Canonical Invariants and Poincaré Integral Invariants

Take a closed rubber band, drop it into phase space, and let Hamiltonian dynamics carry every point of it along its own trajectory. The band stretches, folds, and twists into an unrecognizable tangle — yet one number stitched through it never budges: the loop integral $\oint p\,\mathrm{d}q$ , the total oriented area the band encloses when projected onto the conjugate planes. Henri Poincaré discovered that this number, and a whole tower of higher-dimensional companions, is untouched by every canonical transformation — including time evolution itself. He called them integral invariants, and he built his new celestial mechanics on them. The discovery reframes what canonical transformations are. You already know three tests for canonicity: generating functions, the symplectic condition on the Jacobian, and the fundamental Poisson brackets. Those are local, pointwise criteria. The integral invariants are the same fact expressed globally — a transformation is canonical exactly when it preserves the symplectic area measured by every closed loop. One geometric object, the two-form $\omega=\sum_i \mathrm{d}p_i\wedge\mathrm{d}q_i$ , sits underneath all four tests, and its wedge powers climb dimension by dimension until they reach phase-space volume itself, where the tower tops out in Liouville's theorem. Mastering this article buys you the geometric backbone of everything downstream: why phase-space volume is conserved, why the action $\oint p\,\mathrm{d}q$ is the natural adiabatic invariant, and — through the Poincaré–Cartan form $p\,\mathrm{d}q - H\,\mathrm{d}t$ — the launching point of Hamilton–Jacobi theory.

Key takeaways

Poincaré integral invariants are integrals over closed loops and higher-dimensional surfaces in phase space whose values survive every canonical transformation.
The fundamental one is the relative invariant $\oint_C \sum_i p_i\,\mathrm{d}q_i$ ; by Stokes' theorem it equals the surface integral of the symplectic two-form $\omega=\sum_i \mathrm{d}p_i\wedge\mathrm{d}q_i$ .
Wedge powers $\omega^{\wedge k}/k!$ give a hierarchy of invariants for $k=1,\dots,n$ ; the top one ( $k=n$ ) is phase-space volume, so Liouville's theorem is the last floor of Poincaré's tower.
With time promoted to a coordinate, the Poincaré–Cartan invariant $\oint(\sum_i p_i\,\mathrm{d}q_i - H\,\mathrm{d}t)$ takes the same value on every loop encircling a tube of trajectories — a single statement that encodes both canonicity and Hamilton's equations.
Hamiltonian flow is itself a one-parameter family of canonical transformations, so all the invariants are conserved by the dynamics — the deep reason the phase fluid is incompressible.
Key idea: the two-form $\omega=\sum_i \mathrm{d}p_i\wedge\mathrm{d}q_i$ is the invariant object — a transformation is canonical iff it preserves $\omega$ , and integrating $\omega$ and its wedge powers over phase-space regions yields the tower of Poincaré integral invariants, unchanged by every canonical map and conserved by the dynamics.

The core insight

The two-form $\omega=\sum_i \mathrm{d}p_i\wedge\mathrm{d}q_i$ is the invariant object: a transformation is canonical iff it preserves $\omega$ , and integrating $\omega$ (and its wedge powers) over phase-space regions yields a tower of quantities — the Poincaré integral invariants — unchanged by any canonical map and, since Hamiltonian flow is canonical, conserved by the dynamics.

Where this lives today

Every long-term solar-system integration run today is an act of respect for Poincaré's invariants. Symplectic integrators — the workhorses behind asteroid-hazard propagation, exoplanet stability studies, and the million-body orbit catalogs assembled from Vera Rubin Observatory alerts — are numerical maps built to be exactly canonical per step. They do not conserve the energy exactly; they conserve the symplectic form, and in exchange their energy error stays bounded for billions of steps instead of drifting. That design choice is the integral-invariant philosophy turned into software: protect $\omega$ and the dynamics stays honest. The same discipline now extends to machine learning, where symplectic neural networks constrain learned surrogate models of Hamiltonian systems to preserve the two-form by construction. In accelerator physics the relative invariant has a proper name: emittance. A bunch of $10^{11}$ protons occupies a small loop-bounded region in each $(x,p_x)$ plane, and since every magnet and drift acts on the beam as a canonical map, the enclosed area per plane can be reshaped but its sum never reduced — the hard floor under all beam-focusing design, from the LHC to next-generation light sources. And Gromov's 1985 non-squeezing theorem sharpened Poincaré's point into modern mathematics: a canonical map cannot squeeze a ball through a hole narrower than its $(q_i,p_i)$ shadow, no matter that volume conservation alone would permit it — a classical, exact foreshadowing of the uncertainty principle.

