New integrable problems in rigid body dynamics with quartic integrals

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 will construct two new conditional integrable problems. These cases are combined generalizations of several previously known ones, namely those of Chaplygin and Yehia by the introduction of additional parameters to the structure of each.

Quartic Integral in Rigid Body-Gyrostat Dynamics

“…e present case generalizes a special version of the case introduced by Yehia and Elmandouh in 2016 by adding a new parameter ] [48]. Also, it includes the case announced by Elmandouh in 2015 (K � 0) [3]. Moreover, it generalizes the case presented by Yehia and Elmandouh in 2013 by inserting two arbitrary constants (k � ] � 0) [6].…”

“…e first one is finding the sufficient conditions for the integrability, and this requires the construction of a sufficient number of first integrals of motion. Numerous methods can be utilized to construct the first integrals of motion such as the direct method, Darboux method, and Yehia method (see, e.g., [3][4][5][6][7][8][9][10][11][12][13]). e second one deals with obtaining the necessary conditions of the integrability (see, e.g., [14][15][16][17][18][19][20]), but we must introduce the required number of the integrals to confirm the integrability.…”

“…+ c 2 (cp + 4 dq)c 2 1 Kc 6q 2 − 4k 2 + 4c 1 c 2 pq + k 2 + 8K (− cq − dp)c 3 (cp − dq)c dc 2 + d 2 kK 2 + 3k 2 + p 2 − 2q 2 d − 3pqc K + 2k 3qc 2 − pc 1 d + c 2qc 1 + c 2 q Kc3 …”

The Motion of a Rigid Body with Irrational Natural Frequency

“…In this section, we discuss the problem geometrically to show the orientation of the body at any instant in time. Substituting Equation ( 9) into Euler's angles ψ, θ, and φ in which t has been replaced by t + h, using M 3 = tan θ o , we obtain the Advances in Mathematical Physics following expressions for the angles [11]:…”

The Stability of Certain Motion of a Charged Gyrostat in Newtonian Force Field

“…e first is the integrability problem and the searching for the complete set of the first integrals of the motions. Borisov and Mamaev [5] contain most of those integrable problems up to 2001, and some cases were presented by several authors (see, e.g., [6][7][8][9][10]). e second category regards the problem of study periodic solutions, bifurcation, and chaos in some problems of rigid body-gyrostat (see, e.g., [11][12][13][14]).…”