Hamilton-Jacobi equations and Brane associated Lagrangians

TL;DR

This paper explores the connection between Hamilton-Jacobi formalism and covariant brane Lagrangians, revealing a covariant square root structure and discussing implications for holography and universal equations of motion.

Contribution

It establishes a link between Hamilton-Jacobi equations and brane Lagrangians, introducing a covariant square root formulation and properties of universal field equations.

Findings

Hamiltonian depends only on momenta, not fields.

Associated Lagrangians have a covariant square root form.

Zero Hamilton-Jacobi function relates to lower-dimensional holography.

Abstract

This article seeks to relate a recent proposal for the association of a covariant Field Theory with a string or brane Lagrangian to the Hamilton-Jacobi formalism for strings and branes. It turns out that since in this special case, the Hamiltonian depends only upon the momenta of the Jacobi fields and not the fields themselves, it is the same as a Lagrangian, subject to a constancy constraint. We find that the associated Lagrangians for strings or branes have a covariant description in terms of the square root of the same Lagrangian. If the Hamilton-Jacobi function is zero, rather than a constant, then it is in in one dimension lower, reminiscent of the `holographic' idea. In the second part of the paper, we discuss properties of these Lagrangians, which lead to what we have called `Universal Field Equations', characteristic of covariant equations of motion.