Find the orthogonal trajectory of the system .
ans: .
In polar coordinates, the quantity represents the slope of the tangent of a curve with respect to the radial direction. (Informally, a point on the curve moves an infinitesimal distance dr in the radial direction and rdθ in the transverse direction.) So the orthogonal system should have "radial slope" at that point.
Write as , and differentiate with respect to θ: , where . Substitute and simplify a bit, to get .
The orthogonal trajectory will therefore have . Integrate this: .
To find the integral on the right-hand side, substitute , and it becomes .
That gives the orthogonal trajectory as const., or const.
Finally, the trig identity can be used to write the answer as const.