Volume 31, Number 2
In Memory of Alexey V. Borisov (on his 60th Birthday): Part III (Issue Editors: Ivan Mamaev and Iskander Taimanov)

This paper is concerned with abnormal geodesics on the Carnot group with growth vector $(2, 3, 5, 8, 14)$. Because of a large number of symmetries, this problem reduces to an analysis of the five-dimensional flow. Using the Kovalevskaya method, integrable cases of the resulting system are identified. For these cases, first integrals and explicit solutions are found. It is shown that in the general case the system admits no additional meromorphic first integrals. The paper concludes by discussing some problems regarding the abnormal geodesics on Lie groups.

Equations of motion for a body moving through an ideal fluid when the flow is irrotational and incompressible are obtained taking account of embedded dipoles on the boundary and the Kutta – Chaplygin condition. We develop the embedded dipole model from the complex potential of a dipole on the boundary of a body, oriented so as to preserve no-penetration through the body, using a conformal mapping approach. The resulting hydrodynamic force and moment on the body depend on the dipoles’ strength and position along the body. Using the flat plate as a model geometry, we examine the evolution of the resulting system under the conditions of fixed and time-varying circulation with and without embedded dipoles. We assume two embedded dipoles symmetrically positioned about the center point of the plate, finding that the presence of the dipoles reduces the fluctuations of the angle of attack of the plate. We explore conserved quantities for the system and perform a linear stability analysis, which leads to a constraint on the dipole strength for stability of a plate moving at zero angle of attack with either circulation equal to zero or the Kutta – Chaplygin condition applied.

In this article, we present a classification of toric $b^m$-symplectic manifolds and outline a program for the classification of semitoric singular symplectic manifolds. As a testing ground, we begin with the case of $b$-symplectic manifolds, where a Delzant-type theorem for toric actions has already been established [13]. Building on this foundation, we extend the framework to the broader setting of $b^m$-symplectic manifolds, thereby establishing the first classification result for $b^m$-toric manifolds. This achievement lays the groundwork for a conjectural classification theory of semitoric systems in dimension four on $b^m$-symplectic manifolds. Furthermore, by employing the reduction theory developed in [21], we propose a conjectural classification scheme for higher-dimensional semitoric manifolds, aiming to generalize the Pelayo – Vũ Ngoc classification program [32, 33] to the singular $b^m$-setting.

We present some new Poisson bivectors that are invariants by the flow of the nonholonomic Suslov problem. Two rank-four invariant Poisson bivectors have globally defined Casimir functions and, therefore, define cubic Poisson brackets of the five-dimensional state space with standard symplectic leaves. For the Suslov gyrostat in the potential field we found rank-two Poisson bivectors having only two globally defined Casimir functions and, therefore, we say about formal Hamiltonian description in these cases.

Nonholonomic systems are mechanical systems with ideal linear velocity constraints that are not derivable from position constraints and with dynamics identified by the Lagrange – d’Alembert principle. This paper surveys discrete-time nonholonomic mechanics with applications to numerical integration of nonholonomic systems. This includes an exposition of key elements of discrete mechanics, discrete Lagrange – d’Alembert principle, and exact nonholonomic integrators on vector spaces. Exact variational integrators were introduced and exposed in the context of Lagrangian mechanics by Marsden and West. These integrators sample the trajectories of mechanical systems and are useful for developing practical mechanical integrators.

We propose and study a model for the mechanical system constituted by a chain of $n\geqslant 1$ identical pendula hanging from a viscoelastic string with fixed extrema. The novelty of our approach is to describe the string as a continuous system, specifically, as a one-dimensional viscoelastic Kelvin – Voigt string. The resulting system is a hybrid nonlinear system of coupled PDEs and ODEs. We linearize the system around the attractive equilibrium with pendula and string pointing downwards. The (infinite-dimensional) linearization decouples into a ``vertical'' and a ``horizontal'' subsystems. The former is a viscoelastic version of the well known Rayleigh loaded string, and its point spectrum is known. We thus consider the latter, which describes, at the linear level, the horizontal oscillations of string and pendula. We obtain closed form expressions for the eigenvalue equations and for the eigenfunctions for any value of $n$. Next, we study the point spectrum with a combination of analytical and numerical techniques, adopting a continuation approach from the limiting cases of massless pendula, which involves the well known spectrum of the Kelvin – Voigt string. Finally, we focus on the identification, particularly when $n=2$ and as a function of the parameters, of the eigenvalues closest to the imaginary axis, whose eigenfunction(s) dominate the asymptotic dynamics of the (horizontal) linearized systems and can explain the appearance of synchronization patterns in the chain of pendula.

The problem of the motion of a dynamically symmetric ball with a shifted center of gravity on a horizontal absolutely rough plane is considered, taking into account the unilateral nature of the contact. It is shown that for some values of the parameters and initial conditions, singularities of two types may appear: ambiguity in determining the subsequent motion (separation of the body from the support or continuation of rolling) or the impossibility of determining it within the framework of the model used (the finite time paradox). For a formal description of these situations, the generalized complementarity problem is used. The Littlewood problem of the rolling of a hoop with a point mass is studied in detail. Some cases of paradoxes when imposing differential constraints of various types are also discussed, including the generalization of the Chaplygin sleigh and the “rubber” ball.