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\begin{document}
\title{NONLINEAR EFFECTS IN ACCELERATOR PHYSICS:
FROM SCALE TO SCALE VIA WAVELETS}
\author{A.~Fedorova and  \underline{M.~Zeitlin},
Institute of Problems of Mechanical Engineering,\\
 Russian Academy of Sciences, 199178, Russia, St.~Petersburg,\\
  V.O., Bolshoj pr., 61, e-mail: zeitlin@math.ipme.ru\\
Z.~Parsa, Dept. of Physics, Bldg.~901A, Brookhaven National Laboratory,\\
 Upton, NY 11973-5000, e-mail: parsa@bnl.gov}
\maketitle
\begin{abstract}
This is the first part of our two  papers in which we
present applications of methods from
wavelet analysis to polynomial approximations for
a number of accelerator physics problems.
In the general case we have the solution as
a multiresolution expansion in the base of compactly
supported wavelet basis.
We give extension of our previous results to the case of periodic orbital
particle motion in storage rings. Then we consider more flexible variational
method which is based on biorthogonal wavelet approach.
\end{abstract}
\section{INTRODUCTION}
This is the first part of our two presentation in which we consider
applications of methods from wavelet analysis to nonlinear accelerator
physics problems. This is a continuation of our results from [1], [2],
which is based on approach of two of us from [3], [4] to investigation
of nonlinear problems -- general, with additional structures (Hamiltonian,
symplectic or quasicomplex), chaotic, quasiclassical, quantum, which are
considered in the framework of local(nonlinear) Fourier analysis, or wavelet
analysis. Wavelet analysis is a relatively novel set of mathematical
methods, which gives us a possibility to work with well-localized bases in
functional spaces and with the general type of operators (differential,
integral, pseudodifferential) in such bases.

In [3], [4] we considered application of multiresolution representation to
general nonlinear dynamical system with polynomial type of nonlinearities.
Starting with variational formulation of initial dynamical problem we
construct explicit representation for all dynamical variables in the base of
compactly supported (Daubechies) wavelets. Our solutions are parametrized
by solutions of a number of reduced algebraical problems one from which
is nonlinear with the same degree of nonlinearity and the rest  are
the linear problems which correspond to particular
method of calculation of scalar products of functions from wavelet bases
and their derivatives. In this paper we consider further extension of our
previous results.  In section 2 we consider modification of our previous
construction to the periodic case,
in section 3 we consider generalization of our approach from
[1], [2] to variational formulation in the biorthogonal bases of compactly
supported wavelets.

Our main example is calculation of orbital particle motion in storage rings.
Starting from Hamiltonian which described classical dynamics in storage
rings [5]
$
{\cal H}(\vec{r},\vec{P},t)=c\{\pi^2+m_0^2c^2\}^{1/2}+e\phi
$
and using Serret-Frenet parametrization, truncation of power series expansion
of square root we arrive to the following approximated Hamiltonian for particle
motion in machine coordinates:
\begin{eqnarray}
{\cal H}&=&
   \frac{1}{2}\cdot\frac{[p_x+H\cdot z]^2 + [p_z-H\cdot x]^2}
{[1+f(p_\sigma)]}+p_\sigma - \nonumber\\
& &[1+K_x\cdot x+K_z\cdot z]\cdot f(p_\sigma)+\nonumber\\
& &\frac{1}{2}\cdot[K_x^2+g]\cdot x^2
+\frac{1}{2}\cdot[K_z^2-g]\cdot z^2-\nonumber\\
& &  N\cdot xz+
 \frac{\lambda}{6}\cdot(x^3-3xz^2)+\\
& &\frac{\mu}{24}\cdot(z^4-6x^2z^2+x^4)+ \nonumber\\
& &\frac{1}{\beta_0^2}\cdot\frac{L}{2\pi\cdot h}\cdot\frac{eV(s)}{E_0}\cdot
\cos\left[h\cdot\frac{2\pi}{L}\cdot\sigma+\varphi\right]\nonumber
\end{eqnarray}
Then we use series expansion of function $f(p_\sigma)$
and the corresponding expansion of RHS of Hamiltonian equations of
motions. In the following we take into account only an arbitrary
polynomial ( in terms of dynamical variables) expressions and
neglecting all nonpolynomial types of expressions, i.e. we consider
such approximations of RHS, which are not more than  polynomial
functions in dynamical variables and arbitrary functions of independent
variable $s$.

