On the way to Swarthmore last week, Steve's coat and tie were hanging above the car door across from me. As we went around curves, I watched the tie swing back and forth, and it made me start thinking about how a pendulum would behave in a turning car. Generally, physical laws only apply to inertial frames (that is, perspectives that are not accelerating), so we can't simply measure things in the car; we have to transform those measurements into ones made from a fixed point.

Starting inside the car, let's call the angle of the pendulum above vertical

*θ*, and the length of the pendulum's string

*r*:

In Lagrangian mechanics, the main things we care about are the particle's potential and kinetic energy. In this case, the potential energy is easy, since there's just gravity to worry about:

where

*m*is the mass of the particle, and

*g*is the acceleration due to gravity. The kinetic energy is a little more complicated, since we need to think about the car's motion. In the car's frame, the pendulum's position is

where

*y*is the vertical direction and

*x*the horizontal. We can transform this to a fixed frame velocity using

where

*ω*is the angular velocity of the car, and the

*f*and

*r*subscripts denote the fixed and rotating frames respectively. Plugging in

*a*, we have

where

*R*is the radial direction in the fixed frame, and

*φ*is the angle between

*x*and

*ω.*The kinetic energy then is

The Lagrangian is given by

so plugging in our values gives

The Euler-Lagrange equations for our variables are

so plugging in

*L*gives

We can solve the second equation for

*θdot*and plug it into the first one. Then solving for

*θdotdot*gives

I don't think there's a good way to solve this exactly, but we can find a numerical solution for specific cases. First, we can look at the simplest situation where

*φ*=

*φ*

*dot*= 0.

Pretty standard pendulum motion, but now let's look at something a little more complicated. Suppose

Then we get motion that looks like this

I'm not sure what exactly the qualifications are for chaotic motion, but this may fit. In any case, I think it's pretty cool looking.