The physical picture

Watch a loop of phase points evolve. Each point on the rubber band obeys Hamilton's equations; collectively they smear the band into an ever-stranger shape. Lengths are not preserved. Ordinary geometric area in a tilted plane is not preserved. What is preserved is the symplectic area: project the loop onto the $(q_1,p_1)$ plane, measure the oriented area, do the same for $(q_2,p_2)$ and every other conjugate pair, and add. That sum is Poincaré's relative integral invariant, and the flow cannot change it no matter how violently it shears the band.

Figure

A loop of phase points shears and folds under the flow, but the symplectic area it encloses is exactly conserved

Think of $\omega=\sum_i \mathrm{d}p_i\wedge\mathrm{d}q_i$ as a measuring device that reports oriented area in each conjugate $(q_i,p_i)$ plane and is blind to everything else. Canonical transformations are precisely the coordinate changes this device cannot detect. For one degree of freedom the statement is beautifully concrete: canonical maps of the $(q,p)$ plane are exactly the area-preserving maps. For $n$ degrees of freedom the loop threads $n$ planes at once, and only the sum of the projected areas is protected:

Figure

For two degrees of freedom the loop casts two shadows — only the sum of the two oriented shadow areas is invariant

Because the invariants are integrals, they are global fingerprints of canonicity: robust, coordinate-free, and insensitive to how wildly the map distorts local neighborhoods. They pair naturally with the local tests — the symplectic Jacobian condition and the fundamental Poisson brackets — as two ends of the same fact. And they answer a question the local tests make awkward: why should time evolution conserve phase-space volume? Because the flow is canonical, and volume is the top invariant of the tower.

The core equations

Everything begins with the loop integral. Take any closed curve $C$ in $2n$ -dimensional phase space — not necessarily a trajectory, just a closed chain of phase points — and thread the canonical one-form through it:

I_1=\oint_{C}\sum_{i=1}^{n} p_i\,\mathrm{d}q_i \qquad (1)

where: $C$ — an arbitrary closed curve in phase space; $q_i,p_i$ — canonical coordinates and their conjugate momenta; $n$ — the number of degrees of freedom. This is called a relative integral invariant: the value is protected only for closed curves, because (as we prove below) the integrand itself shifts by an exact differential under a canonical map, and exact differentials integrate to zero only around loops. Stokes' theorem converts the loop integral into something more revealing. The exterior derivative of the one-form $\theta=\sum_i p_i\,\mathrm{d}q_i$ is the symplectic two-form, so the loop integral is secretly measuring the flux of $\omega$ through any surface the loop bounds:

\omega=\sum_{i=1}^{n}\mathrm{d}p_i\wedge\mathrm{d}q_i,\qquad \oint_{C}\theta=\iint_{S}\omega \quad\text{for } \partial S = C \qquad (2)

where: $\omega$ — the fundamental symplectic two-form; $S$ — any surface with boundary $C$ ; $\wedge$ — the antisymmetric exterior product. The two-form is the area-measuring device of the physical picture: $\iint_S\omega$ is the sum of the oriented areas of the shadows $S$ casts on the $n$ conjugate planes. Preserving equation (1) for every loop and preserving $\omega$ are therefore the same demand. Higher invariants stack up by wedging $\omega$ with itself. Each wedge power is a form of twice the dimension, and each integrates to a canonically invariant number:

I_k=\frac{1}{k!}\int_{V_{2k}}\omega^{\wedge k},\qquad k=1,\dots,n \qquad (3)

where: $\omega^{\wedge k}$ — the $k$ -fold wedge power; $V_{2k}$ — any $2k$ -dimensional region of phase space; the $k!$ normalizes the top power into an honest volume element. For $k=1$ this is the surface form of (1). The tower stops at $k=n$ because a $(2n+2)$ -form on a $2n$ -dimensional space vanishes identically. The top of the tower deserves its own line, because it is a theorem you already know. Writing out $\omega^{\wedge n}/n!$ produces exactly the phase-space volume element, so canonical invariance at $k=n$ says:

I_n=\int \mathrm{d}q_1\cdots\mathrm{d}q_n\,\mathrm{d}p_1\cdots\mathrm{d}p_n=\text{const} \qquad (4)

where: $I_n$ — the $2n$ -dimensional volume of any region carried along by a canonical map or by the flow. This is Liouville's theorem, met here not as a stand-alone miracle but as the last member of Poincaré's hierarchy. For one degree of freedom the tower has a single floor and (1) and (4) coincide: area in the phase plane. Finally, promote time to a coordinate. On the extended phase space $(q,p,t)$ the natural object is the Poincaré–Cartan one-form, which glues the Hamiltonian to the canonical one-form with a relative minus sign:

\tilde I=\oint_{C}\Big(\sum_{i=1}^{n} p_i\,\mathrm{d}q_i-H\,\mathrm{d}t\Big) \qquad (5)

where: $H(q,p,t)$ — the Hamiltonian; $C$ — a closed loop in extended phase space encircling a tube of solution curves. The claim — proved in the second derivation — is that every loop around the same tube gives the same $\tilde I$ . Slice the tube at fixed time ( $\mathrm{d}t=0$ on the loop) and (5) collapses to (1): the Poincaré–Cartan invariant contains the relative invariant, and, remarkably, it contains Hamilton's equations too. It is the integrand of the phase-space action, which is why this form reappears as the opening move of Hamilton–Jacobi theory.

