Falling cat problem
Encyclopedia
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 , models the cat as a pair of cylinders (the front and back halves of the cat) capable of changing their relative orientations. 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
, 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
.
In the language of physics, Mongomery'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 , following the work of Shapere and Wilczek
.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
behind the common observation of the cat righting reflex
Cat righting reflex
The cat righting reflex is a cat's innate ability to orient itself as it falls in order to land on its feet. The righting reflex begins to appear at 3-4 weeks of age, and is perfected at 7 weeks. They are able to do this as they have an unusually flexible backbone and no functional clavicle...
: how a free-falling cat
Cat
The cat , also known as the domestic cat or housecat to distinguish it from other felids and felines, is a small, usually furry, domesticated, carnivorous mammal that is valued by humans for its companionship and for its ability to hunt vermin and household pests...
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
Rigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
, 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 , models the cat as a pair of cylinders (the front and back halves of the cat) capable of changing their relative orientations. later described the Kane–Scher model in terms of a connection
Connection (principal bundle)
In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
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
Nonholonomic system
A nonholonomic system in physics and mathematics is a system whose state depends on the path taken to achieve it. Such a system is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space but finally returns to the...
, the study of which is among the central preoccupations of control theory
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The desired output of a system is called the reference...
. 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
Motion planning
Motion planning is a term used in robotics for the process of detailing a task into discrete motions....
.
In the language of physics, Mongomery'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 , following the work of Shapere and Wilczek
Frank Wilczek
Frank Anthony Wilczek is a theoretical physicist from the United States and a Nobel laureate. He is currently the Herman Feshbach Professor of Physics at the Massachusetts Institute of Technology ....
.