A UNIFIED FORMALISM OF THE GENERALIZED LAGRANGIAN MEAN AND THE TRANSFORMED EULERIAN MEAN THEORIES

The generalized Lagrangian mean (GLM) theory and the transformed Eulerian mean (TEM) theory were reconstructed into a unified formalism. Application of a basic postulate that reciprocal translation is possible between Eulerian and Lagrangian information revealed the existence of a dual structure of governing equations having two standard forms which respectively correspond to the Lagrangian and Eulerian approaches. Either of these forms provides a complete equation set for the continuum concept, and can easily be transformed from one to the other. Therefore, whenever a new set of continuum equations is introduced which has either standard form, a new hypothetical continuum can be defined. As in the case of the original equations, the governing equations of the new continuum can also be easily transformed into various forms (other standard forms and the material form). The definition of new dependent variables is essential, since the hypothetical coninuum contains particular arbitrariness due to the definition. Additionally the new governing equations are formulated without any physical restrictions such as initial conditions or spatial symmetry. The GLM theory is reconstructed by selecting a form for new dependent variables and their related additional terms in the Lagrangian mean continuum. In this reconstruction, a new particle representing the original particle group exists at its Lagrangian mean position, and moves with the Lagrangian mean velocity. Similarly, the TEM theory is reconstructed as one choice from the Eulerian mean continuum, and also describes a new particle, being representative of a newly labeled domain moving with the transformed velocity. As a by-product of these reconstructions, the non-acceleration theorem in both theories was also investigated and subsequently extended.