Which equation answers which question

you want	use	notes
the basic invariant of a loop	$\oint_C \sum_i p_i\,\mathrm{d}q_i$ , eq. (1)	closed curves only — relative invariant
a surface/two-form restatement	$\iint_S\omega$ , eq. (2)	Stokes; equivalent to the symplectic condition
invariants of higher-dimensional regions	$I_k=\frac{1}{k!}\int\omega^{\wedge k}$ , eq. (3)	$k=1,\dots,n$ ; each canonically invariant
phase-volume conservation	$I_n$ , eq. (4)	Liouville's theorem — top of the tower
dynamics + canonicity in one object	$\oint(p\,\mathrm{d}q-H\,\mathrm{d}t)$ , eq. (5)	loops on a trajectory tube; gateway to Hamilton–Jacobi
a practical canonicity test	compare $\oint p\,\mathrm{d}q$ with $\oint P\,\mathrm{d}Q$	handy when the map is given geometrically

Derivation AWhy the loop integral survives every canonical transformation

We are proving the invariance statement $\oint_C \sum_i p_i\,\mathrm{d}q_i = \oint_C \sum_i P_i\,\mathrm{d}Q_i$ for an arbitrary closed loop $C$ and any canonical transformation $(q,p)\to(Q,P)$ . The strategy has two prongs that illuminate each other: first the generating-function argument — the two integrands differ by an exact differential, which a closed loop cannot see — and then the Stokes argument, which converts the statement into preservation of the two-form $\omega$ and so ties it to the symplectic condition. The subtlety to watch is the word relative: the one-form itself is NOT invariant, only its closed-loop integral is, and the exactness of the difference is precisely why.

Step 1. Recall the defining property of a canonical transformation in differential form. For any canonical map there is a generating function $F$ with

\sum_i p_i\,\mathrm{d}q_i-\sum_i P_i\,\mathrm{d}Q_i=\mathrm{d}F

Why: this is the exactness statement carried by the canon of generating-function theory — for a time-independent map the difference of the two canonical one-forms is the differential of the single function that generates the transformation. It holds whichever of the four types generates the map.

Step 2. Integrate both sides around the closed loop $C$ :

\oint_C\Big(\sum_i p_i\,\mathrm{d}q_i-\sum_i P_i\,\mathrm{d}Q_i\Big)=\oint_C\mathrm{d}F = 0

Why the right side vanishes: $F$ is a single-valued function of phase-space position, and the integral of an exact differential around any closed path returns the function to its starting value — net change zero. This single line is the whole theorem: the loop integrals in old and new coordinates are equal.

Step 3. Now see the same fact geometrically. Apply Stokes' theorem to the one-form $\theta=\sum_i p_i\,\mathrm{d}q_i$ over a surface $S$ with $\partial S = C$ . Since $\mathrm{d}\theta=\sum_i \mathrm{d}p_i\wedge\mathrm{d}q_i=\omega$ ,

\oint_{C}\theta=\iint_S \omega

Why this reformulation matters: it shifts the invariance from a property of loops to a property of the two-form itself — if a map preserves $\omega$ , every surface integral of $\omega$ is preserved, hence every loop integral of $\theta$ .

Step 4. Run the logic both directions. If the map is canonical, Steps 1–2 show every loop integral is preserved, hence by Step 3 every surface flux of $\omega$ is preserved, hence $\omega$ itself is preserved (shrink the surface to a point to localize). Conversely, if $\omega$ is preserved, Step 3 makes every loop integral invariant. Loop invariance for all loops, preservation of $\omega$ , and canonicity are one and the same property. Note: the exactness $\mathrm{d}\theta=\omega$ is immediate since $\mathrm{d}(p_i\,\mathrm{d}q_i)=\mathrm{d}p_i\wedge\mathrm{d}q_i$ term by term.

Go deeper · Derivation B · 2 minHamiltonian flow preserves the Poincare-Cartan invariant

Derivation BHamiltonian flow preserves the Poincare-Cartan invariant

The goal is the Poincaré–Cartan theorem: the loop integral $\oint_C(\sum_i p_i\,\mathrm{d}q_i - H\,\mathrm{d}t)$ takes the same value on every closed loop drawn around the same tube of trajectories in extended phase space. The strategy is a Stokes argument on the side wall of the tube: the difference of two loop integrals is the flux of the two-form $\mathrm{d}\Theta$ through the wall, and Hamilton's equations make that flux vanish identically — the flow direction is precisely the direction the two-form cannot see. Watch how the equations of motion enter: not as an assumption about the loops (which are arbitrary), but about the wall, which is woven from solution curves.

