A one-frequency parameter-dependent action-angle branch, its transformed Hamiltonian and its drift generator

Let $\epsilon >0$, let $\widehat{J},\widehat{S}\subseteq \mathbb{R}$ be open intervals, and let $J\subseteq \widehat{J}$ and $S\subseteq \widehat{S}$ be intervals with nonempty interiors (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). Let $h:\widehat{J}\times \widehat{S}\to \mathbb{R}$ and $k:\widehat{J}\times \mathbb{R}\times \widehat{S}\to \mathbb{R}$ be of class $C^2$ on these open Euclidean domains ($C^k$ Euclidean maps and diffeomorphisms), and suppose $k$ is $2\pi$-periodic in its second variable on $J\times \mathbb{R}\times S$:

$k(I,\theta +2\pi ,s)=k(I,\theta ,s)\text{for all }(I,\theta ,s)\in J\times \mathbb{R}\times S\mathrm{.}$

The transformed Hamiltonian of the datum is

$K(I,\theta ,s):=h(I,s)+\epsilon k(I,\theta ,s),$

its frequency function is

$\omega (I,s):={\mathrm{\partial }}_{I}h(I,s),$

its angle average is $⟨k⟩(I,s):=\frac{1}{2\pi }{\int }_0^{2\pi }k(I,\theta ,s)d\theta$, and, on the set where $\omega \mathrm{\ne }0$, its drift generator is

$g(I,\theta ,s):=\frac{k(I,\theta ,s)-⟨k⟩(I,s)}{\omega (I,s)}\mathrm{.}$

The branch vector field of the datum is the restriction to $J\times \mathbb{R}\times S$ of the map $\widehat{\mathrm{\Phi }}:\widehat{J}\times \mathbb{R}\times \widehat{S}\to {\mathbb{R}}^3$,

$\widehat{\mathrm{\Phi }}(I,\theta ,s):=(-\epsilon {\mathrm{\partial }}_{\theta }k(I,\theta ,s),{\mathrm{\partial }}_{I}h(I,s)+\epsilon {\mathrm{\partial }}_{I}k(I,\theta ,s),\epsilon ),$

and a branch solution on an interval $T\subseteq \mathbb{R}$ with nonempty interior is a differentiable $(I,\theta ,s):T\to J\times \mathbb{R}\times S$ satisfying $(I,\theta ,s{)}^{\mathrm{\prime }}=\widehat{\mathrm{\Phi }}(I,\theta ,s)$ at every $t\in T$, with derivatives relative to $T$ at included endpoints. On open time intervals this is the first-order system of First-order systems, initial value problems, and solutions on intervals on the open domain $\widehat{J}\times \mathbb{R}\times \widehat{S}$ (Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$). Partial derivatives are those of the ambient Euclidean coordinates.

• Angle interpretation. Since $k$ is $2\pi$-periodic in $\theta$, so are ${\mathrm{\partial }}_{\theta }k$ and ${\mathrm{\partial }}_{I}k$ on $J\times \mathbb{R}\times S$, hence the restriction of $\widehat{\mathrm{\Phi }}$ to $J\times \mathbb{R}\times S$ is $2\pi$-periodic in the second variable: reading the second coordinate in $\mathbb{R}\mathrm{/}2\pi \mathbb{Z}$ gives a well-defined vector field on $J\times (\mathbb{R}\mathrm{/}2\pi \mathbb{Z})\times S$, and if $(I,\theta ,s)$ is a branch solution, so is $(I,\theta +2\pi n,s)$ for every $n\in \mathbb{Z}$. The definition keeps the covering description on $\mathbb{R}$ to avoid quotient notation; all statements about $\theta$ below are invariant under $2\pi$-translations.

• Reading. A datum of this definition is the normal form of a one-frequency Hamiltonian problem in action-angle coordinates after the slow parameter is promoted to a state variable $s$ with $s^{\mathrm{\prime }}=\epsilon$: $h$ is the angle-independent part, $\epsilon k$ the angle-dependent remainder including any contribution of the parameter-dependent chart, and $\omega ={\mathrm{\partial }}_{I}h$ the fast frequency. The definition asserts no existence theorem: whether a given mechanical family admits such a chart with the stated regularity, on which domain, and with $\omega \mathrm{\ne }0$, is a separate hypothesis of any consumer, and the adiabatic statements below are conditional on it.

• Scope. This is pure mathematics: no physical identification of $I$, $\theta$, $s$, $h$, $k$, $\epsilon$ or of the parameter variation with measured data is made here.