Newton's Second Law

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

$$m \frac{\textrm{d}^2\boldsymbol{r}(t)} {\textrm{d} t^2} = \boldsymbol{F}(\boldsymbol{r}, t)$$ (4.1)

where r is a vector indicating a position of the material point in space:

$$\boldsymbol{r} = \left( \begin{array}{c} x \\ y \\ z \end{array} \right)$$ (4.2)

and t is time. The representation of r in terms of x, y, and z assumes a Cartesian system of coordinates. In general, i.e., in non-cartesian systems of coordinates, equation (4.1) applies still, but the derivative d²r(t)/ d t² may have to be specially evaluated so that the changes in the directions of the base vectors, as particle moves from A to B are taken into account. This adds the so called connection symbols to the equations.

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:

$$\boldsymbol{F}(\boldsymbol{r}, t) = - q \boldsymbol{\nabla} V(\boldsymbol{r}),$$ (4.3)

where q is a coupling constant that determines how the material point couples to the force field. Then our Newton equation becomes:

$$m \frac{\textrm{d}^2\boldsymbol{r}(t)} {\textrm{d} t^2} = -q \boldsymbol{\nabla} V(\boldsymbol{r})$$ (4.4)