Volume 20, Number 4
Volume 20, Number 4, 2015
Nonholonomic mechanics + regular articles
Kozlov V. V.
The Dynamics of Systems with Servoconstraints. II
Abstract
This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with leftinvariant kinetic energy and a leftinvariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic righthand sides. As the main example, we consider the rotation of a rigid body with a leftinvariant servoconstraint, which implies that the projection of the body’s angular velocity on some bodyfixed direction is zero.

Sakharov A. V.
Rotation of the Body with Movable Internal Masses Around the Center of Mass on a Rough Plane
Abstract
We consider the motion of a system consisting of a rigid body and internal movable masses on a rough surface. The possibility of rotation of the system around its center of mass due to the motion of internal movable masses is investigated. To describe the friction between the body and the reference surface, a local Amontons–Coulomb law is selected. To determine the normal stress distribution in the contact area between the body and the surface, a linear dynamically consistent model is used. As examples we consider two configurations of internal masses: a hard horizontal disk and two material points, which move parallel to the longitudinal axis of the body symmetry in the opposite way. Motions of the system are analyzed for selected configurations.

Ivanov A. P.
On the Control of a Robot Ball Using Two Omniwheels
Abstract
We discuss the dynamics of a balanced body of spherical shape on a rough plane, controlled by the movement of a builtin shell. These two shells are set in relative motion due to rotation of the two symmetrical omniwheels. It is shown that the ball can be moved to any point on the plane along a straight or (in the case of the initial degeneration) polygonal line. Moreover, any prescribed curvilinear trajectory of the ball center can be followed by an appropriate control strategy as far as the diameter connecting both wheels is nonvertical.

Fasso F., Sansonetto N.
Conservation of Energy and Momenta in Nonholonomic Systems with Affine Constraints
Abstract
We characterize the conditions for the conservation of the energy and of the components of the momentum maps of lifted actions, and of their "gaugelike" generalizations, in timeindependent nonholonomic mechanical systems with affine constraints. These conditions involve geometrical and mechanical properties of the system, and are codified in the socalled reactionannihilator distribution.

Tsiganov A. V.
Killing Tensors with Nonvanishing Haantjes Torsion and Integrable Systems
Abstract
The secondorder integrable Killing tensor with simple eigenvalues and vanishing Haantjes torsion is the key ingredient in construction of Liouville integrable systems of Stäckel type. We present two examples of the integrable systems on threedimensional Euclidean space associated with the secondorder Killing tensors possessing nontrivial torsion. Integrals of motion for these integrable systems are the second and fourthorder polynomials in momenta, which are constructed using a special family of the Killing tensors.

Fortunati A., Wiggins S.
A Kolmogorov Theorem for Nearly Integrable Poisson Systems with Asymptotically Decaying Timedependent Perturbation
Abstract
The aim of this paper is to prove the Kolmogorov theorem of persistence of Diophantine flows for nearly integrable Poisson systems associated to a real analytic Hamiltonian with aperiodic time dependence, provided that the perturbation is asymptotically vanishing. The paper is an extension of an analogous result by the same authors for canonical Hamiltonian systems; the flexibility of the Lie series method developed by A. Giorgilli et al. is profitably used in the present generalization.

Kudryashov N. A., Sinelshchikov D. I.
On the Connection of the Quadratic Lienard Equation with an Equation for the Elliptic Functions
Abstract
The quadratic Lienard equation is widely used in many applications. A connection between this equation and a linear secondorder differential equation has been discussed. Here we show that the whole family of quadratic Lienard equations can be transformed into an equation for the elliptic functions. We demonstrate that this connection can be useful for finding explicit forms of general solutions of the quadratic Lienard equation. We provide several examples of application of our approach.
