The movement of a classical material point is described by
the second law of Newton:
Vector F(r, t) represents a force field. This force field may be calculated by taking into account interactions with other particles, or interactions with electromagnetic waves, or gravitational fields.
The second law of Newton is an idealisation, of course, even if one was to neglect quantum and relativistic effects. There is no reason why only a second time derivative of r should appear in that equation. Indeed if energy is dissipated in the system usually first time derivatives will appear in the equation too. If a material point loses energy due to electromagnetic radiation, third time derivatives will pop up.
But let us assume, for a moment, that our material point
moves in a static force field that can be described
in terms of a gradient of some function V: