Discrete-time systems in quasi-standard form and the $\mathfrak{h}_6$ coalgebra symmetry
Pavel Drozdov, Giorgio Gubbiotti
TL;DR
The paper classifies discrete-time $N$-degree-of-freedom systems in quasi-standard form that admit coalgebra symmetry with respect to the Lie–Poisson algebra $\mathfrak{h}_6$. It proves that such systems collapse to standard form ($\ell_k(\xi)=\xi$) with potentials $V(\boldsymbol{q}) = V(\boldsymbol{\lambda}\cdot \boldsymbol{q}, \boldsymbol{q}^2, \boldsymbol{\lambda}^2)$ and identifies six invariant-bearing potentials: $V_1$, $V_{2,I}$, $V_{2,II}$, plus their singular variants. Through linear and quadratic invariant searches, it derives explicit invariants (e.g., $I_1$, $I_{2,I}$, $I_{2,IIa}, I_{2,IIb}$) and constructs corresponding discrete Lagrangians, establishing a spectrum of dynamical behaviors from maximally superintegrable to quasi-integrable, with several continua limits connecting to known continuous systems. The work advances the discrete coalgebra symmetry program by delivering a complete classification in the $\mathfrak{h}_6$ setting and reveals new discrete quasi-integrable models with potential applications to higher-dimensional integrable dynamics and their continuum limits.
Abstract
In this paper, we characterize all discrete-time systems in quasi-standard form admitting coalgebra symmetry with respect to the Lie--Poisson algebra $\mathfrak{h}_{6}$. The outcome of this study is a family of systems depending on an arbitrary function of three variables, playing the rôle of the potential. Moreover, using a direct search approach, we classify discrete-time systems from this family that admit an additional invariant at most quadratic in the physical variables. We discuss the integrability properties of the obtained cases, their relationship with known systems, and their continuum limits.