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Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, $H = h (I)$ and linear motion on the tori

Physical scope

Action-angle coordinates for a completely integrable datum of a declared hyperregular potential presentation of the framework's force branch: near a compact connected regular fibre and over a ball of regular values on which the integral map is a proper submersion with connected fibres, there are period-one angle and action coordinates of the presentation's two-form with $ω = \sum_{i} d θ_{i} \land d I_{i}$ ; fibrewise-constant functions, in particular the datum and the branch Hamiltonian, depend on the actions alone; and, with zero residual force, Hamiltonian solutions become constant-action, linear-angle flows: these correspond to Cartesian model motions or, in generalized coordinates, to substituted Lagrange solutions; a lower-dimensional immersion alone does not make the latter ambient unconstrained motions. The conclusion is conditional on the Axiom of Countable Choice, on the complete integrability of the datum and on all the fibre hypotheses; it is local over the ball of regular values, asserts no global coordinates on the phase domain, and carries no empirical content. The identification of the coordinates with classical period integrals is a normalization convention, not part of the theorem.

Statement

$(I, θ) : U_{0} ⟶ A \times T^{n}, T^{n} = R^{n} / Z^{n},$

of class $C^{\infty}$ , diffeomorphic onto their image, such that:

(form) the angles are period-one and the presentation's two-form takes the canonical action-angle form $ω = \sum_{i = 1}^{n} d θ_{i} \land d I_{i}$ on $U_{0}$ , in the sense of Action and angle coordinates;
(dimension) each action coordinate $I_{i}$ has dimension $μ ℓ^{2} τ^{- 1}$ (action) and each angle $θ_{i}$ is dimensionless, consistently with the action dimension of the presentation's two-form in its original canonical charts (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework);
(fibrewise constants) every $C^{\infty}$ function on $U_{0}$ that is constant on the fibres of $π$ is a function of $I$ alone; in particular $F_{i} = f_{i} (I)$ for $C^{\infty}$ functions $f_{i}$ and the branch Hamiltonian is $H = h (I)$ with $h = f_{1}$ ;
(linear motion) the solutions of the Hamiltonian system of $h$ in the chart $(I, θ)$ , on the potential branch $d \equiv 0$ , are exactly the momentum-phase curves of Cartesian model motions, or in a generalized-coordinate chart the solutions of its substituted Lagrange equations on $U_{0}$ , by clauses 1-2 of Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential. A lower-dimensional immersion requires a separate ideal-constraint premise to identify these with constrained ambient motions. In either case and they satisfy $\dot{I} = 0, \dot{θ} = \frac{\partial h}{\partial I} (I), θ (t) = θ (0) + t \frac{\partial h}{\partial I} (I (0)) (m o d Z^{n}),$ writing $ν = \partial h / \partial I (I (0))$ , a positive $T$ is a period exactly when $T ν \in Z^{n}$ . The linear-motion formula alone does not assert a periodic-versus-full-density dichotomy: nonperiodic motion can remain in a proper invariant subtorus when the frequency vector has integer relations.
(radian rescaling) with the rescaled coordinates $θ^{r a d} : = 2 π θ$ and $I^{r a d} : = I / (2 π)$ one has $ω = \sum_{i} d θ_{i}^{r a d} \land d I_{i}^{r a d}$ , the angles $θ^{r a d}$ are $2 π$ -periodic, and $H = \tilde{h} (I^{r a d})$ with $\tilde{h} (I^{r a d}) = h (2 π I^{r a d})$ and ${\dot{θ}}^{r a d} = \partial \tilde{h} / \partial I^{r a d}$ ; the components $\partial h / \partial I_{i}$ are cyclic frequencies (cycles per unit time), not angular rates, the angular rates being $2 π \partial h / \partial I_{i}$ .

The statement is conditional on the adopted force branch, the declared presentation, the smooth completely integrable datum, the fibre hypotheses and ${A C}_{ω}$ ; it asserts no global action-angle coordinates on $W_{0}$ , no statement about critical or noncompact fibres and no empirical content.

Facts & Assumptions

Given: ${A C}_{ω}$ , the presentation on its chart, the completely integrable datum $F$ with $F_{1} = H$ on $W_{0}$ , the regular value $b_{0}$ and its compact connected regular fibre $N$ , and the saturated neighbourhood $U$ with the proper submersion $π = F ∣_{U}$ over the ball $B$ with connected fibres.

[F1]

Setting and identification (Completely integrable data of a declared presentation: $n$ smooth phase functions with $F_{1} = H$ , pairwise in involution and independent on the regular locus, Completely integrable Hamiltonian system, Symplectic form and symplectic manifold, The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: $Φ_{X}^{*} λ = \sum P_{a} d q^{a}$ and $Φ_{X}^{*} ω = \sum d q^{a} \land d P_{a}$ ): $(W_{0}, ω)$ is a $2 n$ -dimensional symplectic manifold, and $F$ is a completely integrable Hamiltonian system on it with $F_{1} = H$ ; the fibre $N$ is nonempty, compact, connected and regular, and $π = F ∣_{U}$ is a proper submersion onto the ball $B$ with connected fibres.

