Phys 501, Fall 2013

Topics Covered in Each Lecture

Lecture Number	Date	Content	Corresponding Reading material
1	Sep. 16	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. 18	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, work, kinetic energy, conservative forces	Jose & Saletan pp 05-17
3	Sep. 23	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	Jose & Saletan pp 18-42
4	Sep. 25	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	Jose & Saletan pp 48-62
5	Sep. 30	A precise definition of a manifold, derivation of Lagrange's equations from Newton's equation, equivalent Lagrangians	Jose & Saletan pp 62-69 & 97-102
6	Oct. 02	Singular Lagrangians, 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 69-76
7	Oct. 07	Two-body problem with an internal distance-dependent conservative interaction, Kepler 's problem	Jose & Saletan pp 77-88
8	Oct. 09	Functional variation and functional derivative. Stationary points of a functional and the second functional derivative test, Hamilton's principle and action functional, the second functional derivative of the action functional for a standard Lagrangian	Jose & Saletan pp 108-113
Kurban Bayramı	Oct. 14-18
9	Oct. 21	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; Hamilton's  variational principle in the presence of constraints	Jose & Saletan pp 114-118
10	Oct. 23	Symmetry transformations and Nöther's theorem in Lagrangian mechanics; dissipative forces	Jose & Saletan pp 118-134
Exam 1	Nov. 01
11	Nov. 04	Scattering theory for central forces: scattering cross section and rate, Rutherford cross section for the Coulomb force	Jose & Saletan pp 147-154
12	Nov. 06	Classical Inverse Scattering theory for central forces; verification of the inverse scattering prescription for the Coulomb potential	Jose & Saletan pp 154-157
13	Nov. 11	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
14	Nov. 13	Damped forced oscillator, intensity of the power lost in stationary state, Lorentizan resonanace; Vector bundles, Tangent bundle of the configuration space as the phase space in the Lagrangian mechanics	Jose & Saletan pp 192-196 & 100-103
15	Nov. 18	Hamiltonians Formulation of CM: Hamilton Equations and the classical Hamiltonian, Legendre transformation, cotangent bundle of the configuration space as the phase (state) space, Hamiltonian for special relativistic point particle, relativistic Kepler's problem	Jose & Saletan pp 201-217
16	Nov. 20	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
17	Nov. 25	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
18	Nov. 27	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, application 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.	Fasano & Marmi pp 345-359
19	Dec. 02	Canonical transformations admitting generators, dynamics as a canonical transformation (the Hamiltonian flows), Liouville's Volume Theorem	Jose & Saletan pp 240-257
20	Dec. 04	Hamilton-Jacobi formulation of CM, complete solutions of the Hamilton-Jacobi equation, applications to free particle, time-independent Hamilton-Jacobi equation, application to standard Hamiltonians in one-dimension, identifying the generator of the canonical transformation leading to the Hamilton-Jacobi equation with the classical action function	Jose & Saletan pp 284-290
21	Dec. 09	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 for this system and the condition of their existence. Separation of variables  for solving the time-independent Hamilton-Jacobi equation.	Jose & Saletan pp 290-320
22	Dec. 11	Inertia tensor, Angular Momentum in fixed and body coordinate systems, Principal axes of inertia	Marion & Thornton pp 404-418
Exam 2	Dec. 13
23	Dec. 16	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
24	Dec. 18	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
25	Dec. 23	Elements of Symplectic Manifolds: Tensors on a vector space, dual vector space and dual basis, antisymmetric tensors, p-forms on a differentiable manifold, exterior derivative, closed and exact forms, non-degenerate 2-forms, symplectic forms and symplectic manifolds	Jose & Saletan pp 135-136 & 226-230
26	Dec. 25	Darboux theorem, completely integrable Hamiltonian systems and Liouville's Integrability Theorem.	Jose & Saletan pp 268-274 & 320-324

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