  We present a formalism of the dynamics of internal shocks in relativistic
jets where the source has a time-dependent injection velocity and mass-loss
rate. The variation of the injection velocity produces a two-shock wave
structure, the working surface, that moves along the jet. This new formalism
takes into account the fact that momentum conservation is not valid for
relativistic flows where the relativistic mass lost by radiation must be taken
into account, in contrast to the classic regime. We find analytic solutions for
the working surface velocity and radiated energy for the particular case of a
step function variability of the injection parameters. We model two cases: a
pulse of fast material and a pulse of slow material (with respect to the mean
flow). Applying these models to gamma ray burst light curves, one can determine
the ratio of the Lorentz factors gamma_2 / gamma_1 and the ratio of the
mass-loss rates dot{m_2} / dot{m_1} of the upstream and downstream flows. As an
example, we apply this model to the sources GRB 080413B and GRB 070318 and find
the values of these ratios. Assuming a Lorentz factor gamma_1=100, we further
estimate jet mass-loss rates between dot{m_1} ~ 10^{-5}-1 Msun.yr^{-1}. We also
calculate the fraction of the injected mass lost by radiation. For GRB 070318
this fraction is ~7%. In contrast, for GRB 080413B this fraction is larger than
50%; in this case radiation losses clearly affect the dynamics of the internal
shocks.