\section{VARIATIONAL WAVELET APPROACH FOR PERIODIC TRAJECTORIES}

We start with extension of our approach [1], [2] to the case
of periodic trajectories. The equations of motion corresponding
to Hamiltonian (1) may be formulated as a particular case of
the general system of
ordinary differential equations
$
{dx_i}/{dt}=f_i(x_j,t)$, $  (i,j=1,...,n)$, $0\leq t\leq 1$,
where $f_i$ are not more
than polynomial functions of dynamical variables $x_j$
and  have arbitrary dependence of time.
According to our variational approach [3], [4] we have the
solution in the following form
\begin{eqnarray}
x_i(t)=x_i(0)+\sum_k\lambda_i^k\varphi_k(t),
\end{eqnarray}
where $\lambda_i^k$ are the roots of reduced algebraical systems of equations
with the same degree of nonlinearity and $\varphi_k(t)$
corresponds to useful type of wavelet bases (frames).
It should be noted that coefficients of reduced algebraical system
are the solutions of additional linear problem and
also
depend on particular type of wavelet construction and type of bases.
This linear problem is our second reduced algebraical problem.
Our construction is based on multiresolution approach. Because affine
group of translation and dilations is inside this approach our
method resembles the action of a microscope. We have contribution to
final result from each scale of resolution from the whole
infinite scale of spaces. More exactly, the closed subspace
$V_j$ corresponds to  level j of resolution, or to scale j.
We consider a sequence of
successive approximation by subspaces $V_j$:
$
...V_2\subset V_1\subset V_0\subset V_{-1}\subset V_{-2}\subset ...
$
satisfying the following properties:
%\begin{eqnarray*}
$
\displaystyle\bigcap_{j\in{\bf Z}}V_j=0$,
$\overline{\displaystyle\bigcup_{j\in{\bf Z}}}V_j=L^2({\bf R})$,
$ f(x)\in V_j <=> f(2x)\in V_{j+1}
$
%\end{eqnarray*}.
There is a function $\varphi\in V_0$ such that \{${\varphi_{0,k}(x)=
\varphi(x-k)}_{k\in{\bf Z}}$\} forms a Riesz basis for $V_0$.
We use compactly supported wavelet basis: orthonormal basis
for functions in $L^2({\bf R})$. As usually $\varphi(x)$ is
a scaling function, $\psi(x)$ is a wavelet function, where
$\varphi_i(x)=\varphi(x-i)$. Scaling relation that defines
$\varphi,\psi$ are\\
%\begin{eqnarray*}
$\varphi(x)=\sum\limits^{N-1}_{k=0}a_k\varphi(2x-k)=
\sum\limits^{N-1}_{k=0}a_k\varphi_k(2x),$\\
$\psi(x)=\sum\limits^{N-2}_{k=-1}(-1)^k a_{k+1}\varphi(2x+k)$.\\
%\end{eqnarray*}
Let be  $ f : {\bf R}\longrightarrow {\bf C}$ and the wavelet
expansion is\\
%\begin{eqnarray}
$f(x)=\sum\limits_{\ell\in{\bf Z}}c_\ell\varphi_\ell(x)+
\sum\limits_{j=0}^\infty\sum\limits_{k\in{\bf
Z}}c_{jk}\psi_{jk}(x)$\\
%\end{eqnarray}
The indices $k, \ell$ and $j$ represent translation and scaling, respectively
$
\varphi_{jl}(x)=2^{j/2}\varphi(2^j x-\ell)$,
$\psi_{jk}(x)=2^{j/2}\psi(2^j x-k)$.
The set $\{\varphi_{j,k}\}_{k\in {\bf Z}}$ forms a Riesz basis for $V_j$.
Let $W_j$ be the orthonormal complement of $V_j$ with respect to $V_{j+1}$.
Just as $V_j$ is spanned by dilation and translations of the scaling function,
so are $W_j$ spanned by translations and dilation of the mother wavelet
$\psi_{jk}(x)$.\\
All expansions which we used are based on the following properties:
$
\{\varphi_{jk}\}_{j\geq 0, k\in {\bf Z}}$ is an orthonormal
basis for $L^2({\bf R})$,
%\begin{eqnarray*}
$V_{j+1}=V_j\bigoplus W_j,\quad
   L^2({\bf R})=\overline{V_0\displaystyle\bigoplus^\infty_{j=0} W_j}$,
$\{\varphi_{0,k},\psi_{j,k}\}_{j\geq 0,k\in {\bf Z}}$ is
an orthonormal basis for
$ L^2({\bf R})$,$\quad
 \{\varphi_{j,k},\psi_{\ell,k}; 0\leq j\leq J\leq \ell, k\in{\bf Z}\}$
is an orthonormal basis for $ L^2({\bf R})$.
%\end{eqnarray*}
If in formulae (4) $c_{jk}=0$ for $j\geq J$, then $f(x)$ has an alternative
expansion in terms of dilated scaling functions only
$
f(x)=\sum\limits_{\ell\in {\bf Z}}c_{J\ell}\varphi_{J\ell}(x)
