PHY 411: Lecture Notes and Problems


Lectures Notes:

Homework:

Pylab-Python Code:

Lecture 1 Problem Set 1 pend_vec.py
Introduction, Equations of motion in one degree of freedom from Newton's laws, Hamiltonian and Lagrangian descriptions, Fixed points, Liouville's Theorem Code example to plot stream lines or level curves
Lecture 2 Problem Set 2
Canonical transformations, Poisson brackets, Generating functions, The Symplectic form, The Jacobi constant, Symmetries and Conserved quantities, Noether's theorem
Lecture 3 Problem Set 3 perio.py
Resonances, small divisors, Poincare maps, Surfaces of section, The periodically forced pendulum and the restricted three-body problem Code examples to look at mappings (on the period) of the periodically varying pendulum.
Lecture 4 Problem Set 4 cobweb_sin.py logistic.py lyap.py
One dimensional dynamical systems and interative maps, Bifurcations in one dimensional systems, Periodic orbits in maps, The Logistic map, Period doubling bifurcations, Lyapunov Exponent, Numerical estimates of the Maximal Lyapunov Exponent and related indicators Code examples to make cobweb plots identifying fixed points and periodic orbits of iteratively applied one dimensional maps, plot attracting periodic points of the logistic map or compute and plot Lyapunov exponents for a map.
Lecture 5 Problem Set 5 baker.py
Symbolic dynamics, The Shift Map on Sigma_2, Devany's definition for a chaotic map, Two-dimensional ergodic systems, The Baker map, Topological conjugacy and Topological entropy. Code example to plot the orbits of the Baker map.
Lecture 6 circle_map.py devilstair.py devil_stair_h.py tongue.py
The sine-circle map, Arnold Tonques and the devil staircase. Ways to describe rational and irrational numbers: Diophantine approximation, Continued fractions, Farey sequence, and Pigeonholes. Code examples to plot winding numbers for the circle map and the devil staircase.
Lecture 7
Introduction to numerical integrators of ordinary differential equations. Symplectic integrators, Construction of symplectic integrators using exponentials of evolution operators, Force Gradient Algorith, Extended Phase Space and Regularization
Lecture 8 Problem Set 8
Continuous or infinite dimensional Hamiltonian systems, Examples with canonical coordinate fields, Hydrodynamics in Hamiltonian form, Changing coordinates and constructing a new Poisson bracket, Poisson manifolds, The KdV equation, Infinite conserved quantities, the LAX pair.
Lecture 9
Birkhoff normal form, The Hamilton-Jacobi equation, Hamiltonian Canonical Perturation theory
Lecture 10 Problem Set 10
Infinitesimal Transformations and Lie Groups. Locomotion of deformable bodies. Gauge Transformations of marching jello cubes. Rotations and Rigid Bodies. Euler's equations. Rotational stability of a freely rotating rigid body. Lagrangian and Hamiltonian descriptions of Rigid Bodies. Potential energy on a rigid body from an external force field.
Lecture 11
Equivalent Actions. Geometric view of Celestial mechanics. Maupertuis' principle. Equations of motion as geodesics.




pend_vec.py Code example to plot stream lines or level curves.
perio.py Code example to look at mappings (on the period) of the periodically varying pendulum. Surfaces of section.
cobweb_sin.py Code example to make cobweb plots identifying fixed points and periodic orbits of iteratively applied one dimensional maps.
logistic.py Code example to look at attracting periodic points of the logistic map.
lyap.py Code example to plot either attracting periodic points or Lyapunov exponents for maps like the logistic map.
circle_map.py Code example to plot winding numbers for the circle map.
devilstair.py Code example to plot the devil staircase for the circle map.
devil_stair_h.py Code example to compute winding numbers for the circle map at higher precission.
tongue.py Code example to compute the width of an Arnold tongue for the circle map at higher precission.
baker.py Code example to plot the orbits for the Baker map.
jul.py Code example to look at a Julia set.