Starting with a different action and following a different procedure than the construction of strings with dynamical tensions described by Guendelman [1], a variational procedure of our action leads to a coupled nonlinear system of D + 4 partial differential equations for the D string coordinates X and the quartet of scalar fields including the dilaton and the tension T field. Trivial solutions to this system of complicated equations lead to a constant tension and to the standard string equations of motion. One of the most relevant features of our findings is that the Weyl invariance of the traditional Polyakov string is traded for the invariance under area-preserving diffeomorphisms. The final section is devoted to the physics of maximal proper forces (acceleration), minimal length within the context of Born's Reciprocal Relativity theory [6] and to the Rindler world sheet description of accelerated open and closed strings from a very different approach and perspective than the one undertaken by [7].