Computing with Hamiltonian operators

TL;DR

This paper introduces a new Reduce package, e, for computational analysis of Hamiltonian operators, enabling verification of properties like skew-adjointness, compatibility, and Lie derivatives, with potential applications in Mathematical Physics.

Contribution

The paper presents e, a novel computational tool for Hamiltonian operators, facilitating complex calculations and expanding potential applications in Mathematical Physics.

Findings

e can verify Hamiltonian properties such as skew-adjointness.

It computes compatibility and Lie derivatives of Hamiltonian operators.

The package supports calculations on (variational) multivectors and supermanifolds.

Abstract

Hamiltonian operators are used in the theory of integrable partial differential equations to prove the existence of infinite sequences of commuting symmetries or integrals. In this paper it is illustrated the new Reduce package \cde for computations on Hamiltonian operators. \cde can compute the Hamiltonian properties of skew-adjointness and vanishing Schouten bracket for a differential operator, as well as the compatibility property of two Hamiltonian operators and the Lie derivative of a Hamiltonian operator with respect to a vector field. It can also make computations on (variational) multivectors, or functions on supermanifolds. This can open the way to applications in other fields of Mathematical Physics.