# what symplectic manifolds that show up in real applications are not a cotangent bundle?

I'm reading a bit more about symplectic geometry and going through Ralph Abraham's Foundations of Mechanics. One of the points that's emphasized early on is that Hamiltonian mechanics is naturally posed on a symplectic manifold $$(M, \omega)$$, where $$\omega$$ is a non-degenerate 2-form on $$M$$ such that $$d\omega = 0$$. A rich source of symplectic manifolds comes from taking the cotangent bundle $$T^*Q$$ of some underlying configuration manifold $$Q$$. The cotangent bundle can be given a canonical symplectic structure by taking the exterior derivative of the 1-form $$\theta$$ that looks like $$p\cdot dq$$ in local coordinates. Indeed, pretty much every mechanical system that I can think of is the cotangent bundle of some configuration space:

• for the gravitational $$n$$-body problem, the configuration manifold is $$(\mathbb{R}^3)^n - \Delta$$ where $$\Delta$$ is the set of all points where two or more of the coordinates are equal: $$\Delta = \{(x_1, \ldots, x_n) : x_i = x_j\text{ for some }i, j\}$$.
• for a rotating top, the configuration manifold is SO(3)
• for a mechanical system with holonomic constraints, the configuration manifold is the zero-contour of some smooth function $$g$$, i.e. $$Q = \{q \in \mathbb{R}^n: g(q) = 0\}$$.

What are examples in mechanics where the phase space is not a cotangent bundle? Do these more general symplectic manifolds show up in problems with nonholonomic constraints like a rolling ball?

• Sure I'm aware that $S^{2n}$ or $T^{2n}$ can be given symplectic structures, what I want to know is what mechanical systems have these manifold as phase space? Commented Dec 2, 2021 at 18:05