PHYS 419: Classical Mechanics Lecture Notes POLAR COORDINATES

A vector in two dimensions can be written in Cartesian coordinates as

r = xxˆ + yyˆ (1) where xˆ and yˆ are unit vectors in the direction of Cartesian axes and x and y are the components of the vector, see also the figure. It is often convenient to use coordinate systems other than the Cartesian system, in particular we will often use polar coordinates. These coordinates are specified by r = |r| and the angle φ between r and xˆ, see the figure. The relations between the polar and Cartesian coordinates are very simple:

x = r cos φ y = r sin φ and p y r = x2 + y2 φ = arctan . x

The unit vectors of polar coordinate system are denoted by rˆ and φˆ. The former one is defined accordingly as r rˆ = (2) r Since r = r cos φ xˆ + r sin φ yˆ,

rˆ = cos φ xˆ + sin φ yˆ.

The simplest way to define φˆ is to require it to be orthogonal to rˆ, i.e., to have rˆ · φˆ = 0. This gives the condition

cos φ φx + sin φ φy = 0.

1 The simplest solution is φx = − sin φ and φy = cos φ or a solution with signs reversed. This gives φˆ = − sin φ xˆ + cos φ yˆ.

This vector has unit length φˆ · φˆ = sin2 φ + cos2 φ = 1.

The unit vectors are marked on the figure. With our choice of sign, φˆ points in the direc- tion of increasing angle φ. Notice that rˆ and φˆ are drawn from the position of the point considered. Notice also that due to Eq. (2), the expression for r in terms of rˆ and φˆ is

r = rrˆ.

An expression analogous to Eq. (1) is wrong

r 6= rrˆ + φφˆ.

We will need also the derivatives of vector r expressed in polar coordinates. We have dr r˙ = =r ˙rˆ + rrˆ˙ dt and drˆ rˆ˙ = = −φ˙ sin φ xˆ + φ˙ cos φ yˆ = φ˙ (− sin φ xˆ + cos φ yˆ) = φ˙φˆ dt (notice that in contrast to Cartesian coordinate system, derivatives of unit vectors of the polar system are not zero) so that

r˙ =r ˙rˆ + rφ˙φˆ.

Now get the second derivative

˙ r¨ =r ¨rˆ +r ˙rˆ˙ +r ˙φ˙φˆ + rφ¨φˆ + rφ˙φˆ, so that the only new derivative is that of φˆ:

˙ φˆ = φ˙ (− cos φ xˆ − sin φ yˆ) = −φ˙rˆ.

Grouping terms together, we finally get:

r¨ = (¨r − rφ˙2)rˆ + (rφ¨ + 2r ˙φ˙)φˆ.

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