Step 1. Set up on extended phase space with coordinates $(q_i,p_i,t)$ and the Poincaré–Cartan one-form $\Theta=\sum_i p_i\,\mathrm{d}q_i-H\,\mathrm{d}t$ . Let $C_1$ and $C_2$ be two closed loops encircling the same bundle of solution curves, like two rings slipped over the same rope; the surface $\Sigma$ between them is swept out by the trajectories joining loop to loop.

Step 2. Apply Stokes' theorem to the strip $\Sigma$ , whose boundary is $C_2 - C_1$ :

\oint_{C_2}\Theta-\oint_{C_1}\Theta=\iint_{\Sigma}\mathrm{d}\Theta

Why the difference appears: the two rings bound the strip with opposite orientations, so Stokes delivers exactly the discrepancy we want to kill.

Step 3. Compute the exterior derivative:

\mathrm{d}\Theta=\sum_i \mathrm{d}p_i\wedge\mathrm{d}q_i-\mathrm{d}H\wedge\mathrm{d}t

with $\mathrm{d}H=\sum_i\big(\tfrac{\partial H}{\partial q_i}\,\mathrm{d}q_i+\tfrac{\partial H}{\partial p_i}\,\mathrm{d}p_i\big)+\tfrac{\partial H}{\partial t}\,\mathrm{d}t$ . Why no extra terms: $\mathrm{d}$ is linear and $\mathrm{d}(H\,\mathrm{d}t)=\mathrm{d}H\wedge\mathrm{d}t$ since $\mathrm{d}(\mathrm{d}t)=0$ .

Step 4. Feed the two-form the flow direction. Every point of $\Sigma$ lies on a trajectory with tangent vector $X=\partial_t+\sum_i(\dot q_i\,\partial_{q_i}+\dot p_i\,\partial_{p_i})$ , where $\dot q_i=\partial H/\partial p_i$ and $\dot p_i=-\partial H/\partial q_i$ . Contracting:

\iota_X\,\mathrm{d}\Theta=\sum_i\big(\dot p_i+\tfrac{\partial H}{\partial q_i}\big)\mathrm{d}q_i-\sum_i\big(\dot q_i-\tfrac{\partial H}{\partial p_i}\big)\mathrm{d}p_i = 0

Why it vanishes term by term: each parenthesis is one of Hamilton's equations. This is the memorable heart of the theorem — Hamilton's equations ARE the statement that the flow direction annihilates $\mathrm{d}\Theta$ . One can even define Hamiltonian dynamics this way.

Step 5. Conclude that the wall carries no flux. At every point of $\Sigma$ , one of the two tangent directions is $X$ , and $\mathrm{d}\Theta$ contracted with $X$ gives zero, so the integrand of $\iint_\Sigma \mathrm{d}\Theta$ vanishes pointwise. Hence $\oint_{C_1}\Theta=\oint_{C_2}\Theta$ for any two loops around the tube.

Step 6. Extract the special case that names the theorem. Choose both loops at frozen times ( $\mathrm{d}t=0$ along each): the $H\,\mathrm{d}t$ term drops and the statement becomes $\oint_{C(0)} \sum_i p_i\,\mathrm{d}q_i = \oint_{C(t)} \sum_i p_i\,\mathrm{d}q_i$ — the relative invariant is conserved by time evolution, i.e. the flow is a canonical transformation. Note: the contraction identity $\iota_X\mathrm{d}\Theta=0$ reproduces both of Hamilton's equations, confirming the sign convention in $\Theta$ .

Figure

Two rings around one tube of trajectories — the Poincare-Cartan integral is the same on both

Go deeper · Derivation C · 1 minLoop invariance is exactly the symplectic condition

Derivation CLoop invariance is exactly the symplectic condition

We close the circle of equivalences by proving that preservation of $\oint_C p\,\mathrm{d}q$ for all loops is the same statement as the symplectic condition $M^{\mathsf T}JM=J$ on the Jacobian $M$ of the map. The strategy is to push the two-form $\omega$ through the coordinate change and demand that its matrix representation is untouched; the loop statement then rides along via Stokes. The point of care is directional bookkeeping — differentials transform with the Jacobian of the inverse map, and one inversion at the end untangles it.

Step 1. Package the coordinates as $\xi=(q_1,\dots,q_n,p_1,\dots,p_n)^{\mathsf T}$ and write the two-form with the standard symplectic matrix:

\omega=-\tfrac12\,\mathrm{d}\xi^{\mathsf T}\wedge J\,\mathrm{d}\xi,\qquad J=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix}

Why the packaging works: expanding the quadratic wedge expression reproduces $\sum_i \mathrm{d}p_i\wedge\mathrm{d}q_i$ exactly, with the $\tfrac12$ compensating the double count from antisymmetry.

