On the relation between the canonical Hamilton-Jacobi equation and the De Donder-Weyl Hamilton-Jacobi formulation in general relativity

TL;DR

This paper explores the connection between the canonical Hamilton-Jacobi equation and the De Donder-Weyl formulation in general relativity, demonstrating how one can derive the former from the latter in specific coordinate systems.

Contribution

It establishes a link between two Hamilton-Jacobi formulations in general relativity, providing a method to derive the canonical equation from the De Donder-Weyl approach.

Findings

Canonical Hamilton-Jacobi equation derived from De Donder-Weyl formulation

Relation demonstrated in the context of scalar fields in curved spacetime

Method applicable in Gaussian coordinates

Abstract

We discuss the relation between the canonical Hamilton-Jacobi theory and the De Donder-Weyl Hamilton-Jacobi theory known in the calculus of variations using the examples of a scalar field in curved space-time and general relativity. By restricting ourselves to the Gaussian coordinates we show how the canonical Hamilton-Jacobi equation of general relativity can be derived from the De Donder-Weyl Hamilton-Jacobi formulation of the Einstein equations.