New conditional integrable cases of motion of a rigid body with Kovalevskaya's configuration

Abstract

We consider the general problem of motion of a rigid body about a fixed point under the action of an axisymmetric combination of potential and gyroscopic forces. We introduce two new cases of this problem which are integrable on the zero level of the cyclic integral. The new cases are combined generalizations of several previously known ones, namely those of Kovalevskaya, Yehia, Sokolov, Yehia and Bedweihi and Goriatchev, by the introduction of additional parameters to the structure of each.

New integrable problems in rigid body dynamics with quartic integrals

“…In [28], Yehia has introduced a method for constructing integrable conservative twodimensional mechanical systems whose second integral of motion is polynomial in velocities. This method appeared to be successful in constructing a great number of irreversible systems (involving gyroscopic forces) with a second integral quadratic (see, e.g., [29,30]), cubic [31] and quartic (see, e.g., [23] and [32][33][34][35][36]). In this method, the configuration space is not assumed to be the Euclidean plane.…”

Generalizations of the Kovalevskaya case and quaternions

“…− 4r 0 n 1 + 4c 2 n 2 The family (29) was presented by H. M. Yehia [24], and the additional integral in this case has the form F = L 2 1 − L 2 2 + 4c 1 (L 1 n 3 − L 3 n 1 ) + 4c 2 1 + 8r 0 n 2 − c 2 (n 2 1 − n 2 2 ) n 2 3 2 + +4 L 1 L 2 + 2c 1 (L 2 n 3 − L 3 n 2 ) − 4r 0 n 1 − c 2 n 1 n 2 n 2 3 2 + +4(c + 2c 2 )(L 3 − c − 2c 2 )(L 2 1 + L 2 2 ) + 16(c + 2c 2 )(c 1 n 2 L 1 L 2 − c 1 n 1 L 2 2 − −c 2 1 L 3 − 2r 0 n 3 L 2 ) − 4c 2 (c + 2c 2 ) n 2 3 (4c 1 n 1 + c + 2c 2 − (1 + n 2 3 )L 3 ).…”

“…One of them involves the introduction of additional terms to the system with two degrees of freedom on the algebra e(3), for example, a gyrostatic parameter or terms added to the potential, which do not break the symmetry of the field relative to the fixed axis. These additive terms were examined in detail by Yehia [23,24], Valent [21] and others.…”

On two-dimensional integrable models with a cubic or quartic integral of motion

“…In order to construct the corresponding integrable models, its suffices to substitute this function into the building blocks (29), (30) and (40) for the cubic and quartic cases, respectively, and to express the variables u and s in terms of x and y via (24) and (33). The advantage of our models is that they are given in terms of elementary functions (cf.…”