Hamiltonian vector fields and Poisson brackets

For a smooth real function H on a symplectic manifold, the equation ι_{X_H}ω=dH has a unique smooth solution. In canonical coordinates, contracting a vector (a,b) with dq∧dp gives a dp−b dq. Comparing coefficients with dH therefore gives a=H_p and b=−H_q. A critical point of H gives an equilibrium, but a noncritical level set is typically traversed rather than descended. Hamiltonian motion is not gradient descent: the symplectic pairing rotates the differential into a direction tangent to the energy level.

Cartan’s identity gives L_{X_H}ω=d(ι_{X_H}ω)+ι_{X_H}dω=0. Thus the local Hamiltonian flow preserves ω wherever it exists. It also preserves H because dH(X_H)=ω(X_H,X_H)=0. A time-dependent Hamiltonian still generates symplectic time evolution, but its numerical energy value can change through explicit time dependence. Smoothness alone does not guarantee a complete flow on a noncompact manifold: solutions can escape in finite time, so global time assertions need an additional completeness argument.

Define {f,g}=ω(X_f,X_g), equal in canonical coordinates to the sum of f_qg_p−f_pg_q. Then X_Hf={f,H}. In particular {q_j,p_k}=δ_jk, and an autonomous observable f is conserved precisely when its bracket with H vanishes along the trajectories under consideration. A function with explicit time dependence instead has total derivative f_t+{f,H}. Antisymmetry makes {H,H}=0, recovering energy conservation without another coordinate calculation. The bracket also satisfies the Leibniz rule and Jacobi identity, allowing algebraic manipulation of observables.

If {f,g}=0, their Hamiltonian flows commute wherever both compositions are defined. Under our convention the general vector-field identity is $[X_f,X_g]=-X_{\{f,g\}}$, so the zero-bracket conclusion is unaffected by the minus sign. A nonzero constant bracket also produces commuting vector fields, since a constant has zero Hamiltonian field; translations generated by q and p provide a useful example. Consequently commuting flows alone do not force the bracket itself to vanish on a connected manifold: it can be a nonzero constant.

A vector field X is symplectic when L_Xω=0. Since ω is closed, this says ι_Xω is a closed one-form. It is Hamiltonian only when that one-form is globally exact. On a small contractible chart, closed implies exact, but a global topological obstruction can remain. On the two-torus with angular coordinates q,p and form dq∧dp, translation X=∂q contracts to dp. Its integral around the p-circle is nonzero, so no single-valued global real Hamiltonian has differential dp.

The distinction affects conservation and transformations. Every globally Hamiltonian vector field is symplectic, but a form-preserving isotopy need not be generated by one global Hamiltonian. Nor does preserving the symplectic volume imply preserving the full form in higher dimensions. These are different classes of motion. When deriving a candidate field from data, test the contraction, its closedness and its global periods rather than calling any divergence-free phase-space motion Hamiltonian. The next lesson supplies a canonical global form on a cotangent bundle, where these definitions connect directly to momenta.