## 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.

The files are 0.5-1.0 GB in MP4 format. If your browser is setup with proper A/V player, a lecture will start playing in a separate window immediately. If you are connecting to a low bandwidth or congested network, you might see some artifacts in the videos. If it happens too frequently, you may download the whole file on your local disk and play it offline.

Lecture |
Description |

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. |