Volume 30, Number 5
Special Issue: On the 175th Anniversary of S.V. Kovalevskaya (Issue Editors: Vladimir Dragović, Andrey Mironov, and Sergei Tabachnikov)

January 15, 2025, marks the 175th birthday of S.V. Kovalevskaya, one of the famous female mathematicians whose scientific achievements are considered to be an integral part of the treasury of world mathematics.

This paper provides an original rendition of the heavy top that unravels the mysteries behind S. Kowalewski's seminal work on the motions of a rigid body around a fixed point under the influence of gravity. The point of departure for understanding Kowalewski's work begins with Kirchhoff's model for the equilibrium configurations of an elastic rod in ${\mathbb R}^3$ subject to fixed bending and twisting moments at its ends [17]. This initial orientation to the elastic problem shows, first, that the Kowalewski type integrals discovered by I.V. Komarov and V.B. Kuznetsov [24, 25] appear naturally on the Lie algebras associated with the orthonormal frame bundles of the sphere $S^3$ and the hyperboloid $H^3$ [17] and, secondly, it shows that these integrals of motion can be naturally extracted from a canonical Poisson system on the dual of $so(4,\mathbb C)$ generated by an affine quadratic Hamiltonian $H$ (Kirchhoff – Kowalewski type).
The paper shows that the passage to complex variables is synonymous with the representation of $so(4,\mathbb C)$ as $ sl(2,\mathbb C)\times sl(2,\mathbb C)$ and the embedding of $H$ into $sp(4,\mathbb C)$, an important intermediate step towards uncovering the origins of Kowalewski's integral. There is a quintessential Kowalewski type integral of motion on $sp(4,\mathbb C)$ that appears as a spectral invariant for the Poisson system associated with a Hamiltonian $\mathcal{H}$ (a natural extension of $H$) that satisfies Kowalewski's conditions.
The text then demonstrates the relevance of this integral of motion for other studies in the existing literature [7, 35]. The text also includes a self-contained treatment of the integration of the Kowalewski type equations based on Kowalewski's ingenuous separation of variables, the hyperelliptic curve and the solutions on its Jacobian variety.

We prove the integrability of magnetic geodesic flows of $SO(n)$-invariant Riemannian metrics on the rank two Stefel variety $V_{n,2}$ with respect to the magnetic field $\eta\, d\alpha$, where $\alpha$ is the standard contact form on $V_{n,2}$ and $\eta$ is a real parameter. Also, we prove the integrability of magnetic sub-Riemannian geodesic flows for $SO(n)$-invariant sub-Riemannian structures on $V_{n,2}$. All statements in the limit $\eta=0$ imply the integrability of the problems without the influence of the magnetic field. We also consider integrable pendulum-type natural mechanical systems with the kinetic energy defined by $SO(n)\times SO(2)$-invariant Riemannian metrics. For $n=3$, using the isomorphism $V_{3,2}\cong SO(3)$, the obtained integrable magnetic models reduce to integrable cases of the motion of a heavy rigid body with a gyrostat around a fixed point: the Zhukovskiy – Volterra gyrostat, the Lagrange top with a gyrostat, and the Kowalevski top with a gyrostat. As a by-product we obtain the Lax presentations for the Lagrange gyrostat and the Kowalevski gyrostat in the fixed reference frame (dual Lax representations).

The Korteweg – de Vries – Burgers equation, the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation and the nonlinear differential equation for description surface waves in a convecting fluid are considered. The Cauchy problems for all these partial differential equations are not solved by the inverse scattering transform. Reductions of these equations to nonlinear ordinary differential equations do not pass the Painlevé test. However, there are local expansions of the general solutions in the Laurent series near movable singular points. We demonstrate that the obtained information from the Painlevé test for reductions of nonlinear evolution dissipative differential equations can be used to construct the nonautonomous first integrals of nonlinear ordinary differential equations. Taking into account the found first integrals, we also obtain analytical solutions of nonlinear evolution dissipative differential equations. Our approach is illustrated to obtain the nonautonomous first integrals for reduction of the Korteweg – de Vries – Burgers equation, the modified Korteweg – de Vries – Burgers equation, the dissipative Gardner equation and the nonlinear differential equation for description surface waves in a convecting fluid. The obtained first integrals are used to construct exact solutions of the above-mentioned nonlinear evolution equations with as many arbitrary constants as possible. It is shown that some exact solutions of the equation for description of nonlinear waves in a convecting liquid are expressed via the Painlevé transcendents.

