Non-standard Construction of Hamiltonian Structures and of the Hamilton-Jacobi equation

TL;DR

This paper presents alternative methods for constructing Hamiltonian structures and Hamilton-Jacobi equations without Lagrangians, demonstrating their application to the Euler top and analyzing stability criteria's dependence on the chosen structure.

Contribution

It introduces non-standard Hamiltonian constructions and Hamilton-Jacobi equations, providing explicit solutions for the Euler top and examining stability criteria dependence.

Findings

Alternative Hamilton-Jacobi equations for Euler top are explicitly solved.

Stability criteria depend on the Hamiltonian structure used.

Non-standard constructions can be effective without Lagrangians.

Abstract

Examples of non-standard construction of Hamiltonian structures for dynamical systems and the respective Hamilton-Jacobi (H-J) equations, without using Lagrangians, are presented. Alternative H-J equations for Euler top are explicitly exhibited and solved. We demonstrate that some stability criterion, relating the slope of a Casimir function parametrized by the Lagrange multiplier to critical point type, depends on the used Hamiltonian structure and it is inadequate for this reason.