# Fine distinctions: F vs ma, E vs H, etc

## Main Question or Discussion Point

I would maintain the following and I am curious if others disagree.

When we say "F = ma" this is not to say that the force F is identified with ma. If we define the force function (and how we define what we mean by "force" may be a point of contention) as

$$F=-\nabla V$$

then F is some function of the position and velocity and, maybe, the positions and velocities of other particles. The acceleration, on the other hand, is the second time derivative of the position function. These two quantities are not the same thing. When we say "F = ma" what we mean is that of all possible functions x(t), the physically realizable ones are only those that satisfy

$$\frac{d^2 x}{dt^2}=\frac{F}{m}=-\frac{1}{m}\nabla V$$

Similarly I would maintain that the Hamiltonian is not the same thing as the energy of a system. Rather the statement H = E is a constraint on the allowed states of the system.

This distinction is more clearly seen in quantum theory. When we write

$$H|\psi(t)\rangle = i d/dt |\psi(t)\rangle$$

this is not to say that H = i d/dt. H and i d/dt are two distinct operators. Rather, we are saying that only those time-dependent states $$|\psi(t)\rangle$$ which satisfy the above condition describe our system. Any other function mapping the real numbers (time) to the Hilbert space of states is not physically allowed.

That is, F = ma, H = E, H = i d/dt are meaningful equalities, not tautologies.

I think this is actually an important distinction and failure to see it sometimes results in confusion. But does anyone disagree?