It's a well known fact that classical mechanics isn't a deterministic theory if you only include the positions and masses of various particles as part of the initial conditions. You also need to include the velocities/momenta of the various particles, which are formulated in terms of time-derivatives. This is somewhat surprising because it is natural to think that velocities/momenta aren't really intrinsic to a time, they are just time-derivatives of position.
This trend is also present in quantum physics, where (for example) the Hamiltonian of a physical system needs to be specified for the laws of physics to be deterministic. The Hamiltonian is equivalent to the total energy, where the total energy depends on other "time-derivative properties" like momentum.
Another strange feature of these "extra" quantities that are needed for determinism to work is that they are frame-relative (unlike mass/charge, for example). The velocity (and therefore momentum and kinetic energy) are all frame-relative. Not all time-derivative quantities are frame relative. For example, acceleration is frame-independent in classical physics. But, acceleration doesn’t help in delivering determinism. Surprisingly, the time-derivative quantities that give us determinism are frame-relative ones.
Question: Is it generally true that these kind of "time-derivative" properties that need to be included for determinism to work are always sensitive to an inertial frame? Or, can determinism be formulated in terms of time-derivative properties that not frame-relative? I am interested in this question both in the case of classical mechanics and quantum mechanics.