The Baron and the cat
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Transcript of The Baron and the cat
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The Baron and the The Baron and the cat cat
Joint work with Oded KennethOriginal idea: J. Wisdom
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Baron and catBaron and cat
► A cat can rotate with zero A cat can rotate with zero angular momentumangular momentum
► Can Baron von Can Baron von Munchausen lift himself?Munchausen lift himself?
► Translate with zero Translate with zero momentum?momentum?
Movie: omri Gat
H. Knoerer: Cats always fall on their feet
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Rotations without angular Rotations without angular momentum: Physicsmomentum: Physics
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Rotations without angular Rotations without angular momentum: Mathematicsmomentum: Mathematics
Generators of deformations
The commutator of deformations= a rotation
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Newton’s lawNewton’s law
Lex I: Corpus omne perseverare in statu suo quiescendi velMovendi uniformiter in directum, nisi quatenus a viribusimpressis 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
http://webphysics.davidson.edu/physlet_resources/bu_semester1/c12_cofm_motion.html
If a system experiences no external force, the center-of-mass of the system will remain at rest.
Newton:
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Ambiguous center of mass: Ambiguous center of mass: TopologyTopology
Ambiguous center of mass allows the Baron to move itself
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Ambiguous center of mass: Ambiguous center of mass: GeometryGeometry
Euclidean plane
On a sphere
The three medians of a (Euclidean) triangle intersect at one point
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Swimming trianglesSwimming triangles
Movies: Oren Raz
On sphere
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Swimming trianglesSwimming triangles
Movies: Oren Raz
Inside a torus
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Shape fixes locationShape fixes location
Generically, shape fixes a location
Large rigid bodies generically fits nowhere
Shape: All mutual distances
Location :
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When can rigid bodies move?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 :
Sphere Lobachevski plane
In a symmetric space, shape does not determine location. You may try to swim.
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Killing formsKilling forms
Killing for Euclidean translation
Killing for Euclidean rotation
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Controls: DeformationsControls: Deformations
Vector field for deformations depend on choice of origin: Ambiguity in Killing
Example: Euclidean case
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Main resultMain result
a
Stroke
Deformation spaceb
Swimming distance
Killing 2-form Deformation fields
Space allows for swimming if the Killing 1-form is not closed
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Cats spin, the Baron liesCats spin, the Baron lies
Translations
The Baron lies
Rotations
The cat falls on its feet
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Small swimmers on symmetric Small swimmers on symmetric surfacesurface
Property of spaceSize of stroke
WisdomSwim away from Black holeRelativistic motions
AK:Symmetric spacesNon relativistic bodies
Geometry of aswimmer
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The swimming problemThe swimming problem
Swimming with periodic strokes:
Rigid body motions
Stroke
Shape space
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Conservation of momentumConservation 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
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Killing fields: Translation & Killing fields: Translation & rotationsrotations
Rotation about z axis:Looks like translation near equator
And like rotation at poles
Defining property: No strain
Relation to Riemann
Determined by initial data at a point
translation Rotation
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Rigid body motion: Rigid body motion: CoordinatesCoordinates
1
2
3
Putting coordinates for The analog of Euclidean motions
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Coordinates for rigid body Coordinates for rigid body motionmotion
Symbolic notation for transported shape
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Gauge choice instead of CMGauge choice instead of CM
Euclidean example :
Gauge choice: Analog to choosing cm as fiducial pt
for translations
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Coordinates for deformed Coordinates for deformed shapesshapes
space
Symbolically
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Total spaceTotal space
Coordinates for deformations + Euclidean motions
Motions is legit if it consistent with zero total momentum
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Equation of MotionEquation of Motionleading order withand small
Substitute in conservation law
Gives a linear (system) of equationsfor d
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Main resultMain result
Swimming distance
Killing fields Deformation fieldsShape space
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Role of CurvatureRole of CurvatureKilling field for symmetric spaces can be
Explicitly expressed in terms of the curvature
Swimming in the direction of the k-th Killing field
In a symmetric space
Geometry of embedding space
Stroke and swimmer data
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Application to astrophysics?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 ageof the universe for a significant swimming
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The hero: Oded Kenneth The hero: Oded Kenneth
Thanks to Amos Ori