New solutions of classical problems

A new solution [1, 2] is obtained (1995) for the classical problem of the free rotation of a rigid body about a fixed centre of mass (case of Euler). It is shown that for each inertia tensor all the domain of initial values is divided in two subdomains. It is known that there is no such a system of parameters, which would allow to cover all the domain of initial values by unique map without poles. This fact is confirmed in the work [2], where in each subdomain and at the boundary between them the body rotates about different axes, depending only on the initial values. Stable rotations of the body correspond to the interior points of the subdomains mentioned above, and unstable rotations — to the boundary points. When constructing the solution, the theorem on the representation of the rotation (turn) tensor, described above, plays an essential role. Finally, all characteristics to be found can be expressed via one function, determined by a rapidly convergent series of a quite simple form. For this reason, no problem appears in simulations. The propriety of the determination of axes, about which the body rotates, manifests in the fact that the velocities of precession and proper rotation have a constant sign. Remind that in previously known solutions only the sign of the precession velocity is constant, i.e. in these solutions only one axe of turns is correctly guessed. It follows from the solution [2], that formally stable solutions, however, may be unstable in practice, if a certain parameter is small enough (zero value of the parameter corresponds to the boundary between subdomains). In this case the body may jump from one stable rotational regime to another one under action of arbitrarily small and short loads (a percussion with a small meteorite).

A new solution [3, 4] for the classical problem of the rotation of a rigid body with transversally isotropic inertia tensor is obtained (1996, 2003) in a homogeneous gravity field (case of Lagrange). The solution of this problem from the formal mathematical point of view is known very long ago, and one can find it in many monographs and text-books. However, it is difficult to make a clear physical interpretation of this solution, and some simple types of motion are described by it in an unjustifiably sophisticated way. In the case of a rapidly rotating gyroscope there was obtained practically an exact solution in elementary functions. It was shown [4] that the expression for the precession velocity, found using the elementary theory of gyroscopes, gives an error in the principal term.

It is found (2003), in the frame of the dynamics of rigid bodies, the explanation of the fact that the velocity of the rotation of the Earth is not constant, and the axe of the Earth is slightly oscillating [5]. Usually this fact is explained by the argument that one cannot consider the Earth as an absolutely rigid body. However, if the direction of the dynamic spin slightly differs from the direction of the earth axe, the earth axe will make precession about the vector of the dynamic spin, and, consequently, the angle between the axe of the Earth and the plane of ecliptics will slightly change. In this case the alternation of day and night on the Earth will be determined not by the proper rotation of the Earth about its axe, but by the precession of the axe.

  1. Zhilin P.A. A New Approach to the Analysis of Euler-Poinsot problem // ZAMM. Z. angew. Math. Mech. 75 (1995) SI, 133-134.
  2. Zhilin P.A. A New Approach to the Analysis of Free Rotations of Rigid Bodies // ZAMM. Z. angew. Math. Mech. 76 (1996), 4. P.187-204.
  3. Zhilin P.A. Rotations of Rigid Body with Small Angles of Nutation // ZAMM Z. angew. Math. Mech. 76, (1996). S. 2. P. 711-712.
  4. Zhilin P.A. Rotation of a rigid body with a fixed point: the Lagrange case // Lecture at XXXI Summer School - Conference “Advanced Problems in Mechanics ”. (In book: Zhilin P.A. Advanced Problems in mechanics. Selection of articles. Vol. 1. St. Petersburg. Edition of IPME RAS. 2006. 306 p. (In Russian))
  5. Zhilin P.A. Theoretical mechanics. Fundamental laws of mechanics. Tutorial book. St. Petersburg State Polytechnical University. 2003. 340 p. (In Russian).