Symplectic Forms, Hamiltonian Flows, and Poisson Brackets

A symplectic form turns a function into a direction of motion. On a smooth finite-dimensional manifold $M$, a symplectic form is a two-form $\omega$ that is both nondegenerate and closed. Nondegeneracy makes the equation

$$\iota_{X_f}\omega=\mathrm df$$

solvable for one and only one vector field $X_f$. Closedness then makes the local flow of $X_f$ preserve $\omega$ and makes

$$\{f,g\}=\omega(X_f,X_g)$$

satisfy the Jacobi identity. In canonical coordinates, these geometric statements become Hamilton’s equations and the usual Poisson bracket.

The two hypotheses do different jobs. Dropping closedness leaves unique vector fields but can destroy Jacobi; dropping nondegeneracy can make the Hamiltonian equation insoluble or nonunique. This page separates those roles, fixes one sign convention throughout, and transfers the construction to the equal-time phase space of a real scalar field. Constraint reduction, quantization, and the analytic construction of infinite-dimensional phase spaces are outside its scope.

Required background. Direct Sums, Tensor Products, and Index Structure supplies alternating bilinear maps, duality, and index structure.

Helpful background. Exterior and Graded Algebra, Grassmann Variables, and Berezin Integration supplies wedge-product signs, while Differential Forms, Integration, Orientation, and Stokes Theorem supplies exterior derivatives, pullbacks, and Cartan’s formula.

Symplectic setting and sign convention

We work first on a smooth real manifold $M$ of finite dimension. A two-form $\omega\in\Omega^2(M)$ assigns an alternating bilinear form $\omega_x$ to each tangent space $T_xM$. Contraction puts a vector into the first slot,

$$(\iota_X\omega)(Y)=\omega(X,Y).$$

The exterior derivative is denoted by $\mathrm d$, and $\mathcal L_X$ is the Lie derivative. Later, $\boldsymbol{\delta}$ will instead denote the exterior derivative on a space of fields; that notation prevents it from being confused with the spacetime derivative $\mathrm d$ or with a single directional variation.

All signs on this page follow the package

$$\boxed{ \omega=\sum_i\mathrm dq^i\wedge\mathrm dp_i, \qquad \iota_{X_f}\omega=\mathrm df, \qquad \{f,g\}=\omega(X_f,X_g). }$$

It implies $\{q^i,p_j\}=\delta^i{}_j$ and $\dot f=\partial_t f+\{f,H\}$. Other sources may reverse $\omega$, put a minus sign in the defining equation for $X_f$, or reverse the Poisson bracket. Any of those choices can be consistent, but formulas must be translated as one package rather than mixed line by line.

Closed and nondegenerate two-forms

At $x\in M$, a two-form defines the linear map

$$\begin{aligned} \omega_x^\flat:T_xM&\longrightarrow T_x^*M,\\ v&\longmapsto\iota_v\omega_x. \end{aligned}$$

The form is nondegenerate when $\omega_x^\flat$ is an isomorphism at every point, equivalently,

$$\omega_x(v,w)=0\ \text{for every }w \quad\Longrightarrow\quad v=0.$$

It is closed when $\mathrm d\omega=0$. A symplectic manifold is a pair $(M,\omega)$ satisfying both conditions.

This is not metric geometry. Alternation gives $\omega(v,w)=-\omega(w,v)$ and therefore $\omega(v,v)=0$ for every $v$. There is no norm, angle, or positivity condition. In particular, nondegeneracy does not mean $\omega(v,v)\ne0$ for nonzero $v$; it means that no nonzero vector pairs to zero with every other vector. This sharply contrasts with the symmetric Hessian used in Second Variation, Hessians, and Jacobi Operators.

A nondegenerate alternating form exists only in even dimension. If $\dim M=2n$, then

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

is nowhere zero, so it is a volume form and fixes an orientation. This is the symplectic volume. It is built from $\omega$ without introducing a metric.

