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Adiabatic invariance of the action of a slowly varying one-frequency model: an order- $ε$ drift bound on the $O (1 / ε)$ time scale

Physical scope

Adiabatic invariance of the action coordinate of a declared hyperregular potential presentation of the framework's force branch when the model carries a slow real parameter: if in a one-degree-of-freedom radian action-angle chart the branch Hamiltonian is $h (I, s) + ε k (I, θ, s)$ with nonvanishing frequency $∣ \partial_{I} h ∣ \geq ω_{0}$ and the parameter runs as $s (t) = s_{0} + ε t$ , then along every model solution that stays in the compact chart region the action changes by at most the displayed order- $ε$ bound on $0 \leq t \leq A / ε$ . The statement uses a dimensionless slow parameter, an inverse-time slow rate, energy-dimension h and action-dimension k; it is conditional on C2 open extension data, a uniform fixed canonical chart for the parameter-dependent Hamiltonian on the zero-residual-force branch, the frequency bound and the trapping of the solution; it asserts no exact invariance, no statement across a separatrix or a chart boundary, and no empirical content.

Statement

Adopt the force model with potential presentation $(U, d)$ and mechanical Lagrangian $L = K - U$ of The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions, with the momentum charts and canonical coordinates of 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 $Φ_{X}^{*} ω = \sum d q^{a} \land d P_{a}$ and 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, the Hamiltonian fields of 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 and the branch Hamiltonian $H$ of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions. Fix one momentum chart and let $(I, θ)$ be a fixed, time-independent one-degree-of-freedom canonical chart of the presentation on an open set $U$ of that phase domain, with radian angles and $ω = d θ \land d I$ there, obtained from the period-one chart of Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, $H = h (I)$ and linear motion on the tori by $θ^{r a d} = 2 π θ^{o n e}$ , $I^{r a d} = I^{o n e} / (2 π)$ as in Period-one to radian action-angle rescaling: $θ^{r a d} = 2 π θ$ and $I^{r a d} = I / (2 π)$ preserve $ω$ , suppressing the radian superscripts; $I$ has the action dimension of clause 2 of that theorem. Use the dimensions of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework: $[t] = τ$ , $[I] = μ ℓ^{2} τ^{- 1}$ , $[θ] = [s] = [s_{0}] = [A] = 1$ , $[ε] = [ω_{0}] = τ^{- 1}$ , $[h] = [K_{s}] = μ ℓ^{2} τ^{- 2}$ and $[k] = μ ℓ^{2} τ^{- 1}$ . Thus $ε$ is the prescribed slow parameter rate, $A$ is the dimensionless parameter excursion, and $A / ε$ is a duration. Let $ε, A, ω_{0} > 0$ have these dimensions. Let $\hat{J}, \hat{S} \subseteq R$ be open intervals, and let $J \subseteq \hat{J}$ and $S \subseteq \hat{S}$ be intervals with nonempty interiors (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length), let $K \subseteq J$ and $S_{0} \subseteq S$ be compact intervals, let $s_{0} \in S_{0}$ , and let

$h : \hat{J} \times \hat{S} ⟶ R, k : \hat{J} \times R \times \hat{S} ⟶ R$ be of class $C^{2}$ on these open domains, with $k$ $2 π$ -periodic in its second variable on $J \times R \times S$ ( $C^{k}$ Euclidean maps and diffeomorphisms). Suppose:

(slow datum) for every parameter value $s \in S$ and every $(I, θ) \in J \times R$ in the chart, the branch Hamiltonian of the model restricted to the chart is $K_{s} (I, θ) = h (I, s) + ε k (I, θ, s);$
(nonvanishing frequency) $∣ \partial_{I} h (I, s) ∣ \geq ω_{0}$ for all $(I, s) \in K \times S_{0}$ ;
(parameter path and solution) $s (t) : = s_{0} + ε t$ , and $t \mapsto (I (t), θ (t))$ on $[0, A / ε]$ is the chart expression of a motion of the model (on the potential branch, under the identification of clause 4 of Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, $H = h (I)$ and linear motion on the tori) satisfying $I (t) \in K$ and $s (t) \in S_{0}$ for every $t \in [0, A / ε]$ .

