Let a particle of mass m, constrained to move on a smooth horizontal circle of radius a, be given an initial velocity V , and let it be resisted by the air with a force proportional to the square of its velocity.

Here we have one degree of freedom. Let us take as our coordinate q₁ the angle 𝜃 which the particle has described about the center of its path in the time t.

For the kinetic energy

and we have

Our differential equation is

which reduces to

or

Separating the variables,

Integrating,

and the problem of the motion is completely solved.