This article treats the generalisation to brane dynamics of the covariant canonical variational procedure leading to the construction of a conserved bilinear symplectic current in the manner originally developped by Witten, Zuckerman and others in the context of field theory. After a general presentation, including a review of the relationships between the various (Lagrangian, Eulerian and other) relevant kinds of variation, the procedure is illustrated by application to the particularly simple case of branes of the Dirac-Goto-Nambu type, in which internal fields are absent.