Abstract
In this paper, the problem of the slow spinning motion of a rigid body about a point O, being fixed in space, in the presence of the Newtonian force field and external torque is considered. We achieve the slow spin by giving the body slow rotation with a sufficiently small angular velocity component r 0 about the moving z-axis. We obtain the periodic solutions in a new domain of the angular velocity vector component r 0 ⟶ 0 , define a large parameter proportional to 1 / r 0 , and use the technique of the large parameter for solving this problem. Geometric interpretations of motions will be illustrated. Comparison of the results with the previous works is considered. A discussion of obtained solutions and results is presented.
Highlights
In [1], the problem of rigid body dynamics is considered. e author in [2] gave important space applications to this problem
In [4], the authors introduced a new procedure for solving Euler–Poisson equations. e author in [5] constructed periodic solutions for Euler–Poisson equations utilizing power series expansion containing a small parameter proportional to the inverse of sufficiently high angular velocity component
The authors assumed that the body rotates with a sufficiently large angular velocity component ro about the moving z-axis which moves with the body. e authors achieved a small parameter proportional to 1/r0 and used the small parameter technique to solve the considered problems in the domain (t, ro ⟶ ∞, ε ⟶ 0). e fact of slow motion of that body which must be achieved on a new parameter named the large parameter and must be solved using a new procedure named the large parameter technique was not considered, this motion saves high energy given at the initial moment of the body and can solve the problem in a new domain (t, ro ⟶ 0, ε ⟶ ∞)
Summary
In [1], the problem of rigid body dynamics is considered. e author in [2] gave important space applications to this problem. In [4], the authors introduced a new procedure for solving Euler–Poisson equations (of a rotatory rigid body over a fixed point). E author considered the fast spin motion of a rigid body and achieved a small parameter proportional to the inverse of high angular components about the z-axis. In [9], the author investigated the motion over the fixed point O of a fast spinning heavy solid in a uniform gravity field (the classical problem). He assumed fast spinning of the body, achieved a small parameter, and used Poincare’s method for the solution. The authors assumed that the body rotates with a sufficiently large angular velocity component ro about the moving z-axis which moves with the body. e authors achieved a small parameter proportional to 1/r0 and used the small parameter technique to solve the considered problems in the domain (t, ro ⟶ ∞, ε ⟶ 0). e fact of slow motion of that body which must be achieved on a new parameter named the large parameter and must be solved using a new procedure named the large parameter technique was not considered, this motion saves high energy given at the initial moment of the body and can solve the problem in a new domain (t, ro ⟶ 0, ε ⟶ ∞)
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