Field theory with the Maxima computer algebra system

TL;DR

The paper addresses automating the derivation of Euler–Lagrange field equations using a computer algebra system. It demonstrates this approach with Maxima and its itensor package to manipulate indicial tensors and perform functional differentiation, deriving Maxwell's equations from the Maxwell Lagrangian ${\cal L}_{EM}$ and confirming current conservation. The key contribution is a compact, transparent 15-line workflow that extracts the generally covariant Maxwell equations from the Lagrangian and shows how gauge invariance yields a conserved current. The work highlights itensor's practicality for field-theory calculations and discusses potential extensions to general relativity and other theories, as well as interoperability with ctensor for component-based analyses.

Abstract

The Maxima computer algebra system, the open-source successor to MACSYMA, the first general-purpose computer algebra system that was initially developed at the Massachusetts Institute of Technology in the late 1960s and later distributed by the United States Department of Energy, has some remarkable capabilities, some of which are implemented in the form of add-on, "share" packages that are distributed along with the core Maxima system. One such share package is itensor, for indicial tensor manipulation. One of the more remarkable features of itensor is functional differentiation. Through this, it is possible to use itensor to develop a Lagrangian field theory and derive the corresponding field equations. In the present note, we demonstrate this capability by deriving Maxwell's equations from the Maxwell Lagrangian, and exploring the properties of the system, including current conservation.