## Nonintegrable Constraints in Mechanics

Over the summer of 2022, I led a math undergraduate research program at Cornell University on the subject of nonintegrable constraints in mechanics. Their research project was on studying the long-term behavior of mechanical systems subjected to nonholonomic and impact constraints.

Below are some of their resources and a small summary of their work is to the right.

- Primer on symplectic geometry.
- Primer on invariants in dynamical systems.
- Worksheet on geometric mechanics.
- Worksheet on nonholonomic systems.
- Worksheet on hybrid and impact systems.

###### Chaplygin sleigh with impacts

### L.P.O. Reduction

Given a left-invariant Hamiltonian , Lie-Poisson reduction allows for the (continuous) dymanics to be reduced to with the dynamics

If an impact occurs at a set and is a right coset, the impact system can be reduced to with dynamics

where such that and is a section.

### Hybrid transfer operator

Let be a hybrid system with associated flow . The transfer operator is an induced flow on given by

The long-time properties of this operator encode statistical information about the orbits of the system (e.g. ergodic properties).

Left: An animation of the values of for the bouncing ball subject to dissipation where is a Gaussian distribution.

Right: All trajectories in this system are Zeno. As time progresses, this image displays the initial conditions that “disappear.”