On the generalization of classical Zernike system

We generalize the results obtained recently (Nonlinearity \underline{36} (2023), 1143) by providing a very simple proof of the superintegrability of the Hamiltonian H=\vec{p}\,^{2}+F(\vec{q}\cdot\vec{p}), $\vec{q}, \vec{p}\in\mathbb{R}^{2}$, for any analytic function $F$. The additional integral of motion is constructed explicitly and shown to reduce to a polynomial in canonical variables for polynomial $F$. The generalization to the case $\vec{q}, \vec{p}\in \mathbb{R}^{n}$ is sketched.Comment: 9 pages, no figures; pasections reorganized, references updated, some misprints correcte

More on superintegrable models on spaces of constant curvature

A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalised Kepler potential. The angular components, on the contrary, are given implicitly by a transcendental, in general, equation. In the present note, devoted to the previously less studied models with the radial potential of the generalised Kepler type, a new two-parameter family of relevant angular potentials is constructed in terms of elementary functions. For an appropriate choice of parameters, the family reduces to an asymmetric spherical Higgs oscillator.Comment: 18 page

The covariance of chiral fermions theory

The quasiclassical theory of massless chiral fermions is considered. The effective action is derived using time-dependent variational principle which provides a clear interpretation of relevant canonical variables. As a result their transformation properties under the action of Lorentz group are derived from first principles

A note on the action-angle variables for the rational Calogero-Moser system

A relationship between the action-angle variables and the canonical transformation relating the rational Calogero-Moser system to the free one is discussed.Comment: 6 pages, LaTeX. Acknowledgments are slightly altere