Falling cat problem
The falling cat problem consists of explaining the underlying physics behind the common observation of the cat righting reflex: how a free-falling cat can turn itself right-side-up as it falls, no matter which way up it was initially, without violating the law of conservation of angular momentum.
Although somewhat amusing, and trivial to pose, the solution of the problem is not as straightforward as its statement would suggest. The apparent contradiction with the law of conservation of angular momentum is resolved because the cat is not a rigid body, but instead is permitted to change its shape during the fall. The behavior of the cat is thus typical of the mechanics of deformable bodies.
The solution of the problem, originally due to (Kane & Scher 1969), models the cat as a pair of cylinders (the front and back halves of the cat) capable of changing their relative orientations. Montgomery (1993) later described the Kane–Scher model in terms of a connection in the configuration space that encapsulates the relative motions of the two parts of the cat permitted by the physics. Framed in this way, the dynamics of the falling cat problem is a prototypical example of a nonholonomic system (Batterman 2003), the study of which is among the central preoccupations of control theory. A solution of the falling cat problem is a curve in the configuration space that is horizontal with respect to the connection (that is, it is admissible by the physics) with prescribed initial and final configurations. Finding an optimal solution is an example of optimal motion planning (Arbyan & Tsai 1998; Ge & Chen 2007).
In the language of physics, Montgomery's connection is a certain Yang-Mills field on the configuration space, and is a special case of a more general approach to the dynamics of deformable bodies as represented by gauge fields (Montgomery 1993; Batterman 2003), following the work of Shapere and Wilczek (Shapere and Wilczek 1987).
- Arabyan, A; Tsai, D. (1998), "A distributed control model for the air-righting reflex of a cat", Biol. Cybern. 79: 393–401, doi:10.1007/s004220050488.
- Batterman, R (2003), "Falling cats, parallel parking, and polarized light", Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 34 (4): 527–557.
- Kane, T R; Scher, M P. (1969), "A dynamical explanation of the falling cat phenomenon", Int J Solids Structures 55: 663–670.
- Montgomery, R. (1993), "Gauge Theory of the Falling Cat", in M.J. Enos, Dynamics and Control of Mechanical Systems, American Mathematical Society, pp. 193–218.
- Ge, Xin-sheng; Chen, Li-qun (2007), "Optimal control of nonholonomic motion planning for a free-falling cat", Applied Mathematics and Mechanics 28 (5): 601–607(7), doi:10.1007/s10483-007-0505-z.
- Shapere, Alfred; Wilczek, Frank (1987), "Self-Propulsion at Low Reynolds Number", Physical Review Letters 58: 2051, Bibcode:1987PhRvL..58.2051S, doi:10.1103/PhysRevLett.58.2051.
- Richard Montgomery's homepage, referring to the problem
- Lagrangian Reduction and the Falling Cat Theorem
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