REU 2022 - William Clark

Nonintegrable Constraints in Mechanics

Left to right: Maria Oprea, Kaito Iwasaki, Aden Shaw, Robi Huq, Dora Kassabova, and William Clark

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 $\mathcal{H}:T^*G\to\mathbb{R}$ , Lie-Poisson reduction allows for the (continuous) dymanics to be reduced to $h:\mathfrak{g}^*\to\mathbb{R}$ with the dynamics

$\begin{equation*} \dot{\zeta} = \mathrm{ad}^*_{dh}\zeta \end{equation*}$

If an impact occurs at a set $S\subset G$ and $S = Hg_0$ is a right coset, the impact system can be reduced to $\mathfrak{g}^*\times(H\backslash G)$ with dynamics

$\begin{equation*} \begin{cases} \dot{\zeta} = \mathrm{ad}^*_{dh}\zeta, \quad \dot{q} = d\pi_{\sigma(q)}\left(\ell_{\sigma(q)}\right)_*\mathbb{F}\mathcal{H}(\zeta), & q\not\in \pi(\mathcal{S}) \\ \zeta\mapsto \zeta+\Delta\zeta, \quad q\mapsto q, & q\in \pi(\mathcal{S}), \end{cases} \end{equation*}$

where $\Delta\zeta\in\mathrm{Ann}(\mathfrak{h})$ such that $h(\zeta)=h(\zeta+\Delta\zeta)$ and $\sigma:H\backslash G\to G$ is a section.

Hybrid transfer operator

Let $\mathcal{H} = (M,\mathcal{S},X,\Delta)$ be a hybrid system with associated flow $\varphi_t^\mathcal{H}$ . The transfer operator is an induced flow on $L^1(M,\mu)$ given by

$\begin{equation*} \int_A \, P_t^\mathcal{H}f(x)\mu(dx) = \int_{\varphi_{-t}^\mathcal{H}(A)} \, f(x)\mu(dx), \quad A \; \text{measurable}. \end{equation*}$

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 $P_t^\mathcal{H}f(x)$ for the bouncing ball subject to dissipation where $f$ is a Gaussian distribution.

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