How do non-holonomic constraints work in Hamiltonian formalism?

In Lagrangian formalism, if $(M, g)$ is our configuration manifold, equipped with a Riemannian metric $g\in Hom(TM\otimes TM, \mathbb{R})$ , Lagrangian function $\mathcal{L}: TM\times \mathbb{R}_{+}\rightarrow \mathbb{R}$ is defined as $\mathcal{L}(x(t), \dot{x}(t), t)= \frac{1}{2}mg(\dot{x}(t), \dot{x}(t))-U(x(t), t).$ If our particle flows under the influence of gravity, then $\frac{d}{dt}\biggr(\frac{\partial \mathcal{L}}{\partial \dot{x}}\biggr)-\frac{\partial \mathcal{L}}{\partial x} = 0.$ By using Lagrange multipliers , we could generalize this PDE for holonomic physical systems . Now here's my question: Apparently, there's no canonical way to use Lagrangian mechanics for non-holonomic physical systems . At this point, how does Hamiltonian mechanics behave?