Then for every $t \in [0, A / ε]$ ,

$∣ I (t) - I (0) ∣ \leq ε [2 \sup_{K \times [0, 2 π] \times S_{0}} ∣ g ∣ + A \sup_{K \times [0, 2 π] \times S_{0}} ∣ \partial_{I} g \partial_{θ} k - \partial_{θ} g \partial_{I} k - \partial_{s} g ∣],$ where $g = (k - ⟨ k ⟩) / \partial_{I} h$ is the drift generator and $⟨ k ⟩ (I, s) = \frac{1}{2 π} \int_{0}^{2 π} k (I, θ, s) d θ$ the angle average of $k$ (A one-frequency parameter-dependent action-angle branch, its transformed Hamiltonian and its drift generator). All partial derivatives are those of the ambient open extensions. The generator has dimension $[g] = μ ℓ^{2}$ ; each term inside the second supremum has that same dimension, so both terms multiplied by $ε$ have action dimension. At the included time endpoints, derivatives are relative to the nondegenerate interval $[0, A / ε]$ .

The statement is conditional on the adopted force branch, the declared presentation, the existence of the uniform action-angle chart, the frequency bound and the trapping hypothesis; it asserts no exact invariance of the action, no statement for parameter paths that leave the chart, cross a separatrix or make the frequency vanish, and no empirical content. In particular the order- $ε$ bound is a bound on the stated horizon $[0, A / ε]$ and is not an asymptotic equivalence.

Facts & Assumptions

Given: The presentation, the canonical chart $(I, θ)$ with $ω = d θ \land d I$ , the reals $ε, A, ω_{0}$ , the open extension intervals $\hat{J}, \hat{S}$ , their subintervals $J, S$ and compact sets $K, S_{0}$ , the point $s_{0}$ , the $C^{2}$ functions $h, k$ with $k$ $2 π$ -periodic in $θ$ , the parameter path $s (t) = s_{0} + ε t$ and the solution $(I, θ)$ with the stated trapping.

[F1]

Chart data and dimension (Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, $H = h (I)$ and linear motion on the tori clauses 1, 2 and 4, 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 (a), Dimensions and SI unit conventions of the non-relativistic classical mechanics framework): by Period-one to radian action-angle rescaling: $θ^{r a d} = 2 π θ$ and $I^{r a d} = I / (2 π)$ preserve $ω$ , on $U$ the presentation's two-form is $ω = d θ \land d I$ with radian dimensionless angles and action coordinates of dimension $μ ℓ^{2} τ^{- 1}$ ; in these coordinates the Hamiltonian field of a $C^{2}$ function $K$ has components $X_{K} = (- \partial K / \partial θ, \partial K / \partial I)$ in the ordered coordinates $(I, θ)$ , so the model's motions in the chart satisfy $\dot{I} = - \partial_{θ} K$ , $\dot{θ} = \partial_{I} K$ .

[F2]

The datum of the branch (A one-frequency parameter-dependent action-angle branch, its transformed Hamiltonian and its drift generator): with $K (I, θ, s) = h (I, s) + ε k (I, θ, s)$ , the branch vector field is $(I, θ, s)^{'} = (- ε \partial_{θ} k, \partial_{I} h + ε \partial_{I} k, ε)$ , and a branch solution on an interval is a differentiable curve solving that system.

[F3]

The mathematical drift bound (Adiabatic action drift bound: order- $ε$ change of the action on the $O (1 / ε)$ time scale): for the $C^{2}$ datum $h, k$ with $k$ $2 π$ -periodic in $θ$ , compact intervals $K \subseteq J$ , $S_{0} \subseteq S$ , $s_{0} \in S_{0}$ , reals $ε, A, ω_{0} > 0$ with $∣ \partial_{I} h ∣ \geq ω_{0}$ on $K \times S_{0}$ , and a branch solution on $[0, A / ε]$ with $s (0) = s_{0}$ that stays in $K \times S_{0}$ , the action obeys the displayed order- $ε$ estimate with the drift generator $g = (k - ⟨ k ⟩) / \partial_{I} h$ .

