Recent zbMATH articles in MSC 37J39https://www.zbmath.org/atom/cc/37J392022-04-28T18:15:44.554837ZWerkzeugBilliards with changing geometry and their connection with the implementation of the Zhukovsky and Kovalevskaya caseshttps://www.zbmath.org/1482.370552022-04-28T18:15:44.554837Z"Fomenko, A. T."https://www.zbmath.org/authors/?q=ai:fomenko.anatolii-t"Vedyushkina, V. V."https://www.zbmath.org/authors/?q=ai:vedyushkina.viktoriya-viktorovnaSummary: The paper presents a class of billiards with varying geometry, the so-called force or evolutionary billiards, which enable us to realize, in the sense of Liouville equivalence, the well-known cases of Zhukovsky and Kovalevskaya for certain energy zones. On the corresponding 4-dimensional open phase submanifolds, the indicated systems are implemented for an increase in energy on all successively occurring isoenergy 3-surfaces.Integrability of point-vortex dynamics via symplectic reduction: a surveyhttps://www.zbmath.org/1482.370562022-04-28T18:15:44.554837Z"Modin, Klas"https://www.zbmath.org/authors/?q=ai:modin.klas"Viviani, Milo"https://www.zbmath.org/authors/?q=ai:viviani.miloThe authors consider point-vertex dynamics and their integrability. This dynamics is described via idealized non-smooth solutions to the incompressible Euler equations on two-dimensional manifolds. The aim is to provide a unified treatment for proving integrability results for 2-, 3-, or 4-point-vertices. Part of their goal is to show how the symplectic reduction can provide a broader approach for proving integrability results, especially for point-vertex dynamics.
Euler equations on an orientable Riemannian manifold that govern an incompressible inviscid fluid have the form \(\dot{\mathbf{v}} + \nabla _\mathbf{v} \mathbf{v} = - \nabla p\) with div \(\mathbf{v} = 0\), where \(\mathbf{v}\) is a vector field on a manifold \(M\) incorporating the motion of the fluid's particles, \(p\) is the pressure function and \(\nabla _\mathbf{v}\) is the covariant derivative along \(\mathbf{v}\). \textit{H. Helmholtz} [J. Reine Angew. Math. 55, 25--55 (1858; ERAM 055.1448cj)] showed that the two-dimensional Euler equations have special solutions with a finite number of point-vertices. These solutions are not smooth and are characterized by the vorticity \mbox{curl \(\mathbf{v} = \sum_{i=1} ^ {n} \Gamma _i \delta_{\mathbf{r}_i}\)} where non-zero \(\Gamma_i\) is the strength of the vortex \(i\), \(\mathbf{r}_i\) is its position, and \(\delta_{\mathbf{r}_i}\) is a delta function.
The authors describe point-vertex equations and their Hamiltonian structures on the sphere, the plane, the hyperbolic plane, and the flat torus. Each of these cases has a different symmetry group, and the symplectic reduction is treated separately in each case to establish integrability.
The authors also briefly review nonintegrability results.
They conclude with some observations about how their results pertain to long term predictions for the Euler equations. An appendix offers a kind of visual portrait of point-vortex solutions.
Reviewer: William J. Satzer Jr. (St. Paul)New cases of homogeneous integrable systems with dissipation on tangent bundles of two-dimensional manifoldshttps://www.zbmath.org/1482.370572022-04-28T18:15:44.554837Z"Shamolin, M. V."https://www.zbmath.org/authors/?q=ai:shamolin.m-vSummary: The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of two-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.New cases of homogeneous integrable systems with dissipation on tangent bundles of three-dimensional manifoldshttps://www.zbmath.org/1482.370582022-04-28T18:15:44.554837Z"Shamolin, M. V."https://www.zbmath.org/authors/?q=ai:shamolin.m-vSummary: The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of three-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.New cases of homogeneous integrable systems with dissipation on tangent bundles of four-dimensional manifoldshttps://www.zbmath.org/1482.370592022-04-28T18:15:44.554837Z"Shamolin, M. V."https://www.zbmath.org/authors/?q=ai:shamolin.m-vSummary: The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of four-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.Force evolutionary billiards and billiard equivalence of the Euler and Lagrange caseshttps://www.zbmath.org/1482.370602022-04-28T18:15:44.554837Z"Vedyushkina, V. V."https://www.zbmath.org/authors/?q=ai:vedyushkina.viktoriya-viktorovna"Fomenko, A. T."https://www.zbmath.org/authors/?q=ai:fomenko.anatolii-tSummary: A class of force evolutionary billiards is discovered that realizes important integrable Hamiltonian systems on all regular isoenergy 3-surfaces simultaneously, i.e., on the phase 4-space. It is proved that the well-known Euler and Lagrange integrable systems are billiard equivalent, although the degrees of their integrals are different (two and one).Coadjoint orbits of three-step free nilpotent Lie groups and time-optimal control problemhttps://www.zbmath.org/1482.371012022-04-28T18:15:44.554837Z"Podobryaev, A. V."https://www.zbmath.org/authors/?q=ai:podobryaev.aleksei-vSummary: We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg group and the Engel group. We consider a time-optimal problem on three-step free nilpotent Lie groups with a set of admissible velocities in the first level of the Lie algebra. The behavior of normal extremal trajectories with initial covectors lying in two-dimensional coadjoint orbits is studied. Under some broad conditions on the set of admissible velocities (in particular, in the sub-Riemannian case) the corresponding extremal controls are periodic, constant, or asymptotically constant.