[F2]

The mathematical action-angle theorem (Action-angle coordinates on a saturated proper submersion neighbourhood of a compact connected regular fibre, with the two-form primitive supplied locally, Action and angle coordinates): under the hypotheses of [F1] and ${A C}_{ω}$ , after restricting to $U_{0} = π^{- 1} (B_{0})$ there are period-one action-angle coordinates $(I, θ) : U_{0} \to A \times T^{n}$ , of class $C^{\infty}$ , where $A = I (B_{0})$ is the action image and $B_{0}$ remains in the original integral-value coordinates, with $ω = \sum_{i} d θ_{i} \land d I_{i}$ , and every $C^{\infty}$ function on $U_{0}$ constant on the fibres of $π$ — in particular each $F_{i}$ — is a function of $I$ alone.

[F3]

Hamiltonian equations in canonical coordinates (The Hamiltonian field of a $C^{2}$ function on a phase domain is $J \nabla H$ , and the Poisson bracket is $\sum_{i} (\partial_{q^{i}} f \partial_{p_{i}} g - \partial_{p_{i}} f \partial_{q^{i}} g)$ , The Hamiltonian field of a phase function for a potential presentation: the unique field $X_{f}$ with $ι_{X_{f}} ω = d f$ , and $X_{H} = (H_{p}, - H_{x})$ for the branch Hamiltonian clause (a), Motion of a completely integrable Hamiltonian is linear on invariant tori, Hamiltonian vector field and Hamiltonian function): in a canonical chart $(q, p)$ of a presentation two-form the field $X_{K}$ of a $C^{2}$ function $K$ has the components $X_{K} = (K_{p}, - K_{q})$ , so its integral curves satisfy $\dot{q} = K_{p}$ , $\dot{p} = - K_{q}$ ; in action-angle coordinates with $K = h (I)$ this gives $\dot{I} = 0$ , $\dot{θ} = \partial h / \partial I (I)$ on the torus, with the displayed linear formula.

[F4]

Dimension of the two-form (Canonical coordinates and the canonical form of the two-form for a potential presentation: the coordinate expression $ω = \sum_{a = 1}^{m} d θ^{a} \land d θ^{m + a}$ with $m$ the chart's half-dimension, and the model's own momentum charts clause (b), The canonical one-form and the two-form of a potential presentation: $λ = p \cdot d x$ on the Cartesian momentum phase chart and the induced forms on each generalized-coordinate chart, The two-form of a potential presentation is closed and nondegenerate, and each generalized-coordinate momentum chart carries the same symplectic structure: $Φ_{X}^{*} λ = \sum P_{a} d q^{a}$ and $Φ_{X}^{*} ω = \sum d q^{a} \land d P_{a}$ clause 2, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework): in each canonical chart $(q, p)$ of the presentation, $ω = \sum_{a} d q^{a} \land d p_{a}$ and the momentum $p_{a}$ has dimension $μ ℓ^{2} τ^{- 1} γ_{a}^{- 1}$ while $q^{a}$ has dimension $γ_{a}$ ; hence every coefficient form $d q^{a} \land d p_{a}$ has the constant dimension $μ ℓ^{2} τ^{- 1}$ (action).

[F5]

Radian rescaling (Period-one to radian action-angle rescaling: $θ^{r a d} = 2 π θ$ and $I^{r a d} = I / (2 π)$ preserve $ω$ ): with $θ^{r a d} = 2 π θ$ , $I^{r a d} = I / (2 π)$ one has $ω = \sum_{i} d θ_{i}^{r a d} \land d I_{i}^{r a d}$ , and if $H = h (I)$ then $H = \tilde{h} (I^{r a d})$ with $\tilde{h} (I^{r a d}) = h (2 π I^{r a d})$ and ${\dot{θ}}_{i}^{r a d} = \partial \tilde{h} / \partial I_{i}^{r a d}$ .

[F6]

Model reading (Cartesian model motions on the potential branch and declared generalized-coordinate Lagrange equations integrate their Hamiltonian fields; the forced Cartesian field is Hamiltonian exactly when the residual force has a velocity-independent potential clauses 1-2, Hamilton's equations of the mechanical presentation are equivalent to the model's equation of motion, with residual forces as momentum sources and the coordinate form on every chart clauses 1 and 3, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta): with $d \equiv 0$ , the Hamiltonian curves correspond to Cartesian model motions in a Cartesian chart and to substituted Lagrange solutions in a generalized-coordinate chart. An arbitrary lower-dimensional immersion alone does not identify the latter with ambient unconstrained motions.

