Falling cat problem

A falling cat modeled as two independently rotating parts turns around while maintaining zero net angular momentum

The falling cat problem is a problem that consists of explaining the underlying physics behind the observation of the cat righting reflex: that is, how a free-falling body (cat) can acquire angular momentum such that it is able to right itself as it falls to land on its feet, irrespective of its initial orientation and, without violating the law of conservation of angular momentum.

Although 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.

History

The falling cat problem has elicited interest from famous scientists including George Gabriel Stokes, James Clerk Maxwell, and Étienne-Jules Marey. In a letter to his wife, Katherine Mary Clerk Maxwell, Maxwell wrote, "There is a tradition in Trinity that when I was here I discovered a method of throwing a cat so as not to light on its feet, and that I used to throw cats out of windows. I had to explain that the proper object of research was to find how quick the cat would turn round, and that the proper method was to let the cat drop on a table or bed from about two inches, and that even then the cat lights on her feet." Maxwell's as well as other scientist's fetish with so-called "cat turning" was reputed to have caused the death of many cats after being dropped from windows of multi-storied buildings. Maxwell in particular, was interested in discovering the precise height that a cat needed to be dropped from so that the cat would be incapable of using its righting reflex to land on its feet, an endeavor in which he eventually succeeded.

Whereas the cat-falling problem was regarded as a mere curiosity by Maxwell and Stokes, a more rigorous study of the problem was conducted by Étienne-Jules Marey who applied chronophotography to capture the cat's decent on film using a chronophotagraphic gun. The gun, capable of capturing 12 frames per second, produced images from which Marey made some important deductions. Namely, that as the cat had no rotational motion at the start of its descent the cat was not "cheating" by using the cat handler's hand as a fulcrum. This in itself posed a problem as it implied that it was possible for a body in free fall to acquire angular momentum. Marey also showed that air resistance played no role in facilitating the righting of the cat's body.

Images of a falling cat which appeared in the journal Nature in 1894, captured by a chronophotagraphic gun, a device of Marey's own invention led to important advances in the falling cat problem. Of the cat images that appeared in the summary the author of Nature wrote, "The expression of offended dignity shown by the cat at the end of the first series indicates a want of interest in scientific investigation."

His investigations were subsequently published in Comptes Rendus, and a summary of his findings were published in the journal Nature. The article's summary in Nature appeared thus,

M. Marey thinks that it is the inertia of its own mass that the cat uses to right itself. The torsion couple which produces the action of the muscles of the vertebra acts at first on the forelegs, which have a very small motion of inertia on account of the front feet being foreshortened and pressed against the neck. The hind legs, however, being stretched out and almost perpendicular to the axis of the body, possesses a moment of inertia which opposes motion in the opposite direction to that which the torsion couple tends to produce. In the second phase of the action, the attitude of the feet is reversed, and it is the inertia of the forepart that furnishes a fulcrum for the rotation of the rear.

Despite the publication of the images, many physicists at the time maintained that the cat was still "cheating" by using the handler's hand from its starting position to right itself.

Solution

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).

See also

• Falling Cat (documentary)
• Parallel parking problem

References

• 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" (PDF), Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 34 (4): 527–557, doi:10.1016/s1355-2198(03)00062-5.
• Kane, T R; Scher, M P. (1969), "A dynamical explanation of the falling cat phenomenon", Int J Solids Structures, 5: 663–670, doi:10.1016/0020-7683(69)90086-9.
• Montgomery, R. (1993), "Gauge Theory of the Falling Cat", in M.J. Enos (ed.), Dynamics and Control of Mechanical Systems (PDF), 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.

Further reading

• Richard Montgomery's homepage, referring to the problem
• Lagrangian Reduction and the Falling Cat Theorem