On Phase at a Resonance in Slow-Fast Hamiltonian Systems

Abstract

We consider a slow-fast Hamiltonian system with one fast angle variable (a fast phase) whose frequency vanishes on some surface in the space of slow variables (a resonant surface). Systems of such form appear in the study of dynamics of charged particles in an inhomogeneous magnetic field under the influence of high-frequency electrostatic waves. Trajectories of the system averaged over the fast phase cross the resonant surface. The fast phase makes $\sim\frac{1}{\varepsilon}$ turns before arrival at the resonant surface ($\varepsilon$ is a small parameter of the problem). An asymptotic formula for the value of the phase at the arrival at the resonance was derived earlier in the context of study of charged particle dynamics on the basis of heuristic considerations without any estimates of its accuracy. We provide a rigorous derivation of this formula and prove that its accuracy is $O(\sqrt{\varepsilon})$ (up to a logarithmic correction). This estimate for the accuracy is optimal.

APPENDIX A. PROBABILITY DISTRIBUTION OF A PSEUDOPHASE

Consider some ball $U$ in the set of initial conditions $D_{0}\times\mathbb{S}^{1}$. Fix any $r>1$. According to Proposition 1, solutions of system (2.2) with initial conditions in $U\setminus{\cal V}_{r}$ arrive at the resonance, and ${{\rm mes}}\ {\cal V}_{r}=O(\varepsilon^{r})$. For these solutions one can determine the phase at the resonance $\varphi_{e}$ and the corresponding pseudophase $\Xi$ (4.7). As the phase is determined ${\rm mod}\ 2\pi$, it is natural to consider the pseudophase ${\rm mod}\ 1$. Thus, consider the fractional part $\hat{\Xi}$ of the variable $\Xi$.

Formula (4.8) shows that $\varphi_{e}\ {\rm mod}\ 2\pi$ and $\hat{\Xi}$ are very sensitive to changes in initial conditions. A change of $J_{0}$ of order $\varepsilon$ produces changes of order 1 in $\hat{\varphi}=\varphi_{e}\ {\rm mod}\ 2\pi$ and $\hat{\Xi}$. Therefore, for small $\varepsilon$, it is reasonable to consider $\hat{\varphi}$ and $\hat{\Xi}$ as random values and define their probability distributions.

Let $U^{\varepsilon}_{({\cal\alpha},{\cal\beta})}$ denote the set of initial points in $U$ such that $\hat{\Xi}\in({\cal\alpha},{\cal\beta})\subseteq(0,1)$.

Definition 1 ([13])

The value

$$Pr\big{(}\hat{\Xi}\in({\cal\alpha},{\cal\beta})\big{)}=\lim_{\varepsilon\to 0}\frac{\ {\rm mes}\ U^{\varepsilon}_{({\cal\alpha},{\cal\beta})}}{{\rm mes}\ U}$$

is called the probability of the event $\hat{\Xi}\in({\cal\alpha},{\cal\beta})$.

Therefore, the probability of the event $\hat{\Xi}\in({\cal\alpha},{\cal\beta})$ is the limit (as $\varepsilon\to 0$) of the fraction of phase volume in $U$ that is occupied by initial conditions for trajectories with $\hat{\Xi}\in({\cal\alpha},{\cal\beta})$. This approach to defining a probability follows that taken by V. I. Arnold in [2].

Proposition 2 ([13])

The value $\hat{\Xi}$ has a uniform distribution on the interval $(0,1):$

$$Pr\big{(}\hat{\Xi}\in({\cal\alpha},{\cal\beta})\big{)}={\cal\beta}-{\cal\alpha}.$$

This statement is justified in [13] via computing phase fluxes. It can be obtained as a direct corollary of formula (4.8). Moreover, this formula shows that the uniform distribution of $\hat{\Xi}$ arises on each curve $\{\varphi_{0}={\rm const},y={\rm const},x={\rm const}\}$ in $U$. This is because $d\hat{\Xi}_{a}/dJ_{0}\sim\varepsilon^{-1}$ when $J_{0}$ changes along such a curve. Here $\hat{\Xi}_{a}$ is the principal term of the value $\hat{\Xi}$ given by (4.8). We omit details of the proof.