A treatment is given of the spatial restricted elliptic problem of three bodies interacting under Newtonian gravity. The problem depends on two parameters: the ratio between the masses of the main attracting bodies and the eccentricity of their elliptic orbits. The eccentricity is assumed to be small. Nonlinear equations of motion of the test mass near a triangular libration point are analyzed. It is assumed that the parameters of the problem lie on the curves of third-order resonances corresponding to the planar restricted problem. In addition to these resonances (their number is equal to five), the spatial problem has a resonance that takes place at any parameter values since the the frequency of small linear oscillations of the test mass along the axis perpendicular to the plane of the orbit of the main bodies is equal to the frequency of Keplerian motion of these bodies. In this paper, the normal form of the Hamiltonian function of perturbed motion through fourth-degree terms relative to deviations from the libration point is obtained. Explicit expressions for the coefficients of normal form up to and including the second degree of eccentricity are found.

The motion of a heavy rigid body with suspension point performing high-frequency periodic vibrations of small amplitude is considered. The study is carried out within the framework of an approximate autonomous system written in the form of the modified Euler – Poisson equations, to the right-hand sides of which the components of the vibration moment are added. The question of the existence of two particular motions of the body is resolved, they are permanent rotations and pendulum-type motions. It is shown that permanent rotations of the body can occur in the case of vibration symmetry relative to a vertically located axis. The search for pendulum-type motions is restricted to the case when the axis of these motions is one of the principal inertia axes of the body, as in the case of a heavy rigid body with a fixed point. Two basic variants of vibrations are considered, when the suspension point vibrates along a straight line and along an ellipse. To the latter variant any planar vibrations and a wide class of spatial vibrations of the suspension point are reduced. It is shown that for both basic cases of vibrations, pendulum-type motions are of two types. The motions of the first type are similar to the Mlodzeevsky’s pendulum-type motions of a heavy rigid body with a fixed point. For them, the body’s mass center lies in the principal plane of inertia, and the axis of the pendulum-type motions is perpendicular to this plane. Pendulum-type motions of the second type occur around the principal axis of inertia containing the body’s center of mass. Such motions are absent in the gravitational problem, they are caused by the presence of vibrations. To search for the pendulum-type motions, an approach is proposed that combines the results of the problem of gravitation (without vibration) and that of vibration (ignoring gravity). As an illustration, a number of examples of the interaction of the gravitational field and the vibration field corresponding to both basic variants of vibrations of the body’s suspension point are considered.

For an autonomous dynamical system of $n$ differential equations each integral of motion allows for reduction of the order of equations by 1 and knowledge of $(n − 1)$ integrals is necessary for the system to be integrated by quadratures. The amenability of Hamiltonian systems to being integrated by quadratures is characterised by the Liouville theorem where in $2n$-dimensional phase space only $n$ integrals are sufficient as equations are generated by 1 function — the Hamiltonian.
There are, however, large families of Newton-type differential equations for which knowledge of 2 or 1 integral is sufficient for recovering separability and integration by quadratures. The purpose of this paper is to discuss a tradeoff between the number of integrals and the special structure of autonomous, velocity-independent 2nd order Newton equations $\ddot{\boldsymbol{q}}=\boldsymbol{M}(\boldsymbol{q})$, $\boldsymbol{q}\in\mathbb R^n$, which allows for integration by quadratures.
In particular, we review little-known results on quasipotential and triangular Newton equations to explain how it is possible that 2 or 1 integral is sufficient. The theory of these Newton equations provides new types of separation webs consisting of quadratic (but not orthogonal) surfaces.