Polyakov's Lectures on Modern Classical Dynamics
These lectures were given by Professor Alexander Polyakov when he taught a graduate course on Modern Classical Dynamics at Princeton in the Spring semesters of 2013 and 2014. Vladimir Kirilin, a student in the class, reviewed the videos and provided the notes below. The lectures are presented here for your enjoyment.
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| Lecture 1||
The essential role of symmetries in dynamics. Constraints on the solution following from the symmetries. The principle of least action.
| Lecture 2||
The role of symmetries continued. Noether theorem and conservation laws.
| Lecture 3||
Which equations can and can’t follow from the action and the relation to the energy conservation. Time-reversal symmetry. Fixing the Lagrangian of a free particle based on the symmetries.
| Lecture 4||
The action for the relativistic particle, its symmetries and the conservation laws that follow. Relating optics and mechanics. The Hamiltonian description.
| Lecture 5||
The Hamiltonian description of dynamics. The Poisson bracket.
| Lecture 6||
The Hamiltonian fomalism of dynamics. The Poisson bracket.
| Lecture 7||
Generating functions of the canonical transformation. Constants of motion, Poisson brackets and the motion generated in the phase space. Action-angle variables.
| Lecture 8||
Action-angle variables. Integrability. Topology in the phase space. Ergodicity.
| Lecture 9||
Ergodicity. Statistical mechanics, grand canonical ensemble, thermodynamic quantities. Entropy and the second law. Continuous media.
| Lecture 10||
From discrete to continuous media. Sound waves in a crystal, symmetry breaking. Fluid coordinates.
| Lecture 11||
Equations of motion for the fluids. Incompressible and compressible fluids. Linear approximation.
| Lecture 12||
Symmetries of Euler equations. Special role of time-reversal. Hubble equations from hydrodynamics. Non-linear sound.
| Lecture 13|| |
| Lecture 14||
Singularities in hydrodynamics. Riemann solution for nonlinear sound. Vorticity and irrotational flows.
| Lecture 15||
Vorticity, the passive scalar equation. The streamline approach. Single and multiple vortex solution. Interaction of vortices.
| Lecture 16||
Potential energy of vortex interaction. The Hamiltonian structure of the vortex dynamics. Analogies with two-dimensional electrostatics. Breaking the Galilean invariance. Three-dimensional vortex lines.
| Lecture 17||
Vortex lines in three dimensions. Helicity conservation, linking number. The circulation theorem. Bernoulli equation. Complex potential and the flow around the cylinder.
| Lecture 18||
Conformal maps, Riemann mapping theorem, Zhukovsky functions. Singularities in the velocity field, circulation and the lift force. Continuity equations and the gradient expansion.
| Lecture 19||
Continuity equations. Gradient expansion and constraints on the terms. Shear and bulk viscosity. Navier-Stokes equation. Laminar and turbulent flows.
| Lecture 20||
General instability analysis. Special points: fixed points, limit cycles, strange attractors. Logistic map, period doubling, universality and chaotic behavior.
| Lecture 21||
Necessary conditions for the chaotic behavior: ergodicity and “mixing”. Sensitivity to small perturbations, examples of chaos. Turbulence. Correlation functions. Kolmogorov cascade and exponents.
| Lecture 22||
Solitons and integrability. KdV equation. Toda lattice. Lax representation and extra integrals of motion.