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For $H=\frac{1}{2}\sum p_j^2/h_j^2-\psi$, $l=p_{\varphi }$ and a Stäckel potential, multiplying the separated Hamilton-Jacobi equation by $\lambda -\mu$ gives $2(\tau +\alpha )(\tau +\beta )p_{\tau }^2-(\alpha -\beta )l^2/[2(\tau +\alpha )]-G(\tau )-E\tau =-I$ for both $\tau =\lambda ,\mu$. Eliminating $E$ gives $I=\frac{1}{2}[\mu v_{\lambda }^2+\lambda v_{\mu }^2+(\lambda +\mu +\alpha )v_{\varphi }^2]-[\mu G(\lambda )-\lambda G(\mu )]/(\lambda -\mu )$. Physical components are $v_j=h_j{\dot{q}}_j$ and $p_j=h_j{v}_j$. Separation proves conservation locally; direct Poisson brackets extend it across regular coordinate patches.

A canonical pair consists of a coordinate and its canonical momentum, with Poisson bracket $\{q,p\}=1$.

Canonical quantization promotes canonical variables to operators and their Poisson brackets to canonical commutation relations. Composite observables require an ordering prescription, and quantum anomalies can modify their algebra.

Phase space coordinates are canonical when their Poisson brackets have the form $\{q^I,p_J\}={\delta }_J^I$, with the position-position and momentum-momentum brackets zero.

For a closed string, the currents $J_{\pm }=P\pm T{X}^{\prime }$ have vanishing mixed Poisson brackets. The densities ${\mathcal{H}}_{\pm }=J_{\pm }^2/(4T)$ therefore generate commuting constraint algebras. With opposite Fourier orientations their modes satisfy $\{L_m,L_n\}=-i(m-n)L_{m+n}$ and the identical relation for ${\tilde{L}}_n$. This is a direct sum of Lie algebras consisting of two Witt algebras.

For a finite-dimensional Hamiltonian system, integrability is normally expressed by enough independent first integrals in mutual Poisson bracket involution. In classical field theories, an infinite hierarchy of compatible conserved quantities is the corresponding structure. For Sine-Gordon theory, Bäcklund transformations generate local conservation laws and explicit soliton solutions. Conservation of arbitrary quantities alone should not be confused with proof of their Hamiltonian involution.

With the oscillator symplectic form of an open string, the quadratic generators $L_n={\sum }_k{\alpha }_k\cdot {\alpha }_{n-k}/2$ obey $\{{\alpha }_k,L_n\}=-ik{\alpha }_{k+n}$. Their Poisson brackets close without a central term, so they are first-class constraints. Quantization requires normal ordering and can produce the Virasoro central extension.

A one-dimensional particle with $H=p^2/2+g/x^2$ admits conserved time-translation, dilation and special-conformal charges. Their Poisson brackets form the sl2R Lie algebra. The homogeneity of the inverse-square potential is what allows the symmetry.

The Poisson brackets of first-class constraints close on the constraints. When the constraints are independent and the coefficients are constant, the Jacobi identity for the Poisson bracket makes the coefficients structure constants of a Lie algebra.

The direct product of integrable Hamiltonian systems is integrable. Pull each family's commuting integrals of motion back to the product phase space. Integrals from different factors have zero Poisson bracket because they use disjoint canonical coordinates, and their differentials are independent wherever the differentials within each factor are independent.

A first-class constraint has Poisson brackets with all constraints that vanish on the constraint surface. For a regular constrained Hamiltonian system, first-class constraints generate gauge transformations; each independent constraint and its gauge direction remove one canonical pair.

First integrals are in involution when every pair has zero Poisson bracket. An $n$-degree-of-freedom Hamiltonian system is Liouville integrable when it has $n$ almost-everywhere independent first integrals in involution.

Along a trajectory generated by the Hamiltonian $H$,Thus a time-independent observable is conserved exactly when its Poisson bracket with $H$ vanishes.

On a symplectic manifold, use ${\iota }_{X_f}\omega =-df$ and $\{f,g\}=\omega (X_f,X_g)=X_f(g)$ for the Poisson bracket. Cartan's magic formula gives ${\mathcal{L}}_{X_f}\omega =0$. Taking the interior product of a differential form with $[X_f,X_g]$ then yields $-d\{f,g\}$, proving the homomorphism. Its kernel consists of locally constant functions, since $X_f=0$ precisely when $df=0$. On a connected manifold these are the constants. The quotient of smooth functions by this kernel is therefore the Lie algebra of Hamiltonian vector fields. A different contraction sign requires a compatible bracket sign.

Liouville integrability means having $n$ almost-everywhere independent first integrals in mutual Poisson involution.

An integral of motion is a function on phase space that remains constant along every trajectory. If it has no explicit time dependence, $I(q,p)$ is an integral exactly when its Poisson bracket with the Hamiltonian vanishes: $\{I,H\}=0$.

The dual space ${\mathfrak{g}}^{\ast }$ of a finite-dimensional Lie algebra has this canonical Poisson bracket. For a basis $v^i$, linear coordinates $x^i(\ell )=\ell (v^i)$ give $\{x^i,x^j\}=c_k^{ij}{x}^k$. The Lie algebra's Jacobi identity is precisely the coordinate Jacobi condition for a Poisson bivector. This construction is intrinsic on the dual space; writing the same formula in an arbitrary manifold chart does not by itself provide a compatible global atlas.

The Fourier kinetic term ${\sum }_{k>0}i{\alpha }_{-k}\cdot {\dot{\alpha }}_k/k$ supplies conjugate pairs, with ${\alpha }_{-k}={\alpha }_k^{\ast }$. Inverting its symplectic form yields the displayed Poisson brackets. Nonzero modes are independent of the center-of-mass pair, while ${\alpha }_0=\sqrt{2{\alpha }^{\prime }}p$ is the momentum zero mode. The imaginary factors are compatible with a real action up to a total derivative.