  In this paper we apply the method of Lagrangian descriptors as an indicator
to study the chaotic and regular behavior of trajectories in the phase space of
the classical double pendulum system. In order to successfully quantify the
degree of chaos with this tool, we first derive Hamilton's equations of motion
for the problem in non-dimensional form, showing that they can be written
compactly using matrix algebra. Once the dynamical equations are obtained, we
carry out a parametric study in terms of the system's total energy and the
other model parameters (lengths and masses of the pendulums, and gravity), to
determine the extent of the chaotic and regular regions in the phase space. Our
numerical results show that for a given mass ratio, the maximum chaotic
fraction of phase space trajectories is attained when the pendulums have equal
lengths. Moreover, we give a characterization of the growth and decay of chaos
in the system in terms of the model parameters, and explore the hypothesis that
the chaotic fraction follows an exponential law over different energy regimes.