Step 2. Let $\eta=(Q,P)^{\mathsf T}$ with Jacobian $M=\partial\eta/\partial\xi$ . Differentials transform as $\mathrm{d}\eta = M\,\mathrm{d}\xi$ , so $\mathrm{d}\xi=M^{-1}\mathrm{d}\eta$ , and substituting into Step 1:

\omega=\tfrac12\,\mathrm{d}\eta^{\mathsf T}\wedge\big(M^{-\mathsf T}J M^{-1}\big)\,\mathrm{d}\eta

Why we substitute the inverse: we are re-expressing the OLD form in NEW differentials to compare it with what the new coordinates would natively build.

Step 3. Demand that the re-expressed form equals the native new form $\tfrac12\,\mathrm{d}\eta^{\mathsf T}\wedge J\,\mathrm{d}\eta$ . Since a two-form is determined by its value on all pairs of tangent vectors, equality for every surface (equivalently every loop, by Stokes) forces the matrix identity

M^{-\mathsf T}JM^{-1}=J

Step 4. Invert. Multiplying by $M^{\mathsf T}$ on the left and $M$ on the right gives the symplectic condition in its standard dress:

M^{\mathsf T}JM=J

Why this is the full equivalence: every implication above is reversible — a symplectic Jacobian preserves $\omega$ pointwise, hence all surface integrals, hence all loop integrals; and loop invariance for all loops localizes back to pointwise preservation of $\omega$ .

Step 5. Harvest a corollary for free. Taking determinants of $M^{\mathsf T}JM=J$ gives $(\det M)^2=1$ , and continuity with the identity map selects $\det M=+1$ . The volume element $\mathrm{d}^n Q\,\mathrm{d}^n P=|\det M|\,\mathrm{d}^n q\,\mathrm{d}^n p$ is therefore preserved — Liouville's theorem drops out of the equivalence as the determinant shadow of the symplectic condition. Note: for $n=1$ the symplectic condition reduces to $\det M = 1$ , confirming that one-degree-of-freedom canonical maps are exactly the area-preserving ones.

Limiting cases

Case	Result	Meaning
one degree of freedom	$\oint_C p\,\mathrm{d}q=$ area enclosed by $C$	the relative invariant is literally phase-plane area; canonical maps of the plane = area-preserving maps
top of the tower, $k=n$	$\int\mathrm{d}^n q\,\mathrm{d}^n p=\text{const}$	phase-space volume; Liouville's theorem as the final Poincaré invariant
loop at frozen time	$\oint(p\,\mathrm{d}q-H\,\mathrm{d}t)\to\oint p\,\mathrm{d}q$	with $\mathrm{d}t=0$ the Poincaré–Cartan invariant reduces to the relative invariant
point transformation $Q=Q(q)$	$\sum_i p_i\,\mathrm{d}q_i=\sum_i P_i\,\mathrm{d}Q_i$ exactly	the one-form itself is invariant ( $\mathrm{d}F=0$ ) — configuration-space relabelings are trivially canonical
periodic orbit as the loop	$\oint p\,\mathrm{d}q = 2\pi J$	the invariant evaluated on an orbit is the action variable — the bridge to action-angle theory

Common trapThe integrand is not invariant — only the closed-loop integral is

The single most common misreading of this subject: $\sum_i p_i\,\mathrm{d}q_i$ is not preserved by canonical transformations. It changes by an exact differential $\mathrm{d}F$ — that is the defining property of a generating function — so integrals over open arcs change too. Only around a closed loop does $\oint\mathrm{d}F=0$ wipe out the difference; hence the name relative integral invariant. Two more traps live nearby. First, the higher invariants integrate the wedge power $\omega^{\wedge k}$ — an antisymmetric, orientation-sensitive object — not any naive $k$ -th power of an area; forgetting the antisymmetry double-counts overlapping projections. Second, volume preservation ( $\det M = 1$ ) is strictly weaker than canonicity when $n\ge 2$ : every canonical map preserves volume, but most volume-preserving maps fail to preserve $\omega$ — see the worked example below.

Worked examples

1 · Loop integrals as a canonicity meter (confidence builder)

A harmonic oscillator with $m=1\ \mathrm{kg}$ , $\omega_0=2\ \mathrm{rad/s}$ is on the energy- $E$ orbit with $E=0.5\ \mathrm{J}$ . Take that orbit as the loop $C$ . (a) Compute $\oint_C p\,\mathrm{d}q$ . (b) Test the rescaling map $Q=2q$ , $P=p$ for canonicity with the loop meter. (c) Repair it.

