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Classical Mechanics Lecture Notes POLAR COORDINATES

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Classical Mechanics Lecture Notes POLAR COORDINATES PHYS 419: Classical Mechanics Lecture Notes POLAR COORDINATES A vector in two dimensions can be written in Cartesian coordinates as r = xx^ + yy^ (1) where x^ and y^ are unit vectors in the direction of Cartesian axes and x and y are the components of the vector, see also the ¯gure. It is often convenient to use coordinate systems other than the Cartesian system, in particular we will often use polar coordinates. These coordinates are speci¯ed by r = jrj and the angle Á between r and x^, see the ¯gure. The relations between the polar and Cartesian coordinates are very simple: x = r cos Á y = r sin Á and p y r = x2 + y2 Á = arctan : x The unit vectors of polar coordinate system are denoted by r^ and Á^. The former one is de¯ned accordingly as r r^ = (2) r Since r = r cos Á x^ + r sin Á y^; r^ = cos Á x^ + sin Á y^: The simplest way to de¯ne Á^ is to require it to be orthogonal to r^, i.e., to have r^ ¢ Á^ = 0. This gives the condition cos ÁÁx + sin ÁÁy = 0: 1 The simplest solution is Áx = ¡ sin Á and Áy = cos Á or a solution with signs reversed. This gives Á^ = ¡ sin Á x^ + cos Á y^: This vector has unit length Á^ ¢ Á^ = sin2 Á + cos2 Á = 1: The unit vectors are marked on the ¯gure. With our choice of sign, Á^ points in the direc- tion of increasing angle Á. Notice that r^ and Á^ are drawn from the position of the point considered. Notice also that due to Eq. (2), the expression for r in terms of r^ and Á^ is r = rr^: An expression analogous to Eq. (1) is wrong r 6= rr^ + ÁÁ^: We will need also the derivatives of vector r expressed in polar coordinates. We have dr r_ = =r _r^ + rr^_ dt and dr^ r^_ = = ¡Á_ sin Á x^ + Á_ cos Á y^ = Á_ (¡ sin Á x^ + cos Á y^) = Á_Á^ dt (notice that in contrast to Cartesian coordinate system, derivatives of unit vectors of the polar system are not zero) so that r_ =r _r^ + rÁ_Á^: Now get the second derivative _ rÄ =r Är^ +r _r^_ +r _Á_Á^ + rÁÄÁ^ + rÁ_Á^; so that the only new derivative is that of Á^: _ Á^ = Á_ (¡ cos Á x^ ¡ sin Á y^) = ¡Á_r^: Grouping terms together, we ¯nally get: rÄ = (Är ¡ rÁ_2)r^ + (rÁÄ + 2r _Á_)Á^: 2.
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