Lagrangian Formalism for nonlinear second-order Riccati Systems: one-dimensional Integrability and two-dimensional Superintegrability

TL;DR

This paper develops a Lagrangian framework for nonlinear second-order Riccati systems, demonstrating integrability and superintegrability in one- and two-dimensional cases, with explicit integrals and orbit characterizations.

Contribution

It introduces nonnatural Lagrangians for Riccati systems, proves superintegrability in 2D, and provides explicit integrals and orbit descriptions for nonlinear oscillators.

Findings

Lagrangian description for Riccati equations established

Superintegrability proven for 2D systems

Explicit integrals and nonlinear Lissajous orbits derived

Abstract

The existence of a Lagrangian description for the second-order Riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized Riccati equation. The Lagrangians are nonnatural and the forces are not derivable from a potential. The constant value $E$ of a preserved energy function can be used as an appropriate parameter for characterizing the behaviour of the solutions of these two systems. In the second part the existence of two--dimensional versions endowed with superintegrability is proved. The explicit expressions of the additional integrals are obtained in both cases. Finally it is proved that the orbits of the second system, that represents a nonlinear oscillator, can be considered as nonlinear Lissajous figures