The Lorentz-covariant upgrade of Newton's Second Law sets a particle's mass times the second derivative with respect to its Lorentz-invariant proper time of its observed space-time location equal to the four-force applied to it. Hamiltonian mechanics yields the Lorentz-covariant upgrade of Newton's Second Law when the Hamiltonian is a Lorentz-invariant function of the particle's observed space-time location and conjugate four-momentum, and the time derivatives of these two Lorentz-covariant four-vectors are taken with respect to the particle's Lorentz-invariant proper time. Very simple Lorentz-invariant Hamiltonians that yield the electromagnetic Lorentz Force Law and the gravitational geodesic equation are pointed out, as is the straightforward quantization of Lorentz-covariant Hamiltonian mechanics. The zero-mass limit of the relativistic free-particle Schrödinger equation has the simple second-order wave-equation form which is characteristic of source-free electromagnetism. The relativistic correction to the hydrogen atom without spin is a very weak, short-range complement to the Coulomb potential.