A new family of phase patterns in three-dimensional dynamics of a solid body interacting with matter

Abstract

1. Setting the problem. At the present time, solving the problem of motion of a solid body in a resisting matter completely depends on developing mathematical methods and possible model constraints inherent in the problem. For example, the well-known Kirchhoff problem of motion of a solid body in a perfect incompressible fluid, being quiescent at infinity and executing vortex-free motion [1], deals with only one aspect of the problem. This aspect is related to the problem of integrability of corresponding dynamic systems (in our case, with the existence of the complete set of analytic and meromorphic first integrals). It is easy to note that the Kirchhoff problem is one of the first approximations to the description of the interaction between a body and a medium because, on introducing an arbitrarily small viscosity, the dynamic Kirchhoff systems cease to be conservative. Thus, asymptotic limiting sets appear in the phase space of resulting systems because they become dissipative “as a whole.” Therefore, no complete set of even-continuous first integrals can be asserted [2]. In connection with this comment, we indicate another aspect of the problem, namely, the complete qualitative analysis of dynamic systems (the topology of phase-space splitting into trajectories). It is the aspect that is an object of the present analysis. 2. Initial conditions and choice of variables. We consider a problem of a three-dimensional motion of a dynamically symmetric solid body under the condition that the line of action of the force S applied to the body from the matter does not change its orientation with respect to the body, a part of the body surface having the shape of a flat disk. The matter flows around the disk [3] according to laws of a jet flow around a body [4, 5]. The force S is aligned with the normal to the disk and is a quadratic form of the velocity of the disk center. The gravity force acting on the body is assumed to be negligible compared to the resisting force of the matter. The choice of six dynamic phase variables v , α , β (i.e., spherical coordinates of the velocity vector for the disk center) and p , q , r (the components of the body absolute angular velocity in the coordinate system linked with the body) allows us to consider the sixorder system of dynamic equations as independent. Moreover, because the resisting force admits a group of the body rotation about the axis of dynamic symmetry (which passes through the center of mass and the center of the disk), the longitudinal component of the angular velocity being conserved: p = p 0 = const [3, 6]. 3. Dynamic equations of motion. If ( A , B , B ) are the principal moments of inertia for the body, m and σ are its mass and the distance between the center of mass and the disk, respectively, z 1 = q cos β + r sin β , z 2 = r cos β — q sin β , z i = Z i v , i = 1, 2, = α ’ v , = β ’ v ,