The model example is $\mathbb R^{2n}$ with coordinates $(q^1,\ldots,q^n,p_1,\ldots,p_n)$ and

$$\omega_0=\sum_{i=1}^n\mathrm dq^i\wedge\mathrm dp_i.$$

Its coefficients are constant, so it is closed, and each $(q^i,p_i)$ block is nondegenerate. A compact example is the two-sphere with its area form. That form is closed because every three-form on a two-dimensional manifold vanishes, and it is nondegenerate because it is nowhere zero.

Hamiltonian vector fields and their flows

For $f\in C^\infty(M)$, nondegeneracy gives a unique vector field $X_f$ such that $\iota_{X_f}\omega=\mathrm df$. This is the Hamiltonian vector field of $f$. In canonical coordinates, write

$$X_f=A^i\frac{\partial}{\partial q^i} +B_i\frac{\partial}{\partial p_i}.$$

Then

$$\iota_{X_f}\omega =A^i\mathrm dp_i-B_i\mathrm dq^i =\frac{\partial f}{\partial q^i}\mathrm dq^i +\frac{\partial f}{\partial p_i}\mathrm dp_i,$$

so

$$X_f =\frac{\partial f}{\partial p_i} \frac{\partial}{\partial q^i} -\frac{\partial f}{\partial q^i} \frac{\partial}{\partial p_i}.$$

Only the differential of the function enters. Consequently $X_{f+c}=X_f$ for any componentwise constant $c$, and $X_f=0$ exactly when $f$ is constant on each connected component of $M$. Thus the assignment from Hamiltonians to vector fields is unique but not injective on functions.

For a Hamiltonian $H$, an integral curve $z(t)$ of $X_H$ therefore obeys

$$\dot q^i=\frac{\partial H}{\partial p_i}, \qquad \dot p_i=-\frac{\partial H}{\partial q^i}.$$

The same signs follow independently from the first-order action

$$S[q,p]=\int_{t_i}^{t_f}\mathrm dt\, \bigl(p_i\dot q^i-H(q,p,t)\bigr).$$

Its variation is

$$\begin{aligned} \delta S ={}&\left[p_i\,\delta q^i\right]_{t_i}^{t_f}\\ &+\int_{t_i}^{t_f}\mathrm dt\, \left[ (\dot q^i-H_{,p_i})\delta p_i -(\dot p_i+H_{,q^i})\delta q^i \right]. \end{aligned}$$

Fixed endpoint positions remove the surface term and recover Hamilton’s equations.

Closedness now enters through Cartan’s formula:

$$\begin{aligned} \mathcal L_{X_f}\omega &=\mathrm d(\iota_{X_f}\omega) +\iota_{X_f}(\mathrm d\omega)\\ &=\mathrm d^2f=0. \end{aligned}$$

If $\Phi_t$ is the local flow of $X_f$, then

$$\frac{\mathrm d}{\mathrm dt}\Phi_t^*\omega =\Phi_t^*(\mathcal L_{X_f}\omega)=0,$$

and hence $\Phi_t^*\omega=\omega$ wherever the flow exists. It also preserves $\omega^n/n!$. This is a local-in-time statement. A smooth Hamiltonian vector field need not be complete on a noncompact manifold: for example, $H=q^2p$ on $\mathbb R^2$ gives $\dot q=q^2$, whose positive solution can escape to infinity in finite time.

For an observable $f(q,p,t)$ evolving along $X_H$,

$$\begin{aligned} \frac{\mathrm df}{\mathrm dt} &=\frac{\partial f}{\partial t}+X_H[f]\\ &=\frac{\partial f}{\partial t}+\{f,H\}. \end{aligned}$$

Tong 2015, Chapter 4, §§ 4.1 and 4.3, pp. 80–84 and 93–95, PDF develops these Hamilton equations, canonical brackets, and observable evolution.

An autonomous Hamiltonian is conserved because $\dot H=\{H,H\}=0$. More generally, a time-independent $f$ is conserved exactly when $\{f,H\}=0$ along the motion.

