




There are two directions: the r direction is the direction of a vector
from the origin to the point in question;
a unit vector in this direction has representation:
u_{r}_{ }= i cos+ j sin
thedirection is normal to this:
u = i sin + j cos
The vector r is represented in this coordinate system by r
= ru_{r} , where r = (x^{2}+y^{2})^{1/2};
since we have
r(t) = x(t) i + y(t) j
we obtain
x = rcos , y = rsin
Taking derivatives we find
verify by differentiating yourself that
which gives:
We can compute the second derivative in polar coordinates by continued accurate use of the product and chain rules and
The second and fourth terms here are sometimes referred to in physics as the centrifugal and Coriolis forces. Thus if an object is subject to no external force, so that
you will find that it obeys
,
and
The former causes the radial velocity to grow if there is angular motion: the latter slows down the angular motion if the object is moving away from the origin.