Symplectic Forms and Poisson Brackets

A symplectic form converts covectors into vectors. In mechanics this means that the differential $dF$ of an observable determines a vector field $X_F$, and in particular $dH$ determines Hamiltonian evolution. Closedness makes the induced bracket satisfy the Jacobi identity; nondegeneracy makes the vector field unique. These are finite-dimensional statements here, providing the clean model against which presymplectic and gauge-reduced field theories should be compared.

Required background. Exterior Algebra and Differential Forms supplies forms, wedge products, and contraction, while the Exterior Derivative and Poincaré Lemma supplies closedness and exterior differentiation. Tensor Fields and Natural Operations supplies smooth tensor fields and the bundle maps induced by a nondegenerate two-form.

Helpful background. Calculus and Linear Algebra supplies antisymmetric matrices and invertibility; Variational Mechanics and Symmetries supplies Hamiltonian mechanics in coordinates.

Symplectic forms and Hamiltonian vector fields

A symplectic manifold is an even-dimensional smooth manifold $P$ equipped with a two-form $\Omega$ satisfying

$$d\Omega=0, \qquad \Omega_p(v,w)=0\ \text{for every }w\in T_pP \Longrightarrow v=0.$$

The second condition is nondegeneracy. It says that $\Omega^\flat:T_pP\to T_p^*P$, $v\mapsto\iota_v\Omega$, is an isomorphism. We adopt

$$\iota_{X_F}\Omega=dF, \qquad \{F,G\}=\Omega(X_F,X_G).$$

These signs must be kept together. Other texts may put a minus sign in the definition of $X_F$ and adjust the bracket accordingly. Cannas da Silva introduces symplectic forms in Cannas da Silva 2006, Lecture 1, §§1.2–1.3, pp. 4–7 and develops Hamiltonian vector fields and Poisson brackets in Cannas da Silva 2006, Lecture 18, §§18.1–18.3, pp. 105–109.

On $T^*Q$ with canonical coordinates $(q^i,p_i)$,

$$\Omega=dq^i\wedge dp_i.$$

Writing $X_F=A^i\partial_{q^i}+B_i\partial_{p_i}$ gives

$$\iota_{X_F}\Omega=A^i dp_i-B_i dq^i=dF,$$

so

$$X_F=\frac{\partial F}{\partial p_i}\frac{\partial}{\partial q^i} -\frac{\partial F}{\partial q^i}\frac{\partial}{\partial p_i}, \qquad \{F,G\} =\frac{\partial F}{\partial q^i}\frac{\partial G}{\partial p_i} -\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q^i}.$$

Taking the motion to follow $X_H$ yields $\dot q^i=\partial H/\partial p_i$ and $\dot p_i=-\partial H/\partial q^i$. Moreover $\dot F=X_H[F]=\{F,H\}$, so $H$ itself is conserved when it has no explicit time dependence.

Why closedness gives the Poisson algebra

The bracket is bilinear and antisymmetric because $\Omega$ is. It obeys the Leibniz rule because $d(FG)=F\,dG+G\,dF$, hence $X_{FG}=F X_G+G X_F$. The nontrivial property is Jacobi. Cartan’s identity and $d\Omega=0$ give

$$\mathcal L_{X_F}\Omega =d(\iota_{X_F}\Omega)+\iota_{X_F}d\Omega =d^2F=0.$$

Thus Hamiltonian flows preserve $\Omega$. Moreover,

$$\begin{aligned} \iota_{[X_F,X_G]}\Omega &=\mathcal L_{X_F}(\iota_{X_G}\Omega) -\iota_{X_G}(\mathcal L_{X_F}\Omega)\\ &=d\bigl(X_F[G]\bigr) =-d\{F,G\}. \end{aligned}$$

Nondegeneracy therefore gives the useful intermediate identity

$$[X_F,X_G]=-X_{\{F,G\}}.$$

Evaluating the identity $d\Omega=0$ on three Hamiltonian vector fields and using that relation gives

$$\begin{aligned} 0 &=d\Omega(X_F,X_G,X_H)\\ &=-2\bigl(\{F,\{G,H\}\}+\{G,\{H,F\}\}\\ &\qquad+\{H,\{F,G\}\}\bigr). \end{aligned}$$

Therefore

$$\{F,\{G,H\}\}+\{G,\{H,F\}\}+\{H,\{F,G\}\}=0.$$

In canonical coordinates the same check follows because the inverse matrix of $\Omega$ is constant and mixed partial derivatives commute. The geometric proof explains what survives in noncanonical coordinates.

