polar forms of planar systems

Author: jhobbs

content/applied-math/dynamical-systems/limit-cycles.md

@@ -4,6 +4,10 @@ title: Limit Cycles
 ---
 ## Polar Coordinate Methods
 
+Given a system $\dot{x} = f(x,y), \dot{y} = g(x,y),$ it's sometimes useful, especially for the analysis of limit cycles, to convert to polar coordinates. This can be done the manual way from the start, which leads to needing to solve a system of equations for $\dot{r}$ and $\dot{\theta},$ or we can use these handy formulas, which are equivalent:
+
+$$ \dot{r} = \frac{x \dot{x} + y \dot{y}}{r}, \quad \dot{\theta} = \frac{x \dot{y} - y \dot{x}}{r^2}. $$
+
 ## Conservative and Hamiltonian Systems
 
 <!-- TODO: Hamiltonian systems, energy conservation, potential functions, energy level sets, closed orbits -->