Poisson brackets and the Jacobi identity

The coordinate expression for the bracket fixed above is

$$\{f,g\} =\frac{\partial f}{\partial q^i} \frac{\partial g}{\partial p_i} -\frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q^i}.$$

Thus

$$\{q^i,q^j\}=0, \qquad \{p_i,p_j\}=0, \qquad \{q^i,p_j\}=\delta^i{}_j.$$

Invariantly, bilinearity follows from the linearity of $f\mapsto\mathrm df$ and $(\omega^\flat)^{-1}$. Alternation of $\omega$ gives antisymmetry. The product rule

$$\mathrm d(fg)=f\,\mathrm dg+g\,\mathrm df$$

and uniqueness imply

$$X_{fg}=fX_g+gX_f,$$

which yields the Leibniz identity

$$\{fg,h\}=f\{g,h\}+g\{f,h\}.$$

The nontrivial property is Jacobi. Define

$$J(f,g,h) =\{f,\{g,h\}\} +\{g,\{h,f\}\} +\{h,\{f,g\}\}.$$

The sign convention gives $X_f[u]=-\{f,u\}$. Insert $X_f,X_g,X_h$ into the exterior-derivative formula

$$\begin{aligned} \mathrm d\omega(X_f,X_g,X_h) =\sum_{\mathrm{cyc}} \bigl(&X_f\,\omega(X_g,X_h)\\ &-\omega([X_f,X_g],X_h)\bigr). \end{aligned}$$

The derivative terms sum to $-J$. For the commutator terms,

$$\omega([X_f,X_g],X_h) =-\{f,\{g,h\}\}-\{g,\{h,f\}\},$$

so their cyclic contribution is $2J$. Therefore

$$J(f,g,h)=\mathrm d\omega(X_f,X_g,X_h).$$

Closedness gives the Jacobi identity. Conversely, nondegeneracy lets local differentials $\mathrm df$ generate every tangent direction at a point, so Jacobi for this bracket forces $\mathrm d\omega=0$. For a nondegenerate two-form, closedness and Jacobi are therefore equivalent statements. This is the nondegenerate case of the symplectic–Poisson correspondence proved in Crainic, Fernandes, and Mărcuț 2021, § 2.4.2, PDF; their bracket is the negative of this page’s bracket, but the Jacobiator and the equivalence are unchanged by that overall reversal.

Once $\mathrm d\omega=0$, the relation between brackets of functions and commutators of vector fields is

$$\begin{aligned} \iota_{[X_f,X_g]}\omega &=\mathcal L_{X_f}(\iota_{X_g}\omega) -\iota_{X_g}(\mathcal L_{X_f}\omega)\\ &=\mathrm d(X_f[g]) =-\mathrm d\{f,g\}. \end{aligned}$$

Nondegeneracy then gives the convention-sensitive anti-homomorphism

$$[X_f,X_g]=-X_{\{f,g\}}.$$

These properties make $C^\infty(M)$ a Poisson algebra: its ordinary product is commutative, its bracket is a Lie bracket, and that bracket acts as a derivation of the product.

Canonical coordinates are local

If $Q$ is a configuration manifold, its cotangent bundle $T^*Q$ carries a canonical one-form. In local coordinates,

$$\theta=p_i\,\mathrm dq^i, \qquad \omega=-\mathrm d\theta =\mathrm dq^i\wedge\mathrm dp_i.$$

The definition of $\theta$ is coordinate independent even though this formula uses a chart. Because $\omega$ is exact, it is automatically closed.

Cannas da Silva 2006, Lecture 8, p. 46, PDF proves Darboux’s theorem: every symplectic form has the canonical coordinate expression near each point. It does not provide one global canonical chart, make every symplectic form globally exact, or simplify the Hamiltonian. For example, an area form on $S^2$ cannot be exact: if $\omega=\mathrm d\alpha$, Stokes’ theorem would give $\int_{S^2}\omega=0$, contrary to its nonzero area.

