# Holonomic constraints and phase space as a cotangent bundle

My understanding:

• A physical system on a flat configuration space with holonomic constraints can be reformulated as a system on a curved configuration space with fewer dimensions and no constraints.

• By the Whitney embedding theorem, the converse holds: unconstrainted physical systems on curved configuration spaces can be reformulated as systems on flat configuration spaces of higher dimension with holonomic constraints.

• In both cases, the phase space of the physical system is the cotangent bundle of the configuration space.

• However, not all phase spaces are cotangent bundles, e.g., the phase space of a spinning top with fixed spin is the 2-sphere, which is compact and therefore necessarily not a cotangent bundle.

My question: Can a physical systems whose curved phase space is not a cotangent bundle always be reformulated as a system on a flat configuration space with non-holonomic constraints, and vice versa? In other words, if we equate a constrained system with its curved-space reformulation, are the properties "all constraints are holonomic" and "the phase space is a cotangent bundle" equivalent?

1. Calling a phase space that is not $$T^\ast M$$ for some configuration manifold $$M$$ "curved" is intuitive but likely to lead to confusion - a phase space as a symplectic (or Poisson) manifold does not carry a metric and hence has no notion of curvature in the usual sense of (pseudo-)Riemannian manifolds.
Every "nice" symplectic manifold (one that has finite-dimensional cohomologies) $$(Q,\omega)$$ is a reduction of its non-canonical cotangent bundle $$(T^\ast Q,\mathrm{d}\theta + \tau^\ast\omega)$$, which in turn is a Marsden-Weinstein reduction of a canonical $$\mathbb{R}^{2n}$$ for some $$n$$.
This is proven in Gotay's "$$\mathbb{R}^{2n}$$ is a universal symplectic manifold for reduction" (which is linked in this MO answer).
The first reduction is just setting all the cotangent coordinates of $$T^\ast Q$$ to zero, the second is a rather complicated symplectic reduction described in detail in the paper, which includes as one step the finding of holonomic constraints to present a canonical $$T^\ast P$$ as a reduction of $$T^\ast\mathbb{R}^{2n}$$ via the embedding $$P\to\mathbb{R}^{2n}$$ already mentioned in the question. The complicated step is constructing the reduction of $$T^\ast P$$ to $$T^\ast Q$$.