  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.
