The authors study the Camassa-Holm equation
on the real line with and initial condition . This equation admits two distinct classes of solutions, and the dichotomy between the two classes is associated with wave breaking, which takes place in finite time such that the and norms of the solution remain finite while the spatial derivative becomes pointwise unbounded. This equation is reformulated by means of a different set of variables from Eulerian to Lagrangian coordinates to produce a system of semilinear ordinary differential equations. The existence of solutions, short-time stability and global stability are established. The system is shown to be invariant with respect to a relabeling set . Lastly, the two-direction mappings between the Eulerian variable and the Lagrangian variable is defined.