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Newton's Second Law
The movement of a classical material point is described by the
second law of Newton:
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.
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