  A permanent challenge in physics and other disciplines is to solve partial
differential equations, thereby a beneficial investigation is to continue
searching for new procedures to do it. In this Letter, a novel Monte-Carlo
Metropolis framework is presented for solving the equations of motion in
Lagrangian systems. The implementation lies in sampling the paths space with a
probability functional obtained by using the maximum caliber principle. The
methodology was applied to the free particle and the harmonic oscillator
problems, where the numerically-averaged path obtained from the Monte-Carlo
simulation converges to the analytical solution from classical mechanics, in an
analogous way with a canonical system where energy is minimized by sampling the
state space and computing the average state for each system. Thus, we expect
that this procedure can be general enough to solve other differential equations
in physics and to be a useful tool to calculate the time-dependent properties
of dynamical systems in order to understand the non-equilibrium behavior of
statistical mechanical systems.
