The cylindrical coordinates for r (t) can be simplified by introducing the radial and tangential unit vectors, r ^ ^ = cos ⁡ ⁡ ( θ θ ( t ) ) x ^ ^ + sin ⁡ ⁡ ( θ θ ( t ) ) y ^ ^ , θ θ ^ ^ = − − sin ⁡ ⁡ ( θ θ ( t ) ) x ^ ^ + cos ⁡ ⁡ ( θ θ ( t ) ) y ^ ^ . and their time derivatives from elementary calculus: d r ^ ^ d t = ω ω θ θ ^ ^ . d 2 r ^ ^ d t 2 = d ( ω ω θ θ ^ ^ ) d t = α α θ θ ^ ^ − − ω ω r ^ ^ .