Channel ﬂow
- Lecture 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
- Lecture 35 12/03/2008 Stream function, streamlines, streaklines etc. Reading: D. J. Tritton, Physical Fluid Dynamics, Chapter 6, pp 73-80
- Lecture 36 12/05/2008 Exact solutions. Reading: D. J. Tritton, Physical Fluid Dynamics, Chapter 9, pp 106-112
- Lecture 37 12/08/2008 Stokes ﬂow past a sphere. Reading: D. J. Tritton, Physical Fluid Dynamics, pp 109-111
- Lecture 38 11/26/2008 Dynamical Similarity. Reading: D. J. Tritton, Physical Fluid Dynamics, pp 89-95
- Lecture 39 11/28/2008 Turbulence Reading: D. J. Tritton, Physical Fluid Dynamics, pp 295-311