The probability distribution of $\hat{\varphi}_{e}$ can be determined similarly to that of $\hat{\Xi}$ and obtained from the uniform distribution of $\hat{\Xi}$. Assume for simplicity of the formulation that for all $\varphi\in[0,2\pi]$ we have $|b(y_{*},x_{*})|>|\partial{H_{1}(I_{0},\varphi,y_{*},x_{*},0)}/{\partial\varphi}|$. Then the probability distribution density of $\hat{\varphi}_{e}$ is

$$p(\varphi_{e};I_{0},y_{*},x_{*})=\frac{1}{2\pi}\left(1+\frac{1}{b(y_{*},x_{*})}\frac{\partial H_{1}(I_{0},\varphi_{e},y_{*},x_{*},0)}{\partial\varphi_{e}}\right).$$

Consider as an example the test system from Section 9. Differential equations for $I,\varphi$ form a Hamiltonian system that depends on the slow time $\tau=\varepsilon t$. The pseudophase is defined by formula (9.3):

$$\Xi=\frac{1}{2\pi}\left(\varphi_{e}+\frac{1}{2\tau_{*}}\left(\frac{1}{2}+\tau_{*}^{2}\right)\sin\varphi_{e}\right).$$

Here $\varphi_{e}$ is the value of the phase at the resonance, $\tau_{*}=\sqrt{I_{0}}$, and $I_{0}$ is the initial value of action $I$. Consider some interval of initial actions $[I_{A},I_{B}]$ and $m$ evenly distributed values of $I_{0}$ inside this interval. Similarly, consider some interval of initial phases $[\varphi_{A},\varphi_{B}]\subseteq[0,2\pi]$ and $n$ evenly distributed values of $\varphi_{0}$ inside this interval. Thus, we have $m\times n$ initial conditions for $(I,\varphi)$. If $m,n$ are big enough and $\varepsilon$ is small enough, then the number of initial conditions for which the fractional part of $\Xi$ will be in an interval $(\alpha,\beta)\subseteq(0,1)$ is approximately equal to $(\beta-\alpha)\times m\times n$. The probability distribution density of the phase at the resonance $\varphi_{e}\bmod 2\pi$ is

$$p(\varphi_{e},I_{0})=\frac{1}{2\pi}\left(1+\frac{1}{2\sqrt{I_{0}}}\left(\frac{1}{2}+I_{0}\right)\cos\varphi_{e}\right)$$

provided that the value $I_{0}$ is such that the value $p(\varphi_{e},I_{0})$ given by this formula is positive for all $\varphi_{e}\in[0,2\pi]$. This is satisfied, e. g., if $[I_{A},I_{B}]\subseteq[1/4,1]$. Then the number of initial data for which $\varphi_{e}\ {\rm mod}\ 2\pi$ is in an interval $(\varphi_{\alpha},\varphi_{\beta})\subset(0,2\pi)$ is approximately $P_{(\varphi_{\alpha},\varphi_{\beta})}\times m\times n$, where

$$\displaystyle\begin{aligned} \displaystyle P_{(\varphi_{\alpha},\varphi_{\beta})}&\displaystyle=\frac{1}{({I_{B}}-{I_{A}})}\int_{I_{A}}^{I_{B}}\int_{\varphi_{\alpha}}^{\varphi_{\beta}}p(\varphi,I)d\varphi dI\\ &\displaystyle=\frac{\varphi_{\beta}-\varphi_{\alpha}}{2\pi}+\left[\frac{1}{2}(I_{B}^{1/2}-I_{A}^{1/2})+\frac{1}{3}(I_{B}^{3/2}-I_{A}^{3/2})\right]\frac{\sin{\varphi_{\beta}}-\sin{\varphi_{\alpha}}}{2\pi({I_{B}}-{I_{A}})}.\end{aligned}$$

