Solvable structures in the falling cat problem

Abstract

In this paper, the theory of solvable structures is applied to the falling cat problem, which concerns the ability of a cat (or any similar creature) to reorient its body in midair during a fall to land on its feet. Under some assumptions and simplifications in the falling cat problem, a coupled system formed by two second-order ordinary differential equations involving an arbitrary scalar function is obtained. In order to address the determination of exact solutions, the Lie algebra of point symmetries admitted by the system is computed, which turns out to be four-dimensional. However, the corresponding symmetry generators are not linearly independent pointwise, and therefore, the classical Lie-Bianchi method cannot be directly applied to solve the system. In order to overcome this difficulty, a solvable structure for the vector field associated with the system is computed, which is used to completely integrate the equations of motion. An alternative integration procedure based on a combination of the Noether theorem and the Lie-Bianchi method is also shown.