Several examples of classical superintegrable systems in two-dimensional spac are shown to possess hidden symmetries leading to their linearization. They are those determined 50 years ago in [Phys. Lett. 13, 354 (1965)], and the more recent Tremblay-Turbiner-Winternitz system [J. Phys. A: Math. Theor. 42, 242001 (2009)].
We conjecture that all classical superintegrable systems in two-dimensional space have hidden symmetries that make them linearizable.
We introduce a family of $n$-dimensional Hamiltonian systems which, contain, as special reductions, several superintegrable systems as the Tremblay-Turbiner-Winternitz system, a generalized Kepler potential and the anisotropic harmonic oscillator with Rosochatius terms. We conjecture that there exist special values in the space of parameters, apart from those leading to known cases, for which this new Hamiltonian family is superintegrable.
Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
Variables of separation for classical $GL(3)$ magnetic chain obtained by Sklyanin form a generic positive divisor $D$ of degree $n$ on a genus $g$ non-hyperelliptic algebraic curve. Because $n>g$ this divisor $D$ has unique representative $ρ(D)$ in the Jacobian which can be constructed by using dim$|D|=n-g$ steps of Abel's algorithm. We study properties of the corresponding chain of divisors and prove that the classical $GL(3)$ magnetic chain is a superintegrable system with dim$|D|=2$ superintegrable Hamiltonians.
Superintegrable classical Hamiltonian systems in two-dimensional Euclidean space $E_2$ are explored. The study is restricted to Hamiltonians allowing separation of variables $V(x,y)=V_1(x)+V_2(y)$ in Cartesian coordinates. In particular, the Hamiltonian $\mathcal H$ admits a polynomial integral of order $N>2$. Only doubly exotic potentials are considered. These are potentials where none of their separated parts obey any linear ordinary differential equation. An improved procedure to calculate these higher-order superintegrable systems is described in detail. The two basic building blocks of th…
We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by $$\mathcal H_N = p_1^2 + p_2^2 +\sum_{n=1}^N γ_n(q_1 p_1 + q_2 p_2)^n ,$$ where $q_i$ and $p_i$ are generic canonical variables, $γ_n$ are arbitrary coefficients, and $N\in \mathbb N$. For $N=2$, being both $γ_1,γ_2$ different from zero, this reduces to the classical Zernike system. We prove that $\mathcal H_N$ always provides a superintegrable system (for any value of $γ_n$ and $N$) by obtaining the corresponding constants of the motion explicitly, which turn out to be of higher-order in the momenta…