[F4]

Model reading of the chart evolution (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 clause 1, 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 clause 1, The associated first-order system, motions, maximal motions and the Newtonian flow of a force model): when $d \equiv 0$ , the Cartesian momentum-phase curves of model motions are exactly the integral curves of the branch Hamiltonian field; in a generalized-coordinate immersion, Hamiltonian curves instead correspond to the substituted Lagrange equations, and any reconstructed ambient motion satisfies those equations; in the chart $(I, θ)$ these are the curves of [F1].

Proof

technique · direct; identify the model's motion in the fixed canonical chart with a branch solution of the abstract datum, then invoke the mathematical drift bound

1.1F1F2F4givenalgebra

By [F1] and [F4], the chart expression $(I (t), θ (t))$ of the model's motion satisfies $\dot{I} = - \partial_{θ} K_{s (t)}$ and $\dot{θ} = \partial_{I} K_{s (t)}$ for the branch Hamiltonian of hypothesis 1; substituting $K = h + ε k$ gives $\dot{I} = - ε \partial_{θ} k (I, θ, s (t))$ and $\dot{θ} = \partial_{I} h (I, s (t)) + ε \partial_{I} k (I, θ, s (t))$ . With the parameter path $s (t) = s_{0} + ε t$ from hypothesis 3, the triple $(I, θ, s)$ therefore solves the branch system of [F2] on $[0, A / ε]$ .

2.1givenstep 1.1

For the branch solution of step 1.1, the chain-rule calculation in [F3] gives $\frac{d}{d t} (I + ε g) = - ε^{2} (\partial_{I} g \partial_{θ} k - \partial_{θ} g \partial_{I} k - \partial_{s} g)$ . The sign disappears upon taking absolute values. The compact trapping of hypothesis 3 gives $I (t) \in K$ and $s (t) \in S_{0}$ for every $t \in [0, A / ε]$ , and hypothesis 2 gives $∣ \partial_{I} h ∣ \geq ω_{0}$ on $K \times S_{0}$ ; hence all hypotheses of [F3] are met for the branch solution of step 1.1.

3.1F2F3step 1.1step 2.1

Applying [F3] to the branch solution of step 1.1 yields the displayed estimate for every $t \in [0, A / ε]$ , with the drift generator $g = (k - ⟨ k ⟩) / \partial_{I} h$ of the datum; the hypotheses of [F3] were checked in step 2.1.

4.1step 1.1step 3.1∎

The estimate of step 3.1 is the conclusion; collecting, the model's action coordinate drifts by at most order $ε$ on the $O (1 / ε)$ horizon under the stated hypotheses.

Remarks

What the bound does and does not say. The right-hand side is $ε$ times a constant depending only on the datum through the suprema over the compact set $K \times [0, 2 π] \times S_{0}$ and on the rescaled horizon length $A$ ; it grows linearly in $A$ and is uniform in $ε$ . It is not an exact conservation statement: the exact drift of the action is generally nonzero, and the theorem only bounds it.
Hypotheses are physical model assumptions. The existence of the uniform fixed canonical chart for the parameter-dependent Hamiltonian with nonvanishing frequency and the trapping of the solution are hypotheses about the model family, not consequences of complete integrability of a single Hamiltonian: a family can pass through a separatrix or a critical frequency, where the estimate does not apply and the action need not be approximately constant. The companion mathematical page records the same qualification for the abstract bound.

Adiabatic invariance of the action of a slowly varying one-frequency model: an order- $ε$ drift bound on the $O (1 / ε)$ time scale

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Adiabatic invariance of the action of a slowly varying one-frequency model: an order-εε drift bound on the O(1/ε)O(1/ε) time scale

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources

Adiabatic invariance of the action of a slowly varying one-frequency model: an order- $ε$ drift bound on the $O (1 / ε)$ time scale