Nonlinear Physics: From the Pendulum to Turbulence and Chaos

Abstract

Part 1 Particles: the elements of dynamics - phase space, systems with one degree of freedom, an example - the nonlinear pendulum, two more examples of nonlinear oscillations, Poincare's integral invariants, multidimensional integrable systems, mappings, some remarks in conclusion approximate methods - perturbation theory, the averaging method, adiabatic invariants, charged particles in a magnetic field, linear analogues of adiabatic invariance special methods - nonlinear resonance, the Kolmogorov-Arnold-Moser (KAM), structural properties of phase trajectories, simple bifurcations ergodic theory and chaos - ergodicity and mixing, K-systems, examples, recurrences and periodic orbits chaos in detail - a universal mapping for nonlinear oscillations, overlapping of resonances, formation of a stochastic layer, destruction of the integrals of motion, stochastic attractors, examples of stochastic attractors, general notes on the onset of chaos elements of kinetics - the Fokker-Planck-Kolmogorov equation, kinetics in dissipative mappings, stochastic acceleraton and heating of particles fractal properties of chaos - fractals, fractals and chaos. Part 2 Waves: nonlinear stationary waves - steepening of waves, stationary waves, examples of stationary waves, collision-free shock waves Hamiltonian description of waves - variational principles, resonance interaction of waves, nonlinear wave resonances, interaction of nonlinear waves chaos in wave fields - weakly nonlinear fields, the fermi-pasta-ulam (FPU) problem, turbulence of a weekly nonlinear field, stochastic instability of a nonlinear wave strong turbulence - Lorenz model, convective cells, features of the onset of turbulence, Langmuir turbulence, soliton turbulence exactly integrable wave equations - integration of the KdV equation, integrable equations. Part 3 Examples: motion of particles in wave fields - regular and stochastic dynamics of particles, motion in a magnetic field and the field of a wave packet, the paradox of the disappearance of Landau damping, stochastic web billiards - mixing billiards, nonlinear-ray dynamics nonlinear optics - nonlinear geometrical optics, nonlinear co-operative phenomena structural properties of one dimensional chains - atom chains, spin chains, excitation in chains of molecules perturbations in Kepler's problem - nonlinear dynamics in a coulomb field, excitation and ionization of a hydrogen atom, diffusion of the eccentricity of orbits in the gravitational field of planets, diffusion of comets from the oort cloud. Part 4 Numerical simulation: nonlinear physics in colour - general notes on the pictures diskettes the ATRS program.