Step 1. The orbit $\tfrac{p^2}{2m}+\tfrac12 m\omega_0^2 q^2=E$ is an ellipse with semi-axes $q_{\max}=\sqrt{2E/(m\omega_0^2)}=\sqrt{2\times0.5/(1\times4)}=0.5\ \mathrm{m}$ and $p_{\max}=\sqrt{2mE}=\sqrt{2\times1\times0.5}=1.0\ \mathrm{kg\,m/s}$ .

Step 2. The loop integral is the enclosed area: $\oint_C p\,\mathrm{d}q=\pi\,q_{\max}p_{\max}=\pi\times0.5\times1.0=\boxed{1.571\ \mathrm{J\,s}}$ . (Check against the standard formula: area $=2\pi E/\omega_0=2\pi\times0.5/2=1.571\ \mathrm{J\,s}$ — same number.)

Step 3. Apply $Q=2q$ , $P=p$ . The image of the orbit is an ellipse with semi-axes $Q_{\max}=1.0\ \mathrm{m}$ , $P_{\max}=1.0\ \mathrm{kg\,m/s}$ , so $\oint P\,\mathrm{d}Q=\pi\times1.0\times1.0=3.142\ \mathrm{J\,s}\neq1.571\ \mathrm{J\,s}$ . The loop meter reads double: the map is not canonical. (The Jacobian test agrees: $M=\mathrm{diag}(2,1)$ , $\det M=2\neq1$ .)

Step 4. Repair by rescaling the momentum oppositely: $Q=2q$ , $P=p/2$ . Now $\oint P\,\mathrm{d}Q=\pi\times1.0\times0.5=\boxed{1.571\ \mathrm{J\,s}}$ — the invariant is restored, and indeed $\det M=2\times\tfrac12=1$ with $M^{\mathsf T}JM=J$ . The general lesson: $Q=\alpha q$ , $P=p/\alpha$ is canonical for any $\alpha\neq0$ ; stretching $q$ is fine as long as $p$ shrinks to compensate the area.

2 · The slowly shortened pendulum — an integral invariant doing real work (exam level)

A pendulum of length $L_0=1.00\ \mathrm{m}$ swings with amplitude $\theta_0=5.0°$ . Its string is pulled up through the pivot slowly — over many periods — until $L=0.50\ \mathrm{m}$ . Using the near-invariance of the orbit's loop integral $J=\oint p_\theta\,\mathrm{d}\theta$ , find the new amplitude, the new frequency, and the factor by which the energy grew. ( $g=9.81\ \mathrm{m/s^2}$ .)

Step 1. For small oscillations the pendulum is a harmonic oscillator in $\theta$ with effective mass $mL^2$ and frequency $\omega=\sqrt{g/L}$ . From Example 1's geometry, the orbit's loop integral at energy $E$ is $\oint p_\theta\,\mathrm{d}\theta = 2\pi E/\omega$ .

Step 2. Slow (adiabatic) change means the orbit at every instant is close to the frozen-parameter orbit, and the loop integral — the phase-plane area — is the adiabatic invariant. Hence $E/\omega=\text{const}$ , so $E\propto\omega\propto L^{-1/2}$ .

Step 3. Express the energy through the amplitude: $E=\tfrac12 m g L\,\theta_{\max}^2$ . Then $m g L\,\theta_{\max}^2\propto L^{-1/2}$ forces $\theta_{\max}^2\propto L^{-3/2}$ , i.e. $\theta_{\max}\propto L^{-3/4}$ .

Step 4. Numbers. Amplitude: $\theta_{\max}=5.0°\times(1.00/0.50)^{3/4}=5.0°\times2^{0.75}=5.0°\times1.682=\boxed{8.4°}$ . Frequency: $\omega_0=\sqrt{9.81/1.00}=3.13\ \mathrm{rad/s}$ rises to $\omega=\sqrt{9.81/0.50}=4.43\ \mathrm{rad/s}$ (period $2.01\ \mathrm{s}\to1.42\ \mathrm{s}$ ). Energy: grows by $\omega/\omega_0=\sqrt{2}=\boxed{1.41\times}$ .

Step 5. Where the extra energy comes from: the hand pulling the string does net positive work against the time-averaged string tension (which exceeds $mg$ on a swinging pendulum). The invariant organizes the bookkeeping — energy is not conserved, frequency is not conserved, but their ratio $E/\omega$ , the loop integral over $2\pi$ , is protected to exponential accuracy in the slowness. This is the same $\oint p\,\mathrm{d}q$ that Bohr and Sommerfeld set equal to $nh$ in the old quantum theory — adiabatic invariants were chosen as the quantizable quantities precisely because slow perturbations cannot change them.

