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.


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.”