[A1]

${A C}_{ω}$ is The Axiom of Countable Choice ( ${A C}_{ω}$ ); it is consumed through [F2], whose statement assumes it.

Proof

technique · direct; apply the mathematical action-angle theorem to the presentation's symplectic chart, fix the dimension bookkeeping from the two-form, and read the motion off the canonical Hamiltonian field

1.1A1F1F2

Clauses 1 and 3: by [F1] the hypotheses of [F2] are in force, so after restricting the original base to $B_{0}$ and the saturated neighbourhood to $U_{0}$ there is a $C^{\infty}$ diffeomorphism $(I, θ) : U_{0} \to A \times T^{n}$ onto its image with $ω = \sum_{i} d θ_{i} \land d I_{i}$ and every $C^{\infty}$ function constant on the fibres of $π$ (in particular each $F_{i}$ , and $H = F_{1}$ ) a function of $I$ alone; write $F_{i} = f_{i} (I)$ and $H = h (I) = f_{1} (I)$ .

1.2F2F4algebra

Clause 2: by [F4] the two-form has action dimension; in the coordinates $(I, θ)$ the form is $\sum_{i} d θ_{i} \land d I_{i}$ with the angles period-one and dimensionless by [F2] and Action and angle coordinates, so comparing dimensions term by term gives $[I_{i}] = μ ℓ^{2} τ^{- 1}$ .

2.1F3F6step 1.1algebra

Clause 4: in the canonical chart $(θ, I)$ the field of $h (I)$ has components $X_{h} = (\partial h / \partial I, 0)$ by [F3], so $\dot{I} = 0$ and $\dot{θ} = \partial h / \partial I (I)$ . Integrating on $R^{n} / Z^{n}$ gives the displayed linear formula. By [F6], with $d \equiv 0$ these integral curves are exactly Cartesian momentum-phase motions, or generalized-coordinate substituted Lagrange solutions, as applicable to the original chart. The action-angle change transports that same Hamiltonian field; no ambient-motion assertion follows from an immersion alone. Write $ν = \partial h / \partial I (I (0))$ . The orbit returns after $T > 0$ exactly when $T ν \in Z^{n}$ , by the defining quotient. If an integer vector $k \neq 0$ satisfies $k \cdot ν = 0$ , the angle combination $k \cdot θ (t)$ remains fixed modulo $Z$ , so no full-torus density follows merely from nonperiodicity. For example a frequency vector $(c, c α, 0)$ with $c \neq 0$ and irrational $α$ keeps the third angle fixed; a positive period would make both $T c$ and $T c α$ integers, forcing their ratio $α$ rational, contradiction. This verifies the precise period criterion and the stated qualification.

3.1F5step 1.1step 2.1algebra

Clause 5: [F5] applied to the coordinates of step 1.1 gives the invariance of the two-form under the rescaling, the transformed Hamiltonian $\tilde{h}$ and the rescaled evolution equation; the relation ${\dot{θ}}^{r a d} = 2 π \dot{θ}$ identifies the cyclic frequencies $\partial h / \partial I_{i}$ of clause 4 with the angular rates $2 π \partial h / \partial I_{i}$ in the radian chart.

4.1step 1.1step 1.2step 2.1step 3.1∎

Collection: clause 1 is step 1.1 (form), clause 2 is step 1.2, clause 3 is step 1.1 (fibrewise constants), clause 4 is step 2.1 and clause 5 is step 3.1; this proves the statement.

Remarks

The action coordinates are not unique. The theorem does not fix additive constants in the actions or constant shifts of the angles: if $(I, θ)$ has the stated properties then so does $(I + c, θ)$ for every $c \in R^{n}$ . In particular the theorem does not by itself single out the classical normalization $I_{i} = \oint_{γ_{i}} p d q$ of the sources; that normalization is a convention fixed in explicit models, and the one-dimensional companion examples of The one-dimensional action variable of a family of two-turning-point energy levels verify it directly in the models treated there, where the period-one action is $2 π$ times the radian action $J_{a}$ of that definition.
Global coordinates are not claimed. The conclusion holds over the neighbourhood $B_{0}$ of original regular values with action-coordinate image $A$ , after restricting the saturated neighbourhood; a compact phase domain with nontrivial period-lattice monodromy admits no global single-valued action-angle chart, as the published obstruction examples cited on the companion page show.
Frequencies. In the period-one convention of clause 1 the numbers $\partial h / \partial I_{i}$ are frequencies in cycles per unit time; the angular rates are their $2 π$ -multiples, per clause 5. The two conventions are related by the explicit rescaling of Period-one to radian action-angle rescaling: $θ^{r a d} = 2 π θ$ and $I^{r a d} = I / (2 π)$ preserve $ω$ and must not be mixed.

Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, $H = h (I)$ and linear motion on the tori

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, H=h(I)H=h(I) and linear motion on the tori

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, $H = h (I)$ and linear motion on the tori