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The Baron and the cat

The Baron and the cat. Joint work with Oded Kenneth Original idea: J. Wisdom. Baron and cat. A cat can rotate with zero angular momentum Can Baron von Munchausen lift himself? Translate with zero momentum?. Movie: omri Gat. H. Knoerer: Cats always fall on their feet.

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The Baron and the cat

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  1. The Baron and the cat Joint work with Oded Kenneth Original idea: J. Wisdom

  2. Baron and cat • A cat can rotate with zero angular momentum • Can Baron von Munchausen lift himself? • Translate with zero momentum? Movie: omri Gat H. Knoerer: Cats always fall on their feet

  3. Rotations without angular momentum: Physics

  4. Rotations without angular momentum: Mathematics Generators of deformations The commutator of deformations= a rotation

  5. Newton’s law If a system experiences no external force, the center-of-mass of the system will remain at rest. http://webphysics.davidson.edu/physlet_resources/bu_semester1/c12_cofm_motion.html Newton: Lex I: Corpus omne perseverare in statu suo quiescendi vel Movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare. Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed Wikipedia

  6. Ambiguous center of mass: Topology Ambiguous center of mass allows the Baron to move itself

  7. Ambiguous center of mass: Geometry Euclidean plane The three medians of a (Euclidean) triangle intersect at one point On a sphere

  8. Swimming triangles On sphere Movies: Oren Raz

  9. Swimming triangles Inside a torus Movies: Oren Raz

  10. Shape fixes location Shape: All mutual distances Location: Large rigid bodies generically fits nowhere Generically, shape fixes a location

  11. When can rigid bodies move? Euclidean space is a symmetric space: Homogeneous and isotropic. In a symmetric space if a rigid body fits somewhere, it fits any where. Examples: Lobachevski plane Sphere In a symmetric space, shape does not determine location. You may try to swim.

  12. Killing forms Killing for Euclidean translation Killing for Euclidean rotation

  13. Controls: Deformations Example: Euclidean case Vector field for deformations depend on choice of origin: Ambiguity in Killing

  14. Swimming distance Deformation space sb Stroke sa Deformation fields Killing 2-form Main result Space allows for swimming if the Killing 1-form is not closed

  15. Cats spin, the Baron lies Translations The Baron lies Rotations The cat falls on its feet

  16. Small swimmers on symmetric surface Property of space Size of stroke Geometry of aswimmer Wisdom Swim away from Black hole Relativistic motions AK: Symmetric spaces Non relativistic bodies

  17. t Stroke s s Shape space The swimming problem Swimming with periodic strokes: Rigid body motions

  18. Conservation of momentum A system of particles Killing vector field Swimming equations As many equations as Killing fields: Can’t move in other directions Eq. independent of time parameterization: Geometric

  19. Killing fields: Translation & rotations Defining property: No strain Relation to Riemann Determined by initial data at a point translation Rotation Rotation about z axis: Looks like translation near equator And like rotation at poles

  20. Rigid body motion: Coordinates t3 t2 t1 Putting coordinates for The analog of Euclidean motions

  21. Coordinates for rigid body motion t1 t2 Symbolic notation for transported shape

  22. Gauge choice instead of CM Euclidean example: Gauge choice: Analog to choosing cm as fiducial pt for translations

  23. s space Coordinates for deformed shapes Symbolically

  24. t s s Total space Coordinates for deformations + Euclidean motions Motions is legit if it consistent with zero total momentum

  25. Equation of Motion To leading order with s and t small Substitute in conservation law Gives a linear (system) of equations for dt

  26. t s s Main result Swimming distance Shape space Deformation fields Killing fields

  27. Role of Curvature Killing field for symmetric spaces can be Explicitly expressed in terms of the curvature In a symmetric space Swimming in the direction of the k-th Killing field Stroke and swimmer data Geometry of embedding space

  28. Application to astrophysics? How does swimming affect the motion of galaxies of size x? Hubble constant 75 km/sec/Mpc gives a measure of the curvature Perhaps not surprisingly, it takes the age of the universe for a significant swimming

  29. The hero: Oded Kenneth Thanks to Amos Ori

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