  We develop a monadic framework formalising an operational notion of dynamics,
seen as the setting and evolution of initial value problems, in general
physical theories. We identify in the Eilenberg-Moore category the natural
environment for dynamical systems and characterise Cauchy surfaces abstractly
as automorphisms in the Kleisli category. Our main results formally vindicates
the Aristotelian view that time and change are defined by one another. We show
that dynamics which respect the compositional structure of physical systems
always define a canonical notion of time, and give the conditions under which
they can be faithfully seen as actions of time on physical systems. Finally, we
construct state spaces and path spaces, and show that our framework to be
equivalent to the path space approaches to dynamics. The monadic standpoint is
thus as strong as the established paradigms, but the shift from histories to
dynamics helps shed new light on the nature of time in physics. In the appendix
we present some additional structures of wide applicability, introduce prop-
agators and draft applications to quantum theory, classical mechanics and
network theory.
