The movement of a classical material point is described by
the second law of Newton:

where

(4.2) |

and

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*:

(4.3) |

where

(4.4) |