The Integrable Case of Adler–van Moerbeke. Mechanical interpretation

Theoretical mechanics


Sokolov S. V.

Mechanical Engineering Research Institute of the Russian Academy of Sciences, 4, M. Khariton'evskii per., Moscow, 101990, Russia



The Adler–van Moerbeke integrable case is considered. The case of integrability founded in 1986 by M. Adler and P. van Moerbeke is the most complicated in the rigid body dynamics. It came due to papers A.S. Mishchenko and A. T. Fomenko dedicated to integrability of Euler equations on finite-dimensional Lie groups. As a result a new family of integrable quadratic Hamiltonians with additional integrals of fourth degree on so(4) was appeared. The existence of quartic additional integral is connected with special symmetry so(4) admitting real representation as direct sum of two copies of so(3). Euler equations on Lie algebra so(4) also describe the motion of a rigid body with an ellipsoidal cavity filled by a perfect incompressible vortical fluid around a fixed point in a case of uniform fluid vorticity. These equations as a model of the Earth’s rotation were studied by V. A. Steklov. Modern results for integrable metrics on so(4) and their mechanical interpretation can be found in many reviews.

After Poincare we consider rotations of rigid body with cavity, which contains inviscid incompressible liquid, about fixed axis. Such a case characterizes by additional integral of Lagrange type and corresponds to axially symmetric mass distribution.

We give one of the possible mechanical interpretations for the integrable case under consideration. We explicitly present the most convenient form of additional integral. Put some partial relations between principal values of inertia tensor we reduce our system to Poincare case. Connections with some classical integrable mechanical problems are observed. Conditions of physical realizability for this mechanical model are discussed.


integrable hamiltonian systems, mechanical interpretation, Euler equations


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