Inverse variational problem for equations in the Riccati chain

TL;DR

This paper develops a new approach using Helmholtz theory to construct standard Lagrangian representations for Riccati and higher-order Riccati equations, revealing limitations of traditional Hamiltonization methods.

Contribution

It introduces a symmetry-based method to solve the inverse variational problem for Riccati equations, providing explicit Lagrangians where previous methods failed.

Findings

Standard Lagrangians for Riccati equations are constructed using Helmholtz theory.

Traditional Hamiltonization methods do not apply to Riccati equations.

Explicit Lagrangians are obtained for first and third-order Riccati equations.

Abstract

The nonstandard Lagrangian representations of Ricatti and Riccati-type equations that exist in the literature cannot be obtained using Helmholtz solution of the inverse problem. In this work we consider Riccati and higher-order Riccati equations and construct their standard Lagrangian representation by using a simple variant of the Helmholtz theory. We make use of the self-adjoint form of the linear equations corresponding to odd-order equations in the Riccati chain to provide a symmetry-based approach for the solution of inverse problem. Explicit results presented for Lagrangians of the first and third-order Riccati equations show that one cannot Hamiltonize the Riccati family of equations by the traditional method used in classical mechanics.