Send comments or questions to: ldurand@hep.wisc.edu           

 

PHYSICS 711, CLASSICAL THEORETICAL PHYSICS - DYNAMICS

FALL, 1998

The course is divided into three units with different themes and emphases:

           
Joseph-Louis Lagrange
1764		 William Rowan Hamilton Emmy Noether

UNIT 1: LAGRANGIAN MECHANICS

Week Lec. Date Subjects
1 1 Sept. 2  Chap. 1. Introduction; motion under constraints, types of constraints, d'Alembert's principle.
2 4  Generalized forces, Lagrange's equations, the Lagrangian function.
2 3 Sept. 9  Velocity-dependent forces; examples of Lagrangian problems, shrinking pendulum, motion constrained to a surface; change of variables in the Lagrangian.
Hw. 1 4 11  Change of variables; integrability of Lagrange's equations, Lagrangian constraints; Chap. 2. Variational principles.
3 5 Sept. 14  Variational principles, Euler's variational equations, the brachistochrone. Hamilton's principle, the Euler-Lagrange equations.
6 16  Differential constraints; elimination of holonomic or nonholonomic differential constraints using Lagrange multipliers; integral constraints in variational problems.
Hw. 2 7 18  Examples of constrained problems; free motion with constraints; integrability conditions for differential constraints.
4 8 Sept. 21  Jacobi's formulation of least action, geodesics, connection with dynamics. Constants of the motion, cyclic variables.
9 23  Symmetries and Noether's theorem; examples of symmetries; time translations and the Hamiltonian.
Hw. 3 10 25  The Hamiltonian, distinction from the energy; use of constants of the motion; examples; integrability of dynamical systems. Chap. 3: two-body problems.
5 11 Sept. 28  Two-body problems, constants of the motion, the orbit and time equations.
12 30  Qualitative description of motion, the Kepler problem, scattering; n-body problems, the virial theorem.
Hw. 4 13 Oct. 2  Chap. 6. Small oscillation problems, motions near equilibrium, quadratic Lagrangians, characteristic frequencies of oscillation.
6 14 Oct. 5  Transformation to normal coordinates, examples; symmetries, uniform motions and "zero modes".
15 7  Perturbation methods; motions near a steady motion; stability of motion.
END OF UNIT 1, START UNIT 2
Hw. 5 16 9  Chap. 4. Rotations, direction cosines, representation of rotations by orthogonal matrices, matrix algebra.
     
Leonhard Euler
1736		 Albert Einstein

UNIT 2: SYMMETRY TRANSFORMATIONS, ROTATIONS AND RELATIVITY

Week Lec.   Date   Subjects
7 17 Oct. 12  Transformations of matrices, diagonalization; reflections; examples for rotations. Euler angles.
18 14  HOUR EXAM IN CLASS ON THE MATERIAL IN UNIT 1, CHAPS. 1-3, 6
19 16  Fixed and moving axes, matrix representations of Euler rotations, SU(2) representation of rotations, spinors.
8 20 Oct. 19  SU(2) methods; infinitesimal rotations, angular velocities for Euler rotations, body and space axes.
21  21 Rotating coordinates, centrifugal and Coriolis forces. Examples with Coriolis forces.
Hw. 6 22 23  The groups SU(2) and SO(3), group generators, exponential representations. Chap. 5. Rotating rigid bodies, I, L, T.
9 23 Oct. 26  Calculation of the moment tensor, principal moments of inertia, principal axes; Euler's equations of motion.
24 28  Free rotation, stability of motion near the principal axes; symmetrical rotator, qualitative description of the motion.
Hw. 7 25 30  Symmetries and the use of partially rotating coordinates, examples; Poinsot construction for the asymmetric rotator.
10 26 Nov. 2  Symmetrical top, equations of motion, precession and nutation, examples.
27 4  Chap. 7. Relativity, Lorentz transformations, metric notation, invariants; matrix representations of Lorentz transformations.
Hw. 8 28 6  Boosts, rapidities, and addition of velocities; general Lorentz transformations; covariant vectors, index notation, differential operators and the wave equation.
11 29 Nov. 9  Hyperbolic geometry, the light cone, invariants; particle motion, world lines, the free relativistic Lagrangian; four momentum, four velocity, and kinematics.
30 11  Relativistic particle Lagrangians, conservation laws; electromagnetic interactions, other examples, the no-interaction theorem.
Hw. 9 31 13  The Lorentz group; matrix generators, successive transformations, Thomas precession.
END OF UNIT 2, START UNIT 3

     
Simeon Denis Poisson
1808		 Karl Gustav Jakob Jacobi
1843

UNIT 3: HAMILTONIAN MECHANICS

Week Lec.   Date   Subjects
12 32 Nov. 16  Chap. 8. Legendre transformations and Hamilton's equations of motion; interpretation of Hamilton's equations; the variational principle.
33 18  HOUR EXAM IN CLASS ON THE MATERIAL IN UNIT 2, CHAPS. 4, 5, 7 and lecture material.
34 20  Use of Hamilton's equations, motion near a steady motion, examples.
13 35 Nov. 23  Chap. 9. Canonical transformations: examples, invariance of Hamilton's equations, symplectic transformations.
Hw. 10 36 25  Symplectic structure of Hamiltonian mechanics; symplectic invariants, Poisson and Lagrange brackets, tests for canonical transformations.
  THANKSGIVING RECESS, Nov.26-29
14 37 Nov. 30  The variational principal and generating functions for canonical transformations; examples; infinitesimal canonical transformations.
38 Dec. 2  Time development as a canonical transformation; symmetries in Hamiltonian mechanics, generators of infinitesimal symmetries, Noether's theorem and constants of the motion.
Hw. 11 39 4  Constants of the motion from Hamiltonian symmetry transformations, examples, symmetry algebras, finite transformations, Poisson brackets and differentiasl operators.
15 40 Dec. 7  Chap. 10. Solution of problems by canonical transformation, Hamilton-Jacobi theory, Hamilton's principal function, connection with the action.
41 9  One dimensional problems; solution of the oscillator by the Hamilton-Jacobi method. Hamilton's characteristic function, separation of variables in the Hamilton-Jacobi equation.
Hw. 12 42 11  Separation of variables: examples. Phase plots, action and angle variables.
16 43 Dec. 14  Use of action and angle variables, properties, examples.
END OF UNIT 3
Final Dec. 22  FINAL EXAM TUESDAY, DECEMBER 22, 2:45 pm, EMPHASIS ON UNIT 3
WINTER RECESS DECEMBER 22-JANUARY 11.
SECOND SEMESTER: FIRST CLASS ON TUESDAY, JANUARY 20
PHYSICS 722 STARTS WEDNESDAY, JANUARY 21

© 1997, 1998, Loyal Durand