What we have in physics is experimental data and models attempting to describe the correlations found in the data. Then there is the more or less subjective and cultural meaning given to the models, often implicitly so.
For example, suppose a law of physics showing that Earth orbits around the sun. Let's say such a law, based on the observation of the sky which shows consistent patterns (correlations), based also on earlier models having identified the sun with a specific object located in space (as opposed to, for example, a round patch of light glued to a celestial sphere), describes sun and Earth as being attracted in proportion to the product of their masses and in inverse proportion to the square of their distance. By manipulating this relation of attraction, one can show that Earth perfoms an elliptical orbit around the sun.
Where is the causality here?
Is it that the sun, because it is much more massive than Earth, is the "attractive cause" of Earth motion? That's how some people would make sense of the law, but note that this interpretation is no physics per se, but arises from the way people relate to the physical law by giving it a meaning. And in that case, a physicist could easily correct them by saying that Earth also attracts the sun.
Is causality part of the whole Hamiltonian picture: setting up the initial conditions of sun and Earth at specific points and with specific velocities, the law itself (translated in its potential form) is the cause of the evolving orbital system? Not quite, because, as other answers and comments pointed out, determinism is not causality: if the equations are time-reversible, the final state of the system is as much the cause of the initial state than the other way around, at least physics-wise. For an ordinary person such as myself, since time seems to flow from now to the future, it is the initial state that is the cause for future ones. But again, this is just me, not physics, speaking, trying to make sense of the equations.
Causality implies a narration: this made that happen.
In physics, a narration always comes from an exterior conceptual framework: "I dropped the egg which broke" is a narration consistent with physical laws, but if we wanted to study what it is this sentence talks about, we would have to undertake the description of the egg, the ground, the gravitation field, using statistical quantities and finally invoking the second law of thermodynamics to explain that "because the egg hit the ground it was irreversibly smashed", and during this process some operations of simplification and coarse-graining have to be made, by which narration is injected into the (ideal) bare and raw physical description of the system, which is timeless and eventless, only relational.