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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?

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1 Answer 1

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  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.

  2. Elimination of constraints in the Hamiltonian formalism is not in general as straightforward as simply choosing the constraint submanifold where all constraints vanish. In general, Hamiltonian constraints are often related to gauge theories, and the "real" phase space is not merely the constraint submanifold but the constraint submanifold quotiented by the action of the group of gauge transformations. Mathematically, the process of obtaining the reduced phase space as this quotient is called symplectic (or Marsden-Weinstein) reduction. Trying to "get rid" of gauge-generating constraints ahead of time by choosing gauge fixings for each of them (as discussed in this answer of mine) is not in general possible due to Gribov ambiguities.

  3. With this in mind, the following is true:

    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 $T^\ast Q$.

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