  In this paper, we prove that on any contact manifold, there exists an
arbitrary C^{\infty}-small contactomorphism which does not admit a square root.
In particular, there exists an arbitrary C^{\infty}-small contactomorphism
which is not "autonomous". This result is the first step to study the topology
of non-autonomous contactomorphisms. As an application, we also prove a similar
result for the diffeomorphism group for any smooth manifold.
