## Mathematical Understanding of Nature: Essays on Amazing Physical Phenomena and Their Understanding by MathematiciansThis collection of 39 short stories gives the reader a unique opportunity to take a look at the scientific philosophy of Vladimir Arnold, one of the most original contemporary researchers. Topics of the stories included range from astronomy, to mirages, to motion of glaciers, to geometry of mirrors and beyond. In each case Arnold's explanation is both deep and simple, which makes the book interesting and accessible to an extremely broad readership. Original illustrations hand drawn by the author help the reader to further understand and appreciate Arnold's view on the relationship between mathematics and science. |

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### Contents

Chapter | 1 |

Chapter | 7 |

Rotation of a Liquid | 33 |

What Force Drives a Bicycle Forward? | 37 |

Hooke and Keplerian Ellipses and Their Conformal Transformations | 39 |

The Stability of the Inverted Pendulum and Kapitsas Sewing Machine | 45 |

Space Flight of a Photo Camera Cap | 49 |

The Angular Velocity of a Clock Hand and Feynmans SelfPropagating Pseudoeducation | 51 |

Mean Frequency of Explosions according to Ya B Zeldovich and de Sitters World | 75 |

The Bernoulli Fountains of the Nikologorsky Bridge | 79 |

Shape Formation in a ThreeLiter Glass Jar | 83 |

Lidovs Moon Landing Problem | 87 |

The Advance and Retreat of Glaciers | 91 |

The Ergodic Theory of Geometric Progressions | 99 |

The Malthusian Partitioning of the World | 101 |

Percolation and the Hydrodynamics | 103 |

Planetary Rings | 55 |

Symmetry and Curies Principle | 59 |

Courants Erroneous Theorems | 61 |

IllPosed Problems of Mechanics | 65 |

Rational Fractions of Flows | 69 |

Journey to the Center of the Earth | 71 |

Average Projected Area | 111 |

Inversion in Cylindrical Mirrors in the Subway | 127 |

Adiabatic Invariants | 143 |

Universality of Hacks Exponent for River | 153 |

Rotation of Rigid Bodies and Hydrodynamics | 161 |

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1[Translator’s note adiabatic invariant angle angular velocity average projected area axis beams calculate caustic circle of inversion coefficients complex number computation consider Courant’s cube curvature cylindrical mirror depends described distance dynamical system Earth ellipse ellipsoid equal ergodic Euclidean plane Euclidean space Euler’s theory example field Fields Medal flow force fraction function f galaxies geodesic geodesic curvature geometric progression half-plane homothety Hooke ellipse hyperbolic inertia initial conditions integral intersection Keplerian ellipse Laplace operator Larmor circle length linear math mathematical mathematicians mean value Moon motion observed obtain orbit Ordinary Differential Equations oscillations parameter percolation radius perturbations phase space physical physicists Poincaré position problem pulley refraction Remark rotation Russian satellites smooth Sobolev Solution spatial means special circle sphere stability straight line summand surface symmetry tangent theorem tion torus transformation upper half-plane Uranus V. I. Arnold vector