Another natural way to define the probability distribution in problems with a small parameter is suggested by D. V. Anosov (for a discussion, see [15]). Fix an initial point $(I_{0},\varphi_{0},y_{0},x_{0})\in U$ and calculate the pseudophase $\hat{\Xi}$ for this initial point. Note that $\hat{\Xi}$ depends on $\varepsilon$. Introduce the set

$$V^{\varepsilon_{0}}_{({\cal\alpha},{\cal\beta})}=\{\varepsilon\in(0,\varepsilon_{0}):\hat{\Xi}\in({\cal\alpha},{\cal\beta})\}.$$

Definition 2

The value

$$Pr^{\prime}\big{(}\hat{\Xi}\in({\cal\alpha},{\cal\beta})\big{)}=\lim_{\varepsilon_{0}\to 0}\frac{\ {\rm mes}\ V^{\varepsilon_{0}}_{({\cal\alpha},{\cal\beta})}}{\varepsilon_{0}}$$

is called the probability of the event $\hat{\Xi}\in({\cal\alpha},{\cal\beta})$.

Proposition 3

Probability $Pr^{\prime}$ of the event $\hat{\Xi}\in({\cal\alpha},{\cal\beta})$ does not depend on the choice of the initial point $(I_{0},\varphi_{0},y_{0},x_{0})\in U$ and is given by the formula

$$Pr^{\prime}\big{(}\hat{\Xi}\in({\cal\alpha},{\cal\beta})\big{)}={\cal\beta}-{\cal\alpha}.$$

Thus, two definitions of probability lead to the same result. The proof of Proposition 3 is again based on formula (4.8). We omit the proof. There is an analogous proposition with a complete proof in [17] (see Proposition 3.8 there).

APPENDIX B. ASYMPTOTIC FORMULAS FOR SOME INTEGRALS

Calculate the right-hand side of relation (8.7). We will use $\varphi$ as the integration variable, taking into account that

$$\dot{\varphi}=\sqrt{\varepsilon}P=\sqrt{\varepsilon}\sqrt{2\left({\cal E}-u(\tau)\sin\varphi-\varphi\right)}$$

and ${\cal E}=\dfrac{I_{0}^{2}}{2\varepsilon}+\varphi_{0}+O(\sqrt{\varepsilon})$. We have

$$\displaystyle\begin{aligned} \displaystyle\varepsilon\int_{0}^{t_{e}}u^{\prime}(\varepsilon t)\sin\varphi(t)dt=\sqrt{\varepsilon}\int_{\varphi_{0}}^{\varphi_{e}}\frac{u^{\prime}(\varepsilon t)\sin\varphi}{\sqrt{2\left({\cal E}-u(\varepsilon t)\sin\varphi-\varphi\right)}}d\varphi.\end{aligned}$$

(B.1)

Let us show that replacement of $u^{\prime}(\varepsilon t),u(\varepsilon t),{\cal E}$ with their values at $t=t_{e}$ gives an error $O(\varepsilon)$ in (B.1). Consider the integral on the right-hand side of (B.1) as the sum of integrals from $\varphi_{0}$ to $\varphi_{i}=\varphi_{e}-1$ (denote this integral as $K_{1}$) and from $\varphi_{i}$ to $\varphi_{e}$ (denote this integral as $K_{2}$). Replacement of $u^{\prime}(\varepsilon t),u(\varepsilon t),{\cal E}$ by their values at $t=t_{e}$ in $K_{2}$ gives an error $O(\varepsilon^{3/2}\ln\varepsilon)$ in (B.1). Let ${\cal F}(\gamma,u^{\prime},u,{\cal E})$ denote the integral from $\varphi_{0}$ to $\gamma$ of the integrand in $K_{1}$ in which $u^{\prime},u,{\cal E}$ are considered as constants. Then

$$\displaystyle\begin{aligned} \displaystyle K_{1}&\displaystyle=\int_{\varphi_{0}}^{\varphi_{i}}d{\cal F}-\int_{\varphi_{0}}^{\varphi_{i}}\left(\frac{\partial{\cal F}}{\partial u^{\prime}}\frac{du^{\prime}}{d\varphi}+\frac{\partial{\cal F}}{\partial u}\frac{du}{d\varphi}+\frac{\partial{\cal F}}{\partial{\cal E}}\frac{d{\cal E}}{d\varphi}\right)d\gamma\\ &\displaystyle={\cal F}(\varphi_{i},u^{\prime}|_{\varphi=\varphi_{i}},u|_{\varphi=\varphi_{i}},{\cal E}|_{\varphi=\varphi_{i}})+O(\sqrt{\varepsilon})={\cal F}(\varphi_{i},u^{\prime}(\tau_{e}),u(\tau_{e}),{\cal E}|_{t=t_{e}})+O(\sqrt{\varepsilon}).\end{aligned}$$