There is a related local-to-global distinction for vector fields. A vector field $Y$ is symplectic when $\mathcal L_Y\omega=0$. Since $\mathrm d\omega=0$,

$$\mathcal L_Y\omega=\mathrm d(\iota_Y\omega),$$

so $\iota_Y\omega$ is closed. It is locally exact, making $Y$ locally Hamiltonian. Globally, $Y=X_f$ for a single-valued function $f$ exactly when $\iota_Y\omega$ is exact. On the two-torus, let $q$ and $p$ be angular coordinates and write $\omega=\mathrm dq\wedge\mathrm dp$. Although there are no globally single-valued real functions $q$ and $p$, the one-forms $\mathrm dq$ and $\mathrm dp$ are globally defined. The translation field $Y=\partial_q$ satisfies $\iota_Y\omega=\mathrm dp$. This one-form is closed but not exact, so the flow is symplectic without being globally Hamiltonian.

The closedness hypothesis cannot be discarded from Darboux’s theorem or from the Poisson construction. On $\mathbb R^4$, consider the nondegenerate form

$$\widetilde\omega =\mathrm dq^1\wedge\mathrm dp_1 +e^{q^1}\mathrm dq^2\wedge\mathrm dp_2.$$

Its square is nowhere zero, but

$$\mathrm d\widetilde\omega =e^{q^1}\mathrm dq^1\wedge\mathrm dq^2 \wedge\mathrm dp_2\ne0.$$

The inverse construction still produces

$$\begin{aligned} \{f,g\}_{\sim} ={}&f_{,q^1}g_{,p_1}-f_{,p_1}g_{,q^1}\\ &+e^{-q^1} \left(f_{,q^2}g_{,p_2}-f_{,p_2}g_{,q^2}\right). \end{aligned}$$

However,

$$\begin{aligned} &\{p_1,\{q^2,p_2\}_{\sim}\}_{\sim} +\{q^2,\{p_2,p_1\}_{\sim}\}_{\sim}\\ &\qquad +\{p_2,\{p_1,q^2\}_{\sim}\}_{\sim} =e^{-q^1}\ne0. \end{aligned}$$

This is an almost-Poisson bracket, not a Poisson bracket. The failure equals $\mathrm d\widetilde\omega(X_{p_1},X_{q^2},X_{p_2})$, exactly as the general identity predicts.

Canonical field brackets and time evolution

Consider a real scalar field on a fixed spatial slice $\Sigma$. Assume for now that $\Sigma$ is compact without boundary, is periodic, or that the fields and variations decay sufficiently fast. A phase-space point is a pair $(\phi(\mathbf x),\pi(\mathbf x))$. The field-space one-form and two-form are

$$\begin{aligned} \Theta &=\int_\Sigma\mathrm d^{d-1}x\, \pi(\mathbf x)\,\boldsymbol{\delta}\phi(\mathbf x),\\ \Omega &=-\boldsymbol{\delta}\Theta =\int_\Sigma\mathrm d^{d-1}x\, \boldsymbol{\delta}\phi(\mathbf x) \wedge\boldsymbol{\delta}\pi(\mathbf x). \end{aligned}$$

For tangent vectors $(u_1,v_1)$ and $(u_2,v_2)$, this means

$$\Omega\bigl((u_1,v_1),(u_2,v_2)\bigr) =\int_\Sigma\mathrm d^{d-1}x\, (u_1v_2-v_1u_2).$$

Formally solving $\iota_{X_F}\Omega=\boldsymbol{\delta}F$ gives

$$X_F =\left( \frac{\delta F}{\delta\pi}, -\frac{\delta F}{\delta\phi} \right),$$

and hence the equal-time functional bracket

$$\boxed{ \{F,G\} =\int_\Sigma\mathrm d^{d-1}x\, \left[ \frac{\delta F}{\delta\phi(\mathbf x)} \frac{\delta G}{\delta\pi(\mathbf x)} -\frac{\delta F}{\delta\pi(\mathbf x)} \frac{\delta G}{\delta\phi(\mathbf x)} \right]. }$$

