Inverted pendulum on frictionless cart without applied force

[Number2Pencil]

Homework Statement

As a small verification of a larger problem, I'm trying to determine how the dynamics of an inverted pendulum attached to a frictionless cart would behave.

The cart has no input force applied, so all horizontal movements would be due purely to the centrifugal+normal force of the pendulum, being driven by gravity.

The initial conditions of the system would be that the initial cart speed/position = 0, the initial angular rate of pendulum = 0, and the initial positional angle of the pendulum = 0.01 radians, and just let to fall

Homework Equations

The Attempt at a Solution

After solving through all the kinematic laws, and trying to simulate the system, what I am seeing is that without friction, the pendulum continues to swing perpetually back and forth, always reaching the initial height at a angle symmetrical to the starting angle.

Does this sound feasible?

 

[Sourabh N]

It would help if you provide an image of the system you have in mind. The images from a google search 'inverted pendulum' suggest that while the pendulum is falling, the cart will be accelerated in one direction and while the pendulum is rising, the cart is accelerating in the opposite direction.

If that interpretation is correct, the appearance of the motion of the pendulum will depend on the observer. If you are sitting on the cart, I believe the pendulum will appear to swing in a circle but from the lab frame, it would make a distorted shape (I can't imagine it right away, but it should not be hard to find mathematically)