Phys 501, Fall 2014

Topics Covered in Each Lecture

Lecture Number	Date	Content	Corresponding Reading material
1	Sep. 15	Aim and basic notions of classical mechanics: Kinematical and dynamical aspects, point particles, the definition and mathematical properties of the 3-dimensional Euclidean space R^3, bases and coordinate frames, states of a particle moving in R^3, trajectory of a particle in R^3 and the relevant notions of differential geometry of curves in R^3	Jose & Saletan pp 01-05
2	Sep. 17	Elementary notions and postulates of Classical Mechanics: Isolated particles, Inertial frames, inertial mass and momentum, Newton's second and third law as consequences of the postulates, Newton's equation and the existence, uniqueness and stability of its solution, transformation between inertial frames, general notion of an observable in classical mechanics, momentum, angular momentum, torque	Jose & Saletan pp 05-15
3	Sep. 22	Work, kinetic energy, conservative forces, potential and total energy, using conserved quantities to solve equation of motion in one dimension, many particle systems, conservation of momentum, center of mass, internal and external potential energy, angular momentum in the center of mass frame, non-inertial frames, rotating frames	Jose & Saletan pp 16-42
4	Sep. 24	Lagrangian Formulation of CM: Constrained motion and Lagrange multipliers, generalized coordinates, a local coordinate description of a circle, the basic idea leading to a notion of a manifold, a precise definition of a manifold	Jose & Saletan pp 48-62
5	Sep. 29	Derivation of Lagrange's equations from Newton's equation, equivalent and singular Lagrangians	Jose & Saletan pp 62-70
6	Oct. 01	Energy conservation and invariant quantity for explicitly time-independent Lagrangians, examples: spherical pendulum and charged particle in an electromagnetic field, Free-particle Lagrangian in a rotating frame with a constant angular speed and its application in describing the dynamics of a charged particle in a constant magnetic field	Jose & Saletan pp 70-76
Kurban Bayramı	Oct. 03-07
7	Oct. 08	Two-body problem with an internal distance-dependent conservative interaction, Kepler 's problem (general treatment and details of closed trajectories)	Jose & Saletan pp 77-88
8	Oct. 13	Kepler 's problem (the open trajectories);  Functional variation and functional derivative. Stationary points of a functional.	Jose & Saletan pp 77-88
9	Oct. 15	Hamilton's principle and action functional, the second derivative test for functions of several variables, and the second functional derivative of the action functional.	Jose & Saletan pp 108-113
10	Oct. 20	The differential operator defined by the second functional derivative of the action functional and the role of its spectrum to determine whether a classical path is the minimum or maximum of the action functional	-
11	Oct. 22	Hamilton's  variational principle in the presence of constraints, Symmetry transformations and Nöther's theorem in Lagrangian mechanics	Jose & Saletan pp 114-118 & 124-128
12	Oct. 27	Generalized coordinate transformations+time translations that change the Lagragian by a total time-derivative and the corresponding constants of motion, Conservation of linear and angular momentum as a consequence of Cartesian and angular coordinate translation symmetry (homogeneity and isotropy of space), Energy conservation as a consequence of time-translation symmetry, Application of   Nöther's theorem  for a scaling invariant Lagrangian; angular momentum conservation for a Lagrangian depending on the angular coordinate.	-
Exam 1	Oct. 31	Covers the material of Lectures 1-12.
13	Nov. 03	Scattering theory for central forces: scattering cross section and rate, Rutherford cross section for the Coulomb force	Jose & Saletan pp 147-154
14	Nov. 05	Linear oscillations, normal modes, application of one-dimensional chain of equally spaced identical particles with nearest neighbor harmonic interactions	Jose & Saletan pp 178-185 & 187-192
15	Nov. 10	A chain of 3 one-dimensional oscillators; Damped forced oscillator, intensity of the power lost in stationary state, Lorentizan resonanace	Jose & Saletan pp 192-196