The safest fundamental relation is smeared. For smooth test functions $f$ and $g$, define

$$\Phi[f]=\int_\Sigma\mathrm d^{d-1}x\,f\phi, \qquad \Pi[g]=\int_\Sigma\mathrm d^{d-1}x\,g\pi.$$

Then

$$\{\Phi[f],\Pi[g]\} =\int_\Sigma\mathrm d^{d-1}x\,f(\mathbf x)g(\mathbf x).$$

Only in the distributional shorthand encoded by this identity do we write

$$\{\phi(t,\mathbf x),\pi(t,\mathbf y)\} =\delta^{(d-1)}(\mathbf x-\mathbf y).$$

For the mostly-minus metric convention and natural units, take the following Hamiltonian. Floerchinger 2024, sections “Hamiltonian formalism” and “Poisson brackets” supports the functional derivatives, equal-time bracket, boundary qualification, and scalar evolution below; his covariant formulas use the opposite metric signature, while the expanded positive Hamiltonian agrees.

$$H[\phi,\pi] =\int_\Sigma\mathrm d^{d-1}x\, \left[ \frac12\pi^2 +\frac12(\boldsymbol\nabla\phi)^2 +V(\phi) \right].$$

Its field-space differential is

$$\begin{aligned} \boldsymbol{\delta}H ={}&\int_\Sigma\mathrm d^{d-1}x\, \left[ \pi\,\boldsymbol{\delta}\pi +\boldsymbol\nabla\phi\mathbin{\cdot} \boldsymbol\nabla\boldsymbol{\delta}\phi +V'(\phi)\boldsymbol{\delta}\phi \right]\\ ={}&\int_\Sigma\mathrm d^{d-1}x\, \left[ \pi\,\boldsymbol{\delta}\pi +\bigl(-\boldsymbol\nabla^2\phi+V'(\phi)\bigr) \boldsymbol{\delta}\phi \right], \end{aligned}$$

where the stated boundary conditions remove the integration-by-parts term. Thus

$$\frac{\delta H}{\delta\pi}=\pi, \qquad \frac{\delta H}{\delta\phi} =-\boldsymbol\nabla^2\phi+V'(\phi),$$

and Hamilton’s equations become

$$\dot\phi=\pi, \qquad \dot\pi=\boldsymbol\nabla^2\phi-V'(\phi).$$

Eliminating $\pi$ gives

$$\left(\partial_t^2-\boldsymbol\nabla^2\right)\phi +V'(\phi)=0,$$

or $\Box\phi+V'(\phi)=0$ in the site’s $(+---)$ convention. This recovers the Lagrangian field equation while making equal-time evolution explicit. Tong 2006–2007, § 1.4, pp. 19–20, PDF gives the scalar-field canonical momentum, Hamiltonian, and Hamilton equations in this convention.

The boundary assumption is structural, not cosmetic. Before integration by parts, the variation contains

$$\int_{\partial\Sigma}\mathrm dS\, (\mathbf n\mathbin{\cdot}\boldsymbol\nabla\phi) \boldsymbol{\delta}\phi.$$

If it does not vanish, the displayed $H$ is not functionally differentiable on the proposed phase space. One must restrict the boundary traces, impose suitable falloff, or add a boundary contribution to the Hamiltonian before using $X_H$.

Dimensional analysis supplies a quick consistency check. In $d$ spacetime dimensions,

$$[\phi]=\frac{d-2}{2}, \qquad [\pi]=\frac d2, \qquad [\delta^{(d-1)}]=d-1.$$

Consequently $[\Omega]=0$, as follows from $-(d-1)+[\phi]+[\pi]=0$.

These field-space formulas are formal until a function space and class of differentiable functionals are fixed. In infinite dimensions, nondegeneracy may be only weak: the map $\Omega^\flat$ can be injective without being onto the full continuous dual, so not every formal functional need possess a Hamiltonian vector field. Odzijewicz 2011, § 4, PDF makes this strong-versus-weak distinction precise for Banach symplectic manifolds. The scalar calculation above remains formal: smooth test functions and boundary assumptions make its manipulations meaningful, but do not by themselves fix a phase-space topology or prove existence of every $X_F$.

