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?
