From time to time I’ve been playing with the question whether velocity just is rate of change of position over time in a philosophical elaboration of classical mechanics.
Here’s a thought. It seems that how much kinetic energy an object x has at time t (relative to a frame F, if we like) is a feature of the object at time t. But if velocity is rate of change of position over time, and velocity (together with mass) grounds kinetic energy as per E = m|v|2/2, then kinetic energy at t is a feature of how the object is at time and at nearby times.
This argument suggests that we should take velocity as a primitive property of an object, and then take it that by a law of nature velocity causes a rate of change of position: dx/dt = v.
Alternately, though, we might say that momentum and mass ground kinetic energy as per E = |p|2/2m, and momentum is not grounded in velocity. Instead, on classical mechanics, perhaps we have an additional law of nature according to which momentum causes a rate of change of position over time, which rate of change is velocity: v = dx/dt = p/m.
But in any case, it seems we probably shouldn’t both say that momentum is grounded in velocity and that velocity is nothing but rate of change of position over time.
1 comment:
I’ve sometimes wondered about that. In the Hamiltonian formulation of classical mechanics, with Hamiltonian H = (p^2)/2m + V(x), it’s one of the Hamiltonian equations of motion that dx/dt = ∂H/∂p = p/m. The usual exposition goes from Newton’s laws to the Lagrangian to the Hamiltonian. But that doesn’t say what is fundamental.
In basic single-particle QM, it’s the state that’s fundamental. (Or is it?) Most states don’t have well-defined position or momentum. You can use Schrödinger’s equation to derive d/dt (expectation of X) = (expectation of P)/m. (See Ehrenfest’s theorem. Note that here X and P are operators.) A wave packet with suitably concentrated position and momentum distributions is the closest QM gives to a classical particle. So, for such a wave packet, the likely position and momentum are related approximately as in classical mechanics. At least in this world, QM is surely more fundamental than classical mechanics.
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