Canonical transformations and Darboux coordinates

A diffeomorphism $\Phi:P\to P$ is symplectic when $\Phi^*\Omega=\Omega$. It preserves Hamiltonian vector fields and Poisson brackets under pullback, so it changes the phase-space description without changing the symplectic relations. Infinitesimally, a vector field $Y$ generates local symplectomorphisms when $\mathcal L_Y\Omega=0$. Then $\iota_Y\Omega$ is closed; it is locally $dF$, but globally it need not be exact. Its global obstruction is a class in first de Rham cohomology. Consequently, every symplectic vector field is locally Hamiltonian, while global topology can obstruct a global Hamiltonian.

Darboux’s theorem says that every symplectic form is locally expressible as $dq^i\wedge dp_i$. A concise Moser proof explains why. Choose linear coordinates centered at $p$ so that the constant form $\Omega_0$ equals $\Omega_p$, and set $\Omega_t=\Omega_0+t(\Omega-\Omega_0)$. On a sufficiently small neighborhood every $\Omega_t$ is nondegenerate. The local homotopy formula gives a one-form $\alpha$ with $d\alpha=\Omega-\Omega_0$ and $\alpha_p=0$; solving $\iota_{X_t}\Omega_t=-\alpha$ and integrating $X_t$ gives a local flow $\rho_t$ that fixes $p$ and satisfies

$$\frac{d}{dt}\rho_t^*\Omega_t =\rho_t^*d\left(\iota_{X_t}\Omega_t+\alpha\right)=0.$$

Thus $\rho_1^*\Omega=\Omega_0$, which supplies the desired canonical coordinates. Cannas da Silva develops the local Moser mechanism in Cannas da Silva 2006, Lecture 7, §§7.2–7.3, pp. 42–45, PDF and states Darboux’s theorem in Cannas da Silva 2006, Lecture 8, §8.1, p. 46, PDF. The theorem does not say that a preferred Hamiltonian, metric, polarization, or global canonical chart exists. Symplectic geometry has no local curvature invariant analogous to Riemann curvature, but it can have global topology and nontrivial Hamiltonian group actions.

Symplectic volume and representative examples

On a $2n$-dimensional symplectic manifold, nondegeneracy is equivalent to the top form

$$\frac{\Omega^n}{n!}$$

being nowhere zero. Thus every symplectic manifold is orientable and carries a natural volume form. Hamiltonian flow preserves this volume because it preserves $\Omega$; in canonical mechanics this is Liouville’s theorem. Preservation does not say that trajectories remain close or that the Hamiltonian is bounded below. It is a phase-volume statement, not a stability theorem.

The cotangent bundle $T^*Q$ always has a canonical one-form $\vartheta=p_i\,dq^i$ in local coordinates and canonical symplectic form $\Omega=-d\vartheta=dq^i\wedge dp_i$. Unlike a coordinate formula chosen on an arbitrary space, $\vartheta$ is defined intrinsically by evaluating a covector on the projected tangent vector. This provides a global symplectic structure even when $Q$ has no global coordinates.

A closed oriented surface supplies a contrasting test. Any nowhere-vanishing two-form on it is symplectic because every three-form vanishes, so closedness is automatic. The sphere therefore admits an area-form symplectic structure even though it has no single global canonical chart. These examples separate local Darboux coordinates from global topology and show why exactness of $\Omega$ is not part of the definition.

Degeneracy and scope

If a closed two-form has a nonzero kernel, it is presymplectic. Then $\iota_X\Omega=dF$ may have no solution or many solutions: existence requires $dF$ to annihilate the kernel, and any solution is defined only up to a kernel vector. Quotienting by the kernel can produce a symplectic space only when the quotient is sufficiently regular. These issues are central for constrained theories but absent from the definition of a symplectic manifold.

This page assumes a smooth finite-dimensional manifold. Infinite-dimensional phase spaces require domains, continuity, boundary conditions, and sometimes weak rather than strong nondegeneracy. Quantization is not implied by the existence of $\Omega$.

Exercise: oscillator flow and bracket

On $P=\mathbb R^2$ with $\Omega=dq\wedge dp$ and $H=(p^2+\omega^2q^2)/2$, find $X_H$, verify Hamilton’s equations, and compute $\{q,p\}$ and $\{H,H\}$.

Solution

The defining contraction gives

$$X_H=p\,\partial_q-\omega^2q\,\partial_p.$$

Hence $\dot q=p$ and $\dot p=-\omega^2q$, so $\ddot q+\omega^2q=0$. The coordinate formula gives $\{q,p\}=1$. Antisymmetry gives $\{H,H\}=0$, independently confirming conservation of $H$ along its own flow.

References

• Cannas da Silva, Ana. Lectures on Symplectic Geometry. Lecture Notes in Mathematics 1764, Springer, revised January 2006. Lectures 1, 7–8, and 18, pp. 3–7, 42–46, and 105–109. Open PDF.