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The derivative $D_t={\partial }_t+\{\cdot ,H\}$ is a derivation of the Poisson bracket: $D_t\{f,g\}=\{D_{t}f,g\}+\{f,D_{t}g\}$. The time derivative part obeys the product rule, and the Hamiltonian part obeys it by the Jacobi identity. Therefore explicitly time-dependent conserved quantities also have a conserved Poisson bracket.

For field functionals, a Poisson operator $J$ is an operator whose displayed pairing of variational derivatives defines a Poisson bracket. It must be skew under the integral pairing and satisfy the functional Jacobi identity. A constant skew differential operator, such as $J={\partial }_x$ with periodic or vanishing boundary conditions, gives the simplest examples. The Hamiltonian evolution is $u_t=J(\delta H/\delta u)$.

Prequantization realizes a symplectic manifold using a Hermitian line bundle with connection whose curvature is $i\omega /\hslash$ in the convention ${\iota }_{X_f}\omega =df$. Its existence requires integral symplectic periods. On sections, the Kostant-Souriau prequantum operator represents every classical observable and its Poisson bracket, but the resulting states depend on the full phase space. A polarization in geometric quantization removes this excess of variables.

A canonical transformation preserves the symplectic form, equivalently the canonical Poisson brackets, and therefore the Hamiltonian form of every Hamiltonian system. Preservation of just one selected flow is weaker. For $H=(q_1^2+p_1^2)/2$ on two coordinate pairs, $Q_1=q_1$, $P_1=p_1$, $Q_2=2q_2$, $P_2=p_2$ preserves both that physical Hamiltonian and its equations, but $\{Q_2,P_2\}=2$. Thus it is not canonical. Universally preserving Hamiltonian form instead forces $DFJ(DF{)}^T=J$ by applying the chain rule to arbitrary Hamiltonian gradients.

A quadratic homogeneous polynomial in the fibre variables of the cotangent bundle is a first integral of the geodesic Hamiltonian exactly when its symmetric lowered coefficients form a rank-two Killing tensor. For $v=g^{-1}p$, the Poisson bracket is $\{K,H\}={\nabla }_{(a}{K}_{bc)}{v}^a{v}^b{v}^c$, so the equivalence follows by comparing cubic coefficients at every point.

A collection of Hamiltonian constraints is second class when its Poisson bracket matrix $C_{ab}=\{{\chi }_a,{\chi }_b\}$ is invertible on the constraint surface. These conditions eliminate canonical variables rather than generate a gauge symmetry. Their consistent reduced bracket is the Dirac bracket, $\{F,G{\}}_D=\{F,G\}-\{F,{\chi }_a\}(C^{-1}{)}^{ab}\{{\chi }_b,G\}$. The zero temporal canonical momentum of a massive Proca field and its massive Gauss law form such a pair; imposing an independent temporal canonical commutation relation would contradict the constraint.

An invertible real matrix preserving the standard symplectic form. With $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$, it satisfies $A^{T}JA=J$, equivalently $AJ{A}^T=J$. The first identity describes preservation of the form; the second describes preservation of coordinate Poisson brackets.

For a time-independent Hamiltonian $H$, the explicit time partial derivative of a smooth time-dependent conserved quantity is again a conserved quantity. Here conservation means the identity $D_{t}f=f_t+\{f,H\}=0$ on phase space, with the standard Poisson bracket. Differentiating this identity at fixed canonical coordinates givesBecause $H_t=0$, this is exactly $D_t(f_t)=0$, proving the assertion. Iteration gives the same result for all higher explicit time partial derivatives for which the differentiations are defined. By Hamiltonian energy balance, requiring $H$ itself to be a conserved quantity on all trajectories is equivalent to $H_t=0$.

On the canonical cotangent bundle $T^{\ast }{\mathbb{R}}^n$, $\omega =i\sum d{z}_j\wedge d{\bar{z}}_j$ and $H=z^{†}z$. The unitary group acts by $z\mapsto Uz$, preserving both. For a Hermitian $A$, the generator $-iA$ has Hamiltonian ${\mu }_A=z^{†}Az$. The Poisson bracket satisfies $\{{\mu }_A,{\mu }_B\}={\mu }_{-i[A,B]}$, with no central term. The central generator has moment map $H$; imaginary antisymmetric matrices give ordinary angular momentum, while real symmetric matrices give quadratic observables mixing position and momentum.