Tensor Lagrangians, Lagrangians equivalent to the Hamilton-Jacobi equation and relativistic dynamics

TL;DR

This paper explores tensor Lagrangians, including those equivalent to the Hamilton-Jacobi equation, and their applications to relativistic dynamics and various fundamental fields in physics.

Contribution

It introduces and analyzes tensor Lagrangians that extend beyond standard scalar forms, linking them to Hamilton-Jacobi and relativistic equations.

Findings

Tensor Lagrangians generate massless fields and Dirac fields.

Vector and pseudovector Lagrangians describe electromagnetic fields.

Lagrangians equivalent to Hamilton-Jacobi equations are recast into relativistic forms.

Abstract

We deal with Lagrangians which are not the standard scalar ones. We present a short review of tensor Lagrangians, which generate massless free fields and the Dirac field, as well as vector and pseudovector Lagrangians for the electric and magnetic fields of Maxwell's equations with sources. We introduce and analyse Lagrangians which are equivalent to the Hamilton-Jacobi equation and recast them to relativistic equations.