Vol 71, No 2 (2016)

A mechanical system whose phase space is the tangent bundle of a two-dimensional sphere is studied. The potential nonconservative systems describing a geodesic flow are classified. A multiparameter family of systems possessing a complete set of transcendental first integrals expressed in terms of finite combinations of elementary functions is found. Some examples illustrating the spatial dynamics of a rigid body interacting with a medium are discussed.

The problem of periodic domain initiation in a thin lyotropic nematic liquid crystal layer is studied. This layer has a planar director initial orientation, but the anchoring energy is minimized by the homeotropic one. The periodic structures whose wave vector is perpendicular to the director exist during the director reorientation process from the planar orientation to the homeotropic one when the reorientation wave front appears. It is shown that the divergent terms of the Prank orientation elasticity energy plays an important role in this effect. The saddle-splay Prank constant and the anisotropic anchoring energy coefficient are estimated.

Several optimal control problems are considered for a human motion model in the case of stabilometric tests when the posture of a person changes under a step disturbance. Such a situation is observed in the case of a galvanic vestibular stimulation or a vibration stimulation of the proprioceptors of shin muscles. Stabilometric tests with visual step disturbance are widespread in Russia. A characteristic feature of center of pressure (COP) trajectories for these tests is the presence of swings and overshoots. The test results are compared with the solutions of optimal control problems devoted to the repositioning of an inverted pendulum by a coupling torque. A speed-of-response problem and an optimal stabilization problem with a quadratic quality criterion are considered. The solutions to these optimum problems are compared with the results of stabilometric tests, which allows one to state a hypothesis on the sources of the above effects.