$.
This is a finite wavelet expansion, it can be written solely in
terms of translated scaling functions. We use wavelet $\psi(x)
$, which has $k$ vanishing moments
$
\int x^k \psi(x)dx=0$, or equivalently
$x^k=\sum c_\ell\varphi_\ell(x)$ for each $k$,
$0\leq k\leq K$.
Also we have the shortest possible support: scaling function
$DN$ (where $N$ is even integer) will have support $[0,N-1]$ and
$N/2$ vanishing moments.
There exists $\lambda>0$ such that $DN$ has $\lambda N$
continuous derivatives; for small $N,\lambda\geq 0.55$.
To solve our second associated linear problem we need to
evaluate derivatives of $f(x)$ in terms of $\varphi(x)$.
Let be $
\varphi^n_\ell=d^n\varphi_\ell(x)/dx^n
$.
We derive the wavelet - Galerkin approximation of a
differentiated $f(x)$ as
$
f^d(x)=\sum_\ell c_l\varphi_\ell^d(x)
$
and values $\varphi_\ell^d(x)$ can be expanded in terms of
$\varphi(x)$ [6]:
$
\phi_\ell^d(x)=\sum\limits_m\lambda_m\varphi_m(x)$,$\quad
\lambda_m=\int\limits_{-\infty}^{\infty}\varphi_\ell^d(x)\varphi_m(x)dx
$.
The coefficients $\lambda_m$  are 2-term connection
coefficients. In general we need to find
\begin{eqnarray}
\Lambda^{d_1 d_2 ...d_n}_{\ell_1 \ell_2 ...\ell_n}=
 \int\limits_{-\infty}^{\infty}\prod\varphi^{d_i}_{\ell_i}(x)dx
\end{eqnarray}
For quadratic (Riccati) case we need to evaluate two and three
connection coefficients
\begin{eqnarray*}
\Lambda_\ell^{d_1
d_2}&=&\int^\infty_{-\infty}\varphi^{d_1}(x)\varphi_\ell^{d_2}(x)dx,
\quad d_i\geq 0, \\
\Lambda^{d_1 d_2
d_3}&=&\int\limits_{-\infty}^\infty\varphi^{d_1}(x)\varphi_
\ell^{d_2}(x)\varphi^{d_3}_m(x)dx
\end{eqnarray*}
Now we consider the same objects and procedure of their
calculations but in the base of periodic wavelet functions on
the interval [0,1] and corresponding expansion (2) inside our
variational approach [3], [4]. Periodization procedure
gives us
\begin{eqnarray}
\hat\varphi_{j,k}(x)&\equiv&\sum_{\ell\in Z}\varphi_{j,k}(x-\ell)\\
\hat\psi_{j,k}(x)&=&\sum_{\ell\in Z}\psi_{j,k}(x-\ell)\nonumber
\end{eqnarray}
So, $\hat\varphi, \hat\psi$ are periodic functions on the interval
 [0,1]. Because $\varphi_{j,k}=\varphi_{j,k'}$ if $k=k'mod(2^j)$, we
may consider only $0\leq k\leq 2^j$ and as  consequence our
multiresolution has the form
$\displaystyle\bigcup_{j\geq 0} \hat V_j=L^2[0,1]$ with $\hat V_j=span\{\hat\varphi_{j,k}\}
^{2j-1}_{k=0}$ [7].
Integration by parts and periodicity gives  useful relations between
objects (3) $(d=d_1+d_2)$:
\begin{eqnarray*}
\Lambda^{d_1,d_2}_{k_1,k_2}=(-1)^{d_1}\Lambda^{0,d_2+d_1}_{k_1,k_2},
\Lambda^{0,d}_{k_1,k_2}=\Lambda^{0,d}_{0,k_2-k_1}\equiv
\Lambda^d_{k_2-k_1}
\end{eqnarray*}
So, any 2-tuple can be represent by $\Lambda^d_k$.
Then our second additional linear problem is reduced to the eigenvalue
problem for
$\{\Lambda^d_k\}_{0\leq k\le 2^j}$ by creating a system of $2^j$
homogeneous relations in $\Lambda^d_k$ and inhomogeneous equations.
So, if we have dilation equation in the form
$\varphi(x)=\sqrt{2}\sum_{k\in Z}h_k\varphi(2x-k)$,
then we have the following homogeneous relations
$\Lambda^d_k=2^d\sum_{m=0}^{N-1}\sum_{\ell=0}^{N-1}h_m h_\ell
\Lambda^d_{\ell+2k-m}$, or in such form
$A\lambda^d=2^d\lambda^d$, where $\lambda^d=\{\Lambda^d_k\}_
{0\leq k\le 2^j}$.
Inhomogeneous equations are:
$\sum_{\ell}M_\ell^d\Lambda^d_\ell=d!2^{-j/2}$, where objects
$M_\ell^d(|\ell|\leq N-2)$ can be computed by recursive procedure
$M_\ell^d=2^{-j(2d+1)/2}\tilde{M_\ell^d}$, $\tilde{M_\ell^k}=
<x^k,\varphi_{0,\ell}>=\sum_{j=0}^k {k\choose j} n^{k-j}M_0^j$,
$\tilde{M_0^\ell}=1$. So, we reduced our last problem to standard
linear algebraical problem. Then we use the same methods as in [3], [4].\\
As a result we obtained
for closed trajectories of orbital dynamics described by Hamiltonian (1)
the explicit time solution (2) in the base of periodized wavelets (4).

