A solution for the motion of a body with a fixed point

Substantial condition for the fourth first integral of the rigid body problem

“…Taking equations (13) into consideration, it is easy to prove that equations (5) and (8) are satisfied. Moreover, solving equations (13) for the functions U i ; i = 1, 2, 3, one can obtain system (5) directly.…”

“…Under the present circumstances, we use the contradiction manner in the proof. Let us consider that U i = 0; i = 1, 2, 3; a closer look at system (5) shows that this system will be homogeneous for the function F, i.e.,…”

“…The topic of rigid body motion is considered one of the important topics in classical theoretical mechanics. It emerged as a clear topic after work by many distinguished scientists [1][2][3][4][5][6][7][8][9], in which the rotational motion of a rigid body about one of its fixed points is transformed into quadratures in three well-known cases, the Euler-Poinsot, Lagrange-Poisson and Kovalevskya cases, when the body moves in a uniform field. There are another two associated cases when the body moves under the action of a Newtonian field, known as generalizations of the Euler-Poinsot and Lagrange-Poisson cases.…”

“…There are another two associated cases when the body moves under the action of a Newtonian field, known as generalizations of the Euler-Poinsot and Lagrange-Poisson cases. These cases assume some conditions on the coordinates of the centre of mass and on the principal moments of inertia of the body [3][4][5]. To accomplish the solution of this problem in full generality, we need an additional fourth first integral.…”

Motion of a heavy gyrostat with variable gyrostatic moment in the generalized Steklov case

“…By analogy with [16], we find that the yet unspecified directions of the horizontal axes Oξ and Oη can be chosen in such a way that the components of the fixed hodograph depend on ω 1 , ω 2 , and ω 3 as follows:…”