(B.2)

Combining the estimates for $K_{1}$ and $K_{2}$, we find that replacement of $u^{\prime}(\varepsilon t),u(\varepsilon t),{\cal E}$ with their values at $t=t_{e}$ gives an error $O(\varepsilon)$ in (B.1).

Then the integral can be transformed as follows:

$$\displaystyle\begin{aligned} \displaystyle\varepsilon\int_{0}^{t_{e}}u^{\prime}(\varepsilon t)\sin\varphi(t)dt=\sqrt{\varepsilon}\int_{\varphi_{0}}^{\varphi_{e}}\frac{u^{\prime}(\tau_{e})\sin\varphi}{\sqrt{2\left({\cal E}|_{t=t_{e}}-u(\tau_{e})\sin\varphi-\varphi\right)}}d\varphi+O(\varepsilon)\\ \displaystyle=\sqrt{\varepsilon}\int_{\varphi_{0}-\varphi_{e}}^{0}\frac{u^{\prime}(\tau_{e})\sin(\varphi_{e}+\psi)}{\sqrt{2\left({\cal E}|_{t=t_{e}}-u(\tau_{e})\sin(\varphi_{e}+\psi)-(\varphi_{e}+\psi)\right)}}d\psi+O(\varepsilon)\\ \displaystyle=\sqrt{\varepsilon}\int_{\varphi_{0}-\varphi_{e}}^{0}\frac{u^{\prime}(\tau_{e})\sin(\varphi_{e}+\psi)}{\sqrt{2\left(u(\tau_{e})\sin\varphi_{e}-u(\tau_{e})\sin(\varphi_{e}+\psi)-\psi\right)}}d\psi+O(\varepsilon)\\ \displaystyle=\sqrt{\varepsilon}\int_{\varphi_{0}-\varphi_{a}}^{0}\frac{u^{\prime}(\tau_{*})\sin(\varphi_{a}+\psi)}{\sqrt{2\left(u(\tau_{*})\sin\varphi_{a}-u(\tau_{*})\sin(\varphi_{a}+\psi)-\psi\right)}}d\psi+O(\varepsilon)\\ \displaystyle=\sqrt{\varepsilon}u^{\prime}(\tau_{*})\int_{-\infty}^{0}\frac{\sin(\varphi_{a}+\psi)}{\sqrt{2\left(u(\tau_{*})\sin\varphi_{a}-u(\tau_{*})\sin(\varphi_{a}+\psi)-\psi\right)}}d\psi+O(\varepsilon)\\ \displaystyle=\sqrt{\varepsilon}u^{\prime}(\tau_{*})\int_{-\infty}^{\varphi_{a}}\frac{\sin\varphi}{\sqrt{2\left(u(\tau_{*})\sin\varphi_{a}+\varphi_{a}-u(\tau_{*})\sin\varphi-\varphi\right)}}d\varphi+O(\varepsilon).\end{aligned}$$

(B.3)

