  For the Glimm scheme approximation $u_\epsilon$ to the solution of the system
of conservation laws in one space dimension \begin{equation*} u_t + f(u)_x = 0,
\qquad u(0,x) = u_0(x) \in \mathbb R^n, \end{equation*} with initial data $u_0$
with small total variation, we prove a quadratic (w.r.t. $\mathrm{TV}(u_0)$)
interaction estimate, which has been used in the literature for stability and
convergence results. No assumptions on the structure of the flux $f$ are made
(apart smoothness), and this estimate is the natural extension of the Glimm
type interaction estimate for genuinely nonlinear systems.
  More precisely we obtain the following results: \newline - a new analysis of
the interaction estimates of simple waves; \newline - a Lagrangian
representation of the derivative of the solution, i.e. a map $\mathtt x(t,w)$
which follows the trajectory of each wave $w$ from its creation to its
cancellation; \newline - the introduction of the characteristic interval and
partition for couples of waves, representing the common history of the two
waves; \item a new functional $\mathfrak Q$ controlling the variation in speed
of the waves w.r.t. time. \newline This last functional is the natural
extension of the Glimm functional for genuinely nonlinear systems.
  The main result is that the distribution $D_{tt} \mathtt x(t,w)$ is a measure
with total mass $\leq \mathrm{const} \mathrm{TV}(u_0)^2$.
