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1 Poisson Manifolds
Let M be any n-dimensional manifold - the configuration space of some classical system, for example a particle on M. Then the phase space is the cotangent bundle of M :
T ∗M = {q ∈ M, p ∈ T (^) q∗ M }
Let’s see how this is a Poisson manifold:
Definition 1 A Poisson manifold X is a manifold with a bracket operation
{·, ·}: C∞(X) × C∞(X) → C∞(X)
making the commutative algebra
C∞(X) = {f : X → R : f smooth}
into a Poisson algebra.
Example: M = Rn In this case Rn^ has coordinates xi: Rn^ → R, so for each point in q ∈ Rn^ we get a basis of Tq Rn, namely:
∂ ∂x 1
∂xn
(Picture of R^2 with coordinates x 1 , x 2 , and a tangent plane at q with basis.) These are tangent vectors: given f ∈ C∞(Rn), they act on it to give a number:
∂f ∂xi (q) ∈ R
We also get a basis of T (^) q∗ Rn, namely:
dx 1 ,... , dxn.
(Picture of R^2 with coordinates x 1 , x 2 , and a cotangent space at q with basis.) (Recall, given f ∈ C∞(Rn), we get (df )q ∈ T )q∗Rn^ by:
(df )q (v) = v(f )(q), ∀f ∈ C∞(Rn)
We can call this just “df ” if we are feeling lazy.) Note:
(dxi )(
∂xj
∂xj
xi
= δij
so dxi is the “dual basis” to (^) ∂x∂i. Using this standard basis for T (^) q∗ Rn^ we get an isomorphism
T (^) q∗ Rn^ ∼= Rn
dxi 7 → (0,... , 1 ,... , 0)
with 1 in the nth^ slot. So we get an isomorphism
T ∗Rn^ = {q ∈ Rn, p ∈ T (^) q∗ Rn} ∼= {q ∈ Rn, p ∈ Rn} ∼= Rn^ × Rn
This lets is put coordinates on T ∗Rn, namey
qi, pi: T ∗Rn^ → R, i = 1,... , n
This lets us make T ∗Rn^ into a Poisson manifold:
{F, G} =
∑^ n
i=
∂F
∂pi
∂G
∂qi
∂F
∂qi
∂G
∂pi
(using your homework). More generally, suppose M is any n-dimensional manifold. Given any q ∈ M we can find an open set U 3 x and a chart: φ: U → Rn
This gives coordinates xi ◦ φ on Rn, which we just call xi for short. Copying what we did, we get coordinates qi, pi on T ∗U = {q ∈ U, p ∈ T (^) q∗ U }
and if q ∈ U , then T (^) q∗ U = T (^) q∗ M. How do we make T ∗M into a Poisson manifold? Given F, G ∈ C∞(T ∗M ), we define {F, G} on T ∗U ⊆ T ∗M by:
{F, G} =
∑^ n
i=
∂F
∂pi
∂G
∂qi
∂F
∂qi
∂G
∂pi
Now, alas, we need to check that the Poisson brackets are well-defined on all of T ∗M - i.e., inde- pendent of the choice of chart. But, let’s not. It will be easier to define the Poisson brackets in a coordinate-free way later. First we will develop some more geometry and start understanding what Poisson brackets mean.
2 More Differential Geometry
Given manifolds M and N , a function : M → N is called smooth, or a map, if any of these hold:
- Given any charts φ: U → Rm^ with U ⊆ M , ψ: V → Rn^ with V ⊆ N , this composite
Rm^ → U ⊆ M → N ⊇ V → Rn
is smooth where defined. It’s enough to check this for one chart U containing each point q ∈ M and one chart containing each point f (q) ∈ N.
- Given any smooth curve γ: R → M, f ◦ γ: R → N is a smooth curve in N.
- Given any g ∈ C∞(N ), then g ◦ f ∈ C∞(M ).
We can define a vector field on M in two equivalent ways:
