How do non-holonomic constraints work in Hamiltonian formalism?

Question

In Lagrangian formalism, if $(M, g)$ is our configuration manifold, equipped with a Riemannian metric $g\in Hom(TM\bigotimes TM, \mathbb{R})$, 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 $$\frac{d}{dt}\biggr(\frac{\partial \mathcal{L}}{\partial \dot{x}}\biggr)-\frac{\partial \mathcal{L}}{\partial x} = 0.$$ And by using Lagrangian multiplier, we could generalize this PDE for holonomic physical systems. But apparently, there's no canonical way to use Lagrangian mechanics for non-holonomic physical systems. At this point, how does Hamiltonian mechanics behave?

Answer 1

It depends on whether the first principle for the Lagrangian formulation is

1. a variational principle [i.e. the stationary action principle (SAP)],

2. or not,

cf. e.g. this & this related Phys.SE posts.

1. In the 1st case, one can in principle find a Hamiltonian formulation via a singular Legendre transformation, cf. e.g. my Phys.SE answers here & here.

2. In the 2nd case, it is not entirely clear what the corresponding Hamiltonian formulation is. See also e.g. this related Phys.SE post.