Nontrivial bundle in particle mechanics

Question

We know physical examples of nontrivial bundles in field theory where the base manifold is spacetime. For example in electromagnetism Dirac monopole is an example of a nontrivial vector bundle $A_{\mu}$ over spacetime. On the other hand in particle mechanics the base manifold is just a one dimensional manifold parametrized by time $t$. Let’s call this 1-dimensional manifold as the time manifold in contrast with the spacetime manifold.

Now my question is, do you know examples of physical systems with nontrivial bundles over time manifold, as $\mathbb{R}$, or even compactified as a circle $S^1$?

Answer 1

You can have non trivial line bundles over a circle and this is what happens for a charged particle on a circle with a magnetic flux enclosed. There are also non-trvial bundles in molecular dynamics in which the base space is a circle because we have periodic motion. This is the original context of the Berry phase.

A gauge connection with curvature occurs in the falling cat problem, but I do not know if the tangent bundle over the configuration space of the cat is trivial or not. The paper by Montgomery linked to in the wikipedia article is the best resource.