Then, in a similar way,

$$\displaystyle\begin{aligned} \displaystyle\tau_{e}-\tau_{*}&\displaystyle=\varepsilon\int_{0}^{t_{e}}dt-I_{0}=\sqrt{\varepsilon}\int_{\varphi_{0}}^{\varphi_{e}}\frac{1}{\sqrt{2\left({\cal E}-u(\tau)\sin\varphi-\varphi\right)}}d\varphi-I_{0}\\ &\displaystyle=\sqrt{\varepsilon}\int_{\varphi_{0}}^{\varphi_{e}}\frac{1}{\sqrt{2\left({\cal E}|_{t=t_{e}}-u(\tau_{e})\sin\varphi-\varphi\right)}}d\varphi-I_{0}+O(\varepsilon)\\ &\displaystyle=\sqrt{\varepsilon}\int_{\varphi_{0}}^{\varphi_{e}}-\frac{-1-u(\tau_{e})\cos\varphi+u(\tau_{e})\cos\varphi}{\sqrt{2\left({\cal E}|_{t=t_{e}}-u(\tau_{e})\sin\varphi-\varphi\right)}}d\varphi-I_{0}+O(\varepsilon)\\ &\displaystyle=\sqrt{\varepsilon}\left[-\sqrt{2\left({\cal E}|_{t=t_{e}}-u(\tau_{e})\sin\varphi-\varphi\right)}\right]_{\varphi_{0}}^{\varphi_{e}}\\ &\displaystyle-\sqrt{\varepsilon}\int_{\varphi_{0}}^{\varphi_{e}}\frac{u(\tau_{e})\cos\varphi}{\sqrt{2\left({\cal E}|_{t=t_{e}}-u(\tau_{e})\sin\varphi-\varphi\right)}}d\varphi-I_{0}+O(\varepsilon)\\ &\displaystyle=\sqrt{\varepsilon}\sqrt{2\left({\cal E}|_{t=t_{e}}-u(\tau_{e})\sin\varphi_{0}-\varphi_{0}\right)}-I_{0}\\ &\displaystyle-\sqrt{\varepsilon}\int_{\varphi_{0}}^{\varphi_{e}}\frac{u(\tau_{e})\cos\varphi}{\sqrt{2\left({\cal E}|_{t=t_{e}}-u(\tau_{e})\sin\varphi-\varphi\right)}}d\varphi+O(\varepsilon)\\ &\displaystyle=-\sqrt{\varepsilon}\int_{\varphi_{0}}^{\varphi_{e}}\frac{u(\tau_{e})\cos\varphi}{\sqrt{2\left({\cal E}|_{t=t_{e}}-u(\tau_{e})\sin\varphi-\varphi\right)}}d\varphi+O(\varepsilon)\\ &\displaystyle=-\sqrt{\varepsilon}\int_{\varphi_{0}-\varphi_{e}}^{0}\frac{u(\tau_{e})\cos(\varphi_{e}+\psi)}{\sqrt{2\left({\cal E}|_{t=t_{e}}-u(\tau_{e})\sin(\varphi_{e}+\psi)-(\varphi_{e}+\psi)\right)}}d\psi+O(\varepsilon)\\ &\displaystyle=-\sqrt{\varepsilon}\int_{\varphi_{0}-\varphi_{e}}^{0}\frac{u(\tau_{e})\cos(\varphi_{e}+\psi)}{\sqrt{2\left(u(\tau_{e})\sin\varphi_{e}-u(\tau_{e})\sin(\varphi_{e}+\psi)-\psi\right)}}d\psi+O(\varepsilon)\\ &\displaystyle=-\sqrt{\varepsilon}\int_{\varphi_{0}-\varphi_{a}}^{0}\frac{u(\tau_{*})\cos(\varphi_{a}+\psi)}{\sqrt{2\left(u(\tau_{*})\sin\varphi_{a}-u(\tau_{*})\sin(\varphi_{a}+\psi)-\psi\right)}}d\psi+O(\varepsilon)\\ &\displaystyle=-\sqrt{\varepsilon}u(\tau_{*})\int_{-\infty}^{0}\frac{\cos(\varphi_{a}+\psi)}{\sqrt{2\left(u(\tau_{*})\sin\varphi_{a}-u(\tau_{*})\sin(\varphi_{a}+\psi)-\psi\right)}}d\psi+O(\varepsilon)\\ &\displaystyle=-\sqrt{\varepsilon}u(\tau_{*})\int_{-\infty}^{\varphi_{a}}\frac{\cos\varphi}{\sqrt{2\left(u(\tau_{*})\sin\varphi_{a}+\varphi_{a}-u(\tau_{*})\sin\varphi-\varphi\right)}}d\varphi+O(\varepsilon).\end{aligned}$$

(B.4)

Combining the results of (B.3) and (B.4), we get the required estimates (8.8) and (8.9).