  In this study, we investigate the transition path of a free active Brownian
particle (ABP) on a two-dimensional plane between two given states. The
extremum conditions for the most probable path connecting the two states are
derived using the Onsager--Machlup integral and its variational principle. We
provide explicit solutions to these extremum conditions and demonstrate their
nonuniqueness through an analogy with the pendulum equation indicating possible
multiple paths. The pendulum analogy is also employed to characterize the shape
of the globally most probable path obtained by explicitly calculating the path
probability for multiple solutions. We comprehensively examine a translation
process of an ABP to the front as a prototypical example. Interestingly, the
numerical and theoretical analyses reveal that the shape of the most probable
path changes from an I to a U shape and to the $\ell$ shape with an increase in
the transition process time. The Langevin simulation also confirms this shape
transition. We also discuss further method applications for evaluating a
transition path in rare events in active matter.
