Lectures Notes: |
Homework: |
Pylab-Python Code: |
|---|---|---|
| Lecture 1 | Problem Set 1 |
pend_vec.ipynb
pend_vec.html
phasewrap.ipynb phasewrap.html |
| Introduction, Equations of motion in one degree of freedom from Newton's laws, Hamiltonian and Lagrangian descriptions, Fixed points, Liouville's Theorem | Code examples to plot stream lines or level curves and to look at phase wrapping | |
| Lecture 2 | Problem Set 2 | harm_integrate.ipynb harm_integrate.html |
| Canonical transformations, Poisson brackets, Generating functions, The Symplectic form, The Jacobi constant, Symmetries and Conserved quantities, Noether's theorem | Code example to compare a second order symplectic leapfrog integrator with a second order Runge-Kutta integrator. | |
| Lecture 3 | Problem Set 3 |
perio.ipynb
perio.html
drift_integrate.ipynb drift_integrate.html Keplermap.ipynb Keplermap.html |
| 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, for a drifting Andoyer Hamiltonian and for the 2D Kepler map. | |
| Lecture 4 | Problem Set 4 |
cobweb.ipynb
cobweb.html
logistic.ipynb logistic.html lyap.ipynb lyap.html |
| 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 | baker.ipynb baker.html | |
| 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. | |
| Quantum Chaos | Problem Set 14 | Ripple_and_D_Billiard.ipynb Ripple_and_D_Billiard.html quantum_kicked_rotor.ipynb quantum_kicked_rotor.html |
| Quantum Chaos. Quantization of the kicked rotor. GUI vs Poisson energy level statistics. Anderson localization. Billiards. Wigner and Weyl representations for quantum mechanics in phase space. Coherent states. Discrete versions of coherent states. Semi-classical approximations. The quantized baker map. | Code examples for billiards and quantum chaos | |
| Jeremy's notes | ||
| Jeremy Couturier's lecture notes on Kozai Lidov and Evection resonances | ||
| Lecture 6 |
circle_map.ipynb
circle_map.html
devilstair.ipynb devilstair.html |
|
| 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. | ||
| Lecture 12 | Problem Set 12 | |
| Mechanical Constraints. Holonomic and non-holonomic constraints. Lagrange d'Alembert principle vs Vakonomic dynamics. Geometric view of the falling cat. Control of mechanical systems. | ||
| Pattern formation | FEM code examples | |
| Classification of Fixed points of 2-dimensional dynamical systems. Hopf Bifurcation and birth of a limit cycle. Reaction-Diffusion equations and Turing instability for pattern formation models. Spatial and temporal instability in the Brusellator model. The Swift-Hohenberg and Ginzburg-Landau pattern formation models. | ||
| Active Matter , Notes on boundaries | ||
| Active matter. Self-propelled particles. Continuum models for self-propelled particles (Toner-Tu). Active nematics. Active fluids. Active oscillating materials. | ||