16	Nov. 12	Vector bundles, Tangent bundle of the configuration space as the phase space in the Lagrangian mechanics; Hamiltonians Formulation of CM: Hamilton Equations and the classical Hamiltonian, Legendre transformation	Jose & Saletan pp 100-103 & 201-203
17	Nov. 17	Cotangent bundle of the configuration space as the phase (state) space in the Hamiltonian formulation of CM, Hamilton's equation sof motion written in a unified notion for position and momentum variables; the standard symplectic matrix; Hamiltonian for the special relativistic point particle	Jose & Saletan pp 201-210 & 215-216
18	Nov. 19	Relativistic Kepler's problem	Jose & Saletan pp 210-212
19	Nov. 24	Observables in the Hamiltonian formulation of CM, Poisson bracket and its properties, Lie algebra defined by the Poisson bracket, kinematic and dynamical Lie algebras, Heisenberg and su(1,1) algebras, Hamiltonian dynamical systems	Jose & Saletan pp 217-224
20	Nov. 26	Canonical transformations: Characterization in terms of the invariance of the Poisson brackets, time-independent canonical transformations, local canonical transformations mapping the coordinates and momenta to coordinates and momenta respectively, one-dimensional special case and dilatations, linear canonical transformations and the real symplectic groups Sp(2n,R), application of time-independent linear canonical tranformations to a simple harmonic oscillator	Jose & Saletan pp 231-238
21	Dec. 01	Proof of the fact that an invertible transformation is locally canonical if and only if its Jacobian is a symplectic matrix; Hamiltonian vector fields and integrability condition; local proof of the fact that invariance of the Poisson Bracket implies that an invertible time-dependent transformation of the phase space is canonical; structure of the transformed Hamiltonian under a time-dependent canonical transformation, example: time-dependent dilatation in 1-dim., derivation of Lie's condition for canonicity	Fasano & Marmi pp 345-352
22	Dec. 03	Inertia tensor, Angular Momentum in fixed and body coordinate systems, Principal axes of inertia	Marion & Thornton pp 404-418
23	Dec. 08	Application of the Lie's condition for canonicity for linear canonical transformations in 1.dim; Statement of the theorem: Difference of the Poincare-Cartan formes before and after a canonical transformation is an exact differential; Canonical transformations admitting generators	Fasano & Marmi pp 352-359 and Jose & Saletan pp 240-248
24	Dec. 08	Moments of inertia for different body coordinate systems, further properties of inertia tensor (diagonalization of inertia tensor, transformation properties of inertia tensor, rotation method for diagonalization and orthogonality conditions)	Marion & Thornton pp 419-430
25	Dec. 10	Dynamics as a one-parameter family (subgroup) of canonical transformation, Hamilton-Jacobi formulation of CM, complete solutions of the Hamilton-Jacobi equation, applications to free particle, time-independent Hamilton-Jacobi equation	Jose & Saletan pp 284-288
Exam 2	Dec. 12
26	Dec. 15	Application of Hamilton-Jacobi theory for standard Hamiltonians in one-dimension,  identifying the generator of the canonical transformation leading to the Hamilton-Jacobi equation with the classical action function, Hamiltonian flows,  Liouville's Volume Theorem.	Jose & Saletan pp 288-290
27	Dec. 17	An application of Hamilton-Jacobi formulation for quadratic Hamiltonians in 1-dim, H=p^2/(2m)+a q p+m b^2 q^2/2. The angle and action variables and the condition of their existence.	Jose & Saletan pp 307-320
28	Dec. 19	Euler angles, Euler's equations for a rigid body, Force-free motion of a symmetric top, Motion of a symmetric top with one point fixed	Marion & Thornton pp 431-448
29	Dec. 22	Solving the time-independent Hamilton-Jacobi equation by separation of variables. Liouville's Integrability Theorem, Elements of Symplectic Manifolds: Tensors on a vector space, dual vector space and dual basis	Jose & Saletan pp 290-300 & 320-324
30	Dec. 24	Antisymmetric tensors, p-forms on a differentiable manifold, exterior derivative, closed and exact forms, non-degenerate 2-forms, symplectic manifolds, Darboux theorem	Jose & Saletan pp 135-136, 226-230, & 268-274

Note: The pages from the textbook listed above may not include some of the subjects covered in the lectures.