Lefschetz (1977) discusses the following system of equations $$ \begin{aligned} &\frac{d x}{d t}=-y+x f\left(x^{2}+y^{2}\right) \\ &\frac{d y}{d t}=x+y f\left(x^{2}+y^{2}\right) \end{aligned} $$ Show that this system is equivalent to the polar equation $$ \frac{d r}{d \theta}=r f\left(r^{2}\right) \text {. } $$

Polar coordinates provide an alternative way of representing points in the plane compared to Cartesian coordinates (x, y). Instead of using horizontal and vertical distances, we use:
- Radius (r): the distance from the origin to the point.
- Angle \( \theta \): the angle between the radius and the positive x-axis.

The conversion from Cartesian to polar coordinates involves:
\( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).

This transformation can be utilized in various mathematical problems, especially in differential equations that exhibit radial symmetry.

Differential equations involve derivatives of a function and can describe a multitude of physical and natural phenomena, such as motion, growth, and decay.

• An ordinary differential equation (ODE) involves functions of a single variable and their derivatives.
• Partial differential equations (PDEs) involve multiple variables and their partial derivatives.

In the provided exercise, we deal with an ODE system:\[\frac{d x}{d t}=-y+x f\left(x^{2}+y^{2}\right) \]
\[ \frac{d y}{d t}=x+y f\left(x^{2}+y^{2}\right) \]
These equations can be transformed into a more manageable form using polar coordinates, helping reveal properties of the system that are not easily noticeable in the Cartesian form.

Coordinate transformations change the perspective or basis of a mathematical problem. Here, we switch from Cartesian \( (x, y) \) to polar coordinates \( (r, \theta) \).

To transform:

• Use \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).
• Calculate derivatives in polar form:
• \( \frac{dx}{dt} = \frac{dr}{dt} \cos(\theta) - r \sin(\theta) \frac{d\theta}{dt} \)
• \( \frac{dy}{dt} = \frac{dr}{dt} \sin(\theta) + r \cos(\theta) \frac{d\theta}{dt} \)

Substituting these into the original differential equations allows us to reframe the problem. This way, we obtain simpler, often more insightful forms of the equations that can be easier to solve or analyze.

The Lefschetz system, named after the mathematician Solomon Lefschetz, involves a set of differential equations that describe a particular dynamical system. In this exercise, the system is:

\[\frac{d x}{d t}=-y+x f\left(x^{2}+y^{2}\right) \]
\[ \frac{d y}{d t}=x+y f\left(x^{2}+y^{2}\right) \]
The goal is to show equivalence to a polar coordinate system equation:
\[ \frac{d r}{d \theta}=r f\left(r^{2}\right)\]

By transforming to polar coordinates, we discover that the behavior and evolution of the system can be expressed as a function of the radius and angle, simplifying the problem. The key steps include finding derivatives \( \frac{dr}{dt} \) and \( \frac{d\theta}{dt} \) in polar terms, highlighting underlying symmetries, and leading to simplified, insightful forms of the equations.