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I'm trying to justify the full power of symplectic mechanics yet I keep finding examples of physical systems which are only trivial examples of symplectic mnaifolds, cotangent bundles. What physical systems have phases spaces with are not cotangent bundles?

Since Lagrangian mechanics takes places explicitly on a tangent bundle, I am expected that all examples of non-trivial phases spaces will fail to exhibit the usual duality between Lagrangian and Hamiltonian functions (via Legendre transformation). Is this correct? If so, it would imply that we cannot always use the Lagrangian formalism.

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My favourite non-cotangent-bundle is the classical theory of spin where the Poisson brackets are $$ \{S_i,S_j\} =\epsilon_{ijk} S_k, $$ and the phase space is the two-sphere $S^2$. With spherical polar coordinates, the symplectic form is $$ \omega= J \sin\theta\, d\theta\wedge d\phi. $$

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All examples of phase spaces which are not cotangent bundles must come from systems with gauge symmetries (in the Lagrangian formalism) resp. systems with constraints (in the Hamiltonian formalism), since then the true phase space is not the cotangent bundle of the configuration space itself, but the surface inside it defined by the constraints, with the action of the gauge group quotiented out (i.e. gauge orbits identified to a single point).

A specific example with a non-bundle-like topology - the torus - is given by Qmechanic here.

You can still switch to the Lagrangian formalism by restoring the gauge symmetry, i.e. reversing the quotient and then doing the Legendre transform.

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