Mathematical understanding of nature : essays on amazing physical phenomena and their understanding by mathematicians

Originally published in Russian by MTSNMO, under the title: Mathematicheskoe ponimanie prirody, 2011.

Includes bibliographical references.

Chapter 1. The eccentricity of the Keplerian orbit of Mars
Chapter 2. Rescuing the empennage
Chapter 3. Return along a sinusoid
Chapter 4. The Dirichlet integral and the Laplace operator
Chapter 5. Snell’s law of refraction
Chapter 6. Water depth and Cartesian science
Chapter 7. A drop of water refracting light
Chapter 8. Maximal deviation angle of a beam
Chapter 9. The rainbow
Chapter 10. Mirages
Chapter 11. Tide, Gibbs phenomenon, and tomography
Chapter 12. Rotation of a liquid
Chapter 13. What force drives a bicycle forward?
Chapter 14. Hooke and Keplerian ellipses and their conformal transformations
Chapter 15. The stability of the inverted pendulum and Kapitsa’s sewing machine
Chapter 16. Space flight of a photo camera cap
Chapter 17. The angular velocity of a clock hand and Feynman’s “self-propagating pseudoeducation”
Chapter 18. Planetary rings
Chapter 19. Symmetry (and Curie’s principle)
Chapter 20. Courant’s erroneous theorems
Chapter 21. Ill-posed problems of mechanics
Chapter 22. Rational fractions of flows
Chapter 23. Journey to the center of the earth
Chapter 24. Mean frequency of explosions (according to Ya. B. Zel’dovich) and de Sitter’s world
Chapter 25. The Bernoulli fountains of the Nikologorsky bridge
Chapter 26. Shape formation in a three-liter glass jar
Chapter 27. Lidov’s moon landing problem
Chapter 28. The advance and retreat of glaciers
Chapter 29. The ergodic theory of geometric progressions
Chapter 30. The Malthusian partitioning of the world
Chapter 31. Percolation and the hydrodynamics of the universe
Chapter 32. Buffon’s problem and integral geometry
Chapter 33. Average projected area
Chapter 34. The mathematical notion of potential
Chapter 35. Inversion in cylindrical mirrors in the subway
Chapter 36. Adiabatic invariants
Chapter 37. Universality of Hack’s exponent for river lengths
Chapter 38. Resonances in the Shukhov tower, in the Sobolev equation, and in the tanks of spin-stabilized rockets
Chapter 39. Rotation of rigid bodies and hydrodynamics