Checks, limits, and common pitfalls

Nondegenerate is not positive. A symplectic form is alternating, so $\omega(v,v)=0$ for every $v$. Test its kernel, not a quadratic sign.

Closed is not exact. Exactness implies closedness, but the area form on $S^2$ shows that the converse can fail globally. Darboux coordinates are local and do not remove topology.

A sign convention is a package. With the convention on this page, $X_q=-\partial_p$, $X_p=\partial_q$, $\{q,p\}=1$, and $[X_f,X_g]=-X_{\{f,g\}}$. A source using $\iota_{X_f}\omega=-\mathrm df$ will display a different-looking but potentially consistent collection of signs.

Presymplectic equations need compatibility. On $\mathbb R^3$, let

$$\omega=\mathrm dx\wedge\mathrm dy.$$

It is closed but has kernel spanned by $\partial_z$. For $X=a\partial_x+b\partial_y+c\partial_z$, the equation $\iota_X\omega=\mathrm dH$ requires $\partial_zH=0$. When this condition fails there is no solution; when it holds, $a$ and $b$ are fixed but $c$ is arbitrary. Gauge directions and systematic constraint reduction begin from this failure of nondegeneracy and belong to the later constraint treatment.

Preservation is not completeness. Hamiltonian flow preserves $\omega$ on its interval of existence. A separate global hypothesis is needed to claim that the flow exists for every real time.

Exercises

1. Check the canonical sign triangle. On $\mathbb R^2$ with $\omega=\mathrm dq\wedge\mathrm dp$, find $X_q$, $X_p$, and $\{q,p\}$.

Solution

Since

$$\iota_{\partial_q}\omega=\mathrm dp, \qquad \iota_{\partial_p}\omega=-\mathrm dq,$$

the equations $\iota_{X_q}\omega=\mathrm dq$ and $\iota_{X_p}\omega=\mathrm dp$ give

$$X_q=-\partial_p, \qquad X_p=\partial_q.$$

Therefore

$$\{q,p\}=\omega(-\partial_p,\partial_q)=1.$$

The three results test the whole sign package at once.

2. Diagnose the nonclosed example. For $\widetilde\omega$ on $\mathbb R^4$ above, verify its nondegeneracy and evaluate the Jacobiator of $(p_1,q^2,p_2)$.

Solution

Its top exterior power is

$$\widetilde\omega^2 =2e^{q^1}\mathrm dq^1\wedge\mathrm dp_1 \wedge\mathrm dq^2\wedge\mathrm dp_2,$$

which is nowhere zero. Thus the form is nondegenerate. The only nonzero term in the cyclic Jacobiator is

$$\{p_1,\{q^2,p_2\}_{\sim}\}_{\sim} =\{p_1,e^{-q^1}\}_{\sim} =e^{-q^1}.$$

The induced bracket therefore fails Jacobi precisely because $\mathrm d\widetilde\omega\ne0$.

3. Separate existence from uniqueness. For $\omega=\mathrm dx\wedge\mathrm dy$ on $\mathbb R^3$, solve $\iota_X\omega=\mathrm dH$ for (a) $H=z$ and (b) $H=x^2+y^2$.

Solution

Writing $X=a\partial_x+b\partial_y+c\partial_z$ gives

$$\iota_X\omega=a\,\mathrm dy-b\,\mathrm dx.$$

For $H=z$, the right-hand side is $\mathrm dz$, so no solution exists. For $H=x^2+y^2$, comparison with $2x\,\mathrm dx+2y\,\mathrm dy$ gives $a=2y$ and $b=-2x$, while $c$ is arbitrary. Hence

$$X=2y\,\partial_x-2x\,\partial_y+c\,\partial_z.$$

Degeneracy caused nonexistence in the first case and nonuniqueness in the second.