Origins · 1 minHenri Poincaré and how this was discovered

Henri Poincaré

1854–1912 · Sorbonne, Paris

Poincaré was the last universalist — a mathematician who worked in nearly every branch of the subject while also reshaping celestial mechanics, and the integral invariants are his signature move: extract from an unsolvable dynamical problem the quantities that cannot change. His prize memoir of 1890 on the three-body problem — famously corrected after a printing was recalled at his own expense — introduced integral invariants alongside the recurrence theorem and the first glimpse of homoclinic chaos, all tools for reasoning about dynamics you cannot integrate. The three volumes of Les méthodes nouvelles de la mécanique céleste (1892–1899) systematized them. For Poincaré the invariants were the skeleton of mechanics: what survives when formulas fail. Twentieth-century mathematics agreed — his two-form $\omega$ became the defining object of symplectic geometry, and his invariants its first theorems.

History

1838

— Liouville proves that Hamiltonian flow preserves phase-space volume, the result Poincaré will later place at the top of his tower of invariants.

1890

— Poincaré's prize memoir on the three-body problem introduces integral invariants and the recurrence theorem as tools for unsolvable dynamics.

1899

— The final volume of Les méthodes nouvelles de la mécanique céleste gives the systematic theory of integral invariants of all orders.

1917

— Ehrenfest's adiabatic hypothesis singles out the loop invariant

\oint p\,\mathrm{d}q

as the quantity old quantum theory should quantize.

1985

— Gromov's non-squeezing theorem reveals that the two-form invariants constrain canonical maps far more tightly than volume conservation — the "symplectic camel" cannot pass through the eye of a needle.

Going further

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §9.4 (Poincaré integral invariants) and §9.8 · Landau & Lifshitz, Mechanics, 3rd ed., §49 (Liouville's theorem) and the canonical-invariants discussion · Arnold, Mathematical Methods of Classical Mechanics, Ch. 9 (integral invariants, Poincaré–Cartan) · Morin, Introduction to Classical Mechanics, Hamiltonian chapter · Taylor, Classical Mechanics, Ch. 13 (Hamiltonian mechanics) Key structural identities underlying this article: $\mathrm{d}(\sum p_i\,\mathrm{d}q_i)=\sum\mathrm{d}p_i\wedge\mathrm{d}q_i$ (the Stokes equivalence); $\iota_X\mathrm{d}\Theta=0$ via Hamilton's equations (the Poincaré–Cartan theorem); $(\det M)^2=1$ from $M^{\mathsf T}JM=J$ (Liouville volume invariance); and the standard structural facts $J^2=-I$ , $\det J=1$ , and the Bohr–Sommerfeld form $\oint p\,\mathrm{d}q=nh$ . Worked-example numbers on this page ( $\oint p\,\mathrm{d}q = 1.571\ \mathrm{J\,s}$ for the SHO orbit; $\theta_{\max}=8.4°$ , $\omega = 4.43\ \mathrm{rad/s}$ , energy factor $1.41$ for the shortened pendulum) follow directly from the boxed formulas.

Builds on: Definition of Canonical Transformations · Symplectic Condition · Generating Functions and the Four Types · Fundamental Poisson Brackets Leads to: Symplectic Group and Phase-Space Geometry · Poisson-Bracket Formulation of Dynamics · Poisson Brackets as a Test for Canonicity

Is the one-form p dq itself invariant under canonical transformations?

No — and internalizing this is half the subject. Under a canonical map the one-form changes by an exact differential: $\sum p_i\,\mathrm{d}q_i-\sum P_i\,\mathrm{d}Q_i=\mathrm{d}F$ , where $F$ is the generating function. An integral over an open arc therefore changes by $F(\text{end})-F(\text{start})$ , which is generally nonzero. Only a closed loop kills the exact piece, because the round trip returns $F$ to its starting value. That is exactly what the qualifier relative in "relative integral invariant" records: invariance relative to closed chains only. The two-form $\omega=\mathrm{d}\theta$ , by contrast, is absolutely invariant — pointwise, no loops needed.

What is the actual difference between a relative and an absolute integral invariant?

A relative invariant is an integral that is preserved only when taken over closed chains (loops, closed surfaces); an absolute invariant is preserved over arbitrary regions. The loop integral $\oint p\,\mathrm{d}q$ is relative because its integrand shifts by $\mathrm{d}F$ . Its Stokes partner $\iint_S\omega$ is absolute: $\omega$ itself is preserved pointwise, so the flux through any surface — closed or not — is unchanged. The pattern generalizes: each relative invariant of degree $k$ corresponds via Stokes to an absolute invariant of degree $k+1$ . Poincaré organized his whole hierarchy this way, and Cartan later showed the extended-phase-space form $p\,\mathrm{d}q-H\,\mathrm{d}t$ makes the time-dependent bookkeeping automatic.

Is the loop invariant the same thing as the action variable J?

