Recent satellite observations and dynamical studies have demonstrated the existence of filamentary structures in chemical tracer fields in the stratosphere. It is also evident that such features are often below the spatial resolution of the highest-resolution Eulerian models that have been used up to the present time. These observations have motivated the development of a novel Chemical Lagranigan Model of the Stratosphere (CLaMS) that is based on a Lagrangian transport of tracers. The description of CLaMS is divided into two parts: Part 1 (this paper) concentrates on the Lagrangian dynamics, i.e., on the calculation of trajectories and on a completely new mixing algorithm based on a dynamically adaptive grid, while part 2 describes the chemical integration and initialization procedure. The mixing of different air masses in CLaMS is driven by the large-scale horizontal flow deformation and takes into account the mass exchange between the nearest neighbors determined by Delaunay triangulation. Here we formulate an isentropic, i.e., two-dimensional version of the model and verify the mixing algorithm using tracer distributions measured during the space shuttle CRISTA-1 experiment where highly resolved stratospheric structures were observed in early November 1994. A comparison of the measured Southern Hemispheric N2O distribution with CLaMS results allows the intensity of simulated mixing to be optimized. The long-term robustness of the transport scheme is investigated in a case study of the 1996--1997 Northern Hemisphere polar vortex. This study further provides a dynamical framework for investigations of chemical arctic ozone destruction discussed in part 2.