Hamilton-Jacobi formalism and wave equations for strings

We show that the Hamilton-Jacobi equation for Nambu-Goto strings in three dimensions is given by 1/2${\mathrm{F}}_{\mathrm{\mu }\nu }$ ${\mathrm{F}}^{\mathrm{\mu }\nu }$=-(2πα’ ${)}^{\mathrm{-}2}$, where ${\mathrm{F}}_{\mathrm{\mu }\nu }$(x)=${\mathrm{\partial }}_{\mathrm{\mu }}$ ${\mathrm{A}}_{\nu }$-${\mathrm{\partial }}_{\nu }$ ${\mathrm{A}}_{\mathrm{\mu }}$. We also find string wave equations satisfied by functionals of lines (paths), which reduce to the Hamilton-Jacobi equation in the classical limit. In more dimensions these string wave equations describe a restricted class of Nambu-Goto strings.