Classical Mechanics (Kibble & Berkshire) · Problem 8.5

The energy principle — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · phase-paths-polar-form

Problem

Phase paths in polar form Show that the system of equations

\dot{x}_1 = F_1(x_1, x_2, t), \qquad \dot{x}_2 = F_2(x_1, x_2, t)

can be written in polar coordinates in the form

\dot{r} = \frac{x_1 F_1 + x_2 F_2}{r}, \qquad \dot{\theta} = \frac{x_1 F_2 - x_2 F_1}{r^2},

where

x_1 = r\cos\theta

and

x_2 = r\sin\theta

. A dynamical system satisfies the equations

\dot{x} = -x + y, \qquad \dot{y} = -x - y.

Convert this system into polar form and find the polar equations of the phase paths. Show that every phase path encircles the origin infinitely many times in the clockwise direction. Show further that every phase path terminates at the origin. Sketch the phase diagram.

Intuitive explanation

The key insight is that polar coordinates $(r, \theta)$ naturally capture the distance from and angle around a fixed point—perfect for understanding spiraling behavior. To convert the Cartesian equations of motion, differentiate the transformation relations $x_1 = r\cos\theta$ and $x_2 = r\sin\theta$ with respect to time, then solve for $\dot{r}$ and $\dot{\theta}$ by eliminating the mixed terms. The numerators encode how the vector field $(F_1, F_2)$ points radially and tangentially relative to the current position.

Intuitive explanation

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