Helmoltz problem for the Riccati equation from an analogous Friedmann equation

TL;DR

This paper presents a novel approach to solving the Riccati differential equation by drawing an analogy with a specific Friedmann universe, leading to insights into its Lagrangian and Hamiltonian formulations.

Contribution

It introduces an innovative analogy between the Riccati equation and a Friedmann universe to derive a Lagrangian and Hamiltonian for the Riccati equation.

Findings

Derived a Lagrangian for the Riccati equation

Established an analogy with a Friedmann universe

Provided a framework for physical systems described by Riccati

Abstract

We report a solution of the inverse Lagrangian problem for the first order Riccati differential equation by means of an analogy with the Friedmann equation of a suitable Friedmann-Lema\^itre-Robertson-Walker universe in general relativity. This analogous universe has fine-tuned parameters and is unphysical, but it suggests a Lagrangian and a Hamiltonian for the Riccati equation and for the many physical systems described by it.