Knowlpedia

Definition

Poisson bracket on a symplectic manifold

The Lie bracket on smooth functions induced by the inverse of a symplectic form.

Definition

Let (M,ω)(M,\omega) be a . For f,g∈C∞(M,R)f,g\in C^\infty(M,\mathbb R), let Xf,XgX_f,X_g be their under the convention ιXfω=df\iota_{X_f}\omega=df. The Poisson bracket is

{f,g}=ω(Xf,Xg)=Xg(f)=−Xf(g).\{f,g\}=\omega(X_f,X_g)=X_g(f)=-X_f(g).

This formula produces another smooth real-valued function. It fixes all signs: in Darboux coordinates with ω=∑idqi∧dpi\omega=\sum_i dq_i\wedge dp_i,

{f,g}=∑i(∂f∂qi∂g∂pi−∂f∂pi∂g∂qi),\{f,g\}=\sum_i\left( \frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i} -\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i} \right),

so {qi,pj}=δij\{q_i,p_j\}=\delta_{ij}.

Poisson-algebra laws

The bracket is real-bilinear and antisymmetric, satisfies the Jacobi identity, and is a derivation in each argument:

{f,gh}={f,g}h+g{f,h}.\{f,gh\}=\{f,g\}h+g\{f,h\}.

Therefore C∞(M)C^\infty(M), with pointwise multiplication and this bracket, is a Poisson algebra. These facts are stated in Cannas da Silva, Lecture 18.3, Definition 18.5 and Theorem 18.6.

Vector fields and dynamics

With the plus-sign Hamiltonian convention, the correspondence from functions to is a anti-homomorphism:

X{f,g}=−[Xf,Xg].X_{\{f,g\}}=-[X_f,X_g].

If HH generates a Hamiltonian trajectory γ\gamma, then

ddtf(γ(t))={f,H}.\frac{d}{dt}f(\gamma(t))=\{f,H\}.

Thus {f,H}=0\{f,H\}=0 precisely when ff is constant along the ; this is Cannas da Silva, Theorem 18.9.

Conventions
References
  1. Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2008. DOI record. Relevant: Lecture 18.3, Definition 18.5, Theorem 18.6, and Lecture 18.4, Theorem 18.9.
  2. Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., AMS Chelsea, 2008. DOI record. Relevant: §3.3.