4. Transfer the bracket to a scalar field. Using the smeared functionals $\Phi[f]$ and $\Pi[g]$, compute their bracket. Then use the free massive Hamiltonian, $V(\phi)=m^2\phi^2/2$, to recover the field equation.

Solution

The functional derivatives are

$$\frac{\delta\Phi[f]}{\delta\phi}=f, \qquad \frac{\delta\Pi[g]}{\delta\pi}=g,$$

with the crossed derivatives zero. Therefore

$$\{\Phi[f],\Pi[g]\} =\int_\Sigma\mathrm d^{d-1}x\,fg.$$

For $V'(\phi)=m^2\phi$, Hamilton’s equations give

$$\dot\phi=\pi, \qquad \dot\pi=\boldsymbol\nabla^2\phi-m^2\phi.$$

Taking one more time derivative of the first equation and substituting the second yields

$$(\partial_t^2-\boldsymbol\nabla^2+m^2)\phi=0,$$

the Klein–Gordon equation in the mostly-minus convention.

Synthesis and continuation

A symplectic form converts covectors into vectors. Its nondegeneracy turns $\mathrm df$ into a unique Hamiltonian vector field; its closedness makes that field preserve the geometry and turns the induced antisymmetric derivation into a Poisson bracket. Darboux coordinates expose the universal local form, while nonexact examples, incomplete flows, and presymplectic kernels show why local canonical formulas do not settle global or constrained questions.

For a scalar field, the same sign package gives the equal-time functional bracket and reproduces the mostly-minus Euler–Lagrange equation, provided the Hamiltonian is differentiable under the chosen boundary conditions. This is the finite-dimensional geometric core needed before symmetries, moment maps, constraints, and reduction can be developed.

The physical meaning of the scalar data and the choice of its initial-data space are developed in Hamiltonian Initial Data and Phase Space.

References

• Ana Cannas da Silva, Lectures on Symplectic Geometry — Open PDF, revised January 2006, Lecture Notes in Mathematics 1764, Springer, doi:10.1007/978-3-540-45330-7, Lectures 1, 2, 8, and 18, especially pp. 3–12, 46, and 105–109. This is the structural source for symplectic forms, the cotangent-bundle construction, Darboux coordinates, Hamiltonian flows, and Poisson brackets.
• Marius Crainic, Rui Loja Fernandes, and Ioan Mărcuț, Lectures on Poisson Geometry — Open PDF, Graduate Studies in Mathematics 217, American Mathematical Society, 2021, doi:10.1090/gsm/217, §§ 1.1 and 2.4.2, especially pp. 3–4 and 32–33. This source supports the nondegenerate-bivector/closed-form equivalence. Its Poisson bracket has the opposite overall sign from the convention on this page.
• Stefan Floerchinger, “Quantum Field Theory 1, Lecture 03: Classical Field Theory”, Friedrich Schiller University Jena, updated April 9, 2024, sections “Hamiltonian formalism,” “Functional differentiation,” and “Poisson brackets.” This source supports the functional bracket, boundary conditions, and scalar-field evolution; its spacetime metric convention is mostly plus.
• Anatol Odzijewicz, “Hamiltonian and Quantum Mechanics” — Open PDF, Geometry & Topology Monographs 17 (2011) 385–472, doi:10.2140/gtm.2011.17.385, § 4, especially pp. 399–400. This source supports the distinction between strong and weak symplectic structures in infinite dimensions.
• David Tong, Classical Dynamics — Open PDF, Cambridge Part III lecture notes, 2015, Chapter 4, §§ 4.1 and 4.3, pp. 80–84 and 93–95. These notes provide the route through Hamilton’s equations, canonical brackets, observable evolution, and conservation laws.
• David Tong, Lectures on Quantum Field Theory — Open PDF, Cambridge Part III lecture notes, 2006–2007, § 1.4, pp. 19–20. This source supplies the scalar-field canonical momentum, Hamiltonian, and Hamilton equations used in the QFT-facing calculation.