They are the same integral evaluated on different loops. The invariant $\oint_C p\,\mathrm{d}q$ is defined for any closed curve $C$ — a purely kinematic chain of phase points. The action variable of action-angle theory is the special case where $C$ is an actual periodic orbit (or a fundamental loop on an invariant torus), conventionally normalized as $J=\tfrac{1}{2\pi}\oint p\,\mathrm{d}q$ . The canonical invariance of the general loop integral is what makes the action variable well-defined and coordinate-independent — and its adiabatic near-invariance (worked example 2) is why slowly driven systems keep $E/\omega$ fixed while both $E$ and $\omega$ drift.

If canonical maps preserve phase-space volume, why is volume preservation not enough to make a map canonical?

For one degree of freedom it is enough — area and symplectic area coincide. But for $n\ge2$ canonicity demands that the two-form $\omega$ be preserved, i.e. the sum of oriented shadow areas of every little parallelogram, which is vastly more restrictive than preserving one number (the total volume). A counting argument makes it stark: volume preservation is a single scalar condition ( $\det M=1$ ), while $M^{\mathsf T}JM=J$ imposes $n(2n-1)$ independent conditions — six versus one already for $n=2$ . Gromov's non-squeezing theorem is the geometric shape of the gap: you can construct volume-preserving maps that squeeze a ball through an arbitrarily thin cylinder in a conjugate plane, but no canonical map can do it.

Does time evolution really count as a canonical transformation? Nothing is being "transformed"…

It does, and the identification is load-bearing. Fix a duration $\tau$ and define the map $g_\tau$ that sends each initial condition $(q_0,p_0)$ to the phase point $(q(\tau),p(\tau))$ reached by flowing along Hamilton's equations. That map — from phase space to itself — satisfies the symplectic condition (the Poincaré–Cartan derivation on this page proves its loop invariance directly), so it is canonical with the Hamiltonian's flow as generator. Every conservation statement then cascades: loops keep their $\oint p\,\mathrm{d}q$ , volumes keep their measure (Liouville), and any region's higher invariants ride along. This is the cleanest answer to "why is the phase fluid incompressible": because motion itself is a canonical re-coordinatization.

Why does the Poincare-Cartan form have a minus sign in front of H dt?

The sign makes $\Theta=p\,\mathrm{d}q-H\,\mathrm{d}t$ the integrand of the phase-space action $\int(p\,\dot q-H)\,\mathrm{d}t$ — the Legendre-transformed Lagrangian — so its stationarity reproduces Hamilton's equations. Geometrically, the minus sign is what makes the contraction $\iota_X\mathrm{d}\Theta$ vanish along the flow: flip it and the Step-4 cancellation in the derivation fails, the tube wall carries flux, and the theorem dies. A tidy way to remember it: in extended phase space, $-H$ plays the role of the momentum conjugate to the coordinate $t$ , so $\Theta$ is just " $\sum p\,\mathrm{d}q$ " with the time pair included on the same footing.

Do the integral invariants survive when the Hamiltonian depends on time?

Yes — this is precisely what the Poincaré–Cartan construction buys. For time-dependent $H$ the flow map from time $t_1$ to $t_2$ is still canonical, so the frozen-time loop integral $\oint p\,\mathrm{d}q$ is still conserved, and the extended-phase-space invariant $\oint(p\,\mathrm{d}q-H\,\mathrm{d}t)$ handles even loops that wander in time. What time dependence destroys is energy conservation, not symplecticity: $H$ itself changes along trajectories, but $\omega$ has no idea. This split — geometry protected, energy not — is exactly what the adiabatic-invariant worked example exploits: pull the pendulum string and $E$ climbs, but the loop area $2\pi E/\omega$ holds.

What actually breaks if the system is dissipative?

Everything in this article, immediately and diagnosably. Add a drag force and the flow is no longer Hamiltonian: phase-space volume contracts onto attractors (violating the top invariant), loop areas shrink ( $\tfrac{\mathrm{d}}{\mathrm{d}t}\oint p\,\mathrm{d}q<0$ for linear damping), and no canonical structure survives. This failure is useful in both directions. Numerically, monitoring a "conserved" loop area is a sharp detector of hidden dissipation — physical or algorithmic — in a simulation. Theoretically, it marks the boundary of the formalism: statistical mechanics built on Liouville's theorem applies to closed Hamiltonian systems, and open, damped systems need different tools (Fokker–Planck operators, contact geometry) precisely because Poincaré's invariants are gone.

Canonical Invariants and Poincaré Integral Invariants

Where this lives today

The physical picture

The core equations

Which equation answers which question

Limiting cases

Worked examples

1 · Loop integrals as a canonicity meter (confidence builder)

2 · The slowly shortened pendulum — an integral invariant doing real work (exam level)

Going further

Before this

Where this leads