  We address the problem of determining the least improbable deviations leading
to an unsafe rare event in a weakly perturbed mechanical system with
probabilistic initial conditions. These deviations are obtained as the solution
to a variational problem formulated using rigorous approximation techniques
grounded in the principles of large deviations theory. These types of results
have been extended to accommodate stochastic uncertainty in the initial states,
which is a common assumption in mechanical systems. Furthermore, we demonstrate
the applicability of the method by solving the problem for a rare collision
event between two space objects, i.e. a high-dimensional and non-linear
problem, resulting in the most likely sample paths leading to the realization
of the unsafe rare event. The solution is validated against the necessary
conditions for optimality derived from the maximum principle. Access to these
unsafe sample paths offers relevant information regarding the dangerous
configurations of rare events and can be used to design control strategies to
reduce the probability of realization.