\section{VARIATIONAL APPROACH IN BIORTHOGONAL WAVELET BASES}
Now we consider further generalization of our variational wavelet approach.
In [3], [4] we consider different types of variational principles
which give us weak solutions of our nonlinear problems.
But because integrand of variational functionals is represented
by bilinear form (scalar product) it seems more reasonable to
consider wavelet constructions which take into account all advantages of
this structure.

As an example let us consider action functional for loops in the phase space
$
F(\gamma)=\displaystyle\int_\gamma pdq-\int_0^1H(t,\gamma(t))dt
$.
The critical points of $F$ are such loops $\gamma$, which solve
the Hamiltonian equations associated with the Hamiltonian $H$
and hence are periodic orbits. So, $(M,\omega)$ is symplectic
manifolds, $H: M \to R $, $H$ is Hamiltonian, $X_H$ is
unique Hamiltonian vector field defined  by
$
\omega(X_H(x),\upsilon)=-dH(x)(\upsilon)$, $ \upsilon\in T_xM,
\quad x\in M$,
where $ \omega$ is the symplectic structure.
A T-periodic solution $x(t)$ of the Hamiltonian equations
$
\dot x=X_H(x) \quad \mbox{ on $M$}
$
is a solution, satisfying the boundary conditions $x(T)$ $=x(0), T>0$.
Let us consider the loop space $\Omega=C^\infty(S^1, R^{2n})$,
where $S^1=R/{\bf Z}$, of smooth loops in $R^{2n}$.
Let us define a function $\Phi: \Omega\to R $ by setting
$$
\Phi(x)=\displaystyle\int_0^1\frac{1}{2}<-J\dot x, x>dt-
\int_0^1 H(x(t))dt, \quad x\in\Omega
$$
The critical points of $\Phi$ are the periodic solutions of $\dot x=X_H(x)$.
Computing the derivative at $x\in\Omega$ in the direction of $y\in\Omega$,
we find
\begin{eqnarray*}
\Phi'(x)(y)&=&\frac{d}{d\epsilon}\Phi(x+\epsilon y)\vert_{\epsilon=0}
=\\
&&\displaystyle\int_0^1<-J\dot x-\bigtriangledown H(x),y>dt
\end{eqnarray*}
Consequently, $\Phi'(x)(y)=0$ for all $y\in\Omega$ iff the loop $x$ satisfies
the equation
$-J\dot x(t)-\bigtriangledown H(x(t))=0$,
i.e. $x(t)$ is a solution of the Hamiltonian equations, which also satisfies
$x(0)=x(1)$, i.e. periodic of period 1.

We started with two hierarchical sequences of approximations spaces [8]:
%\begin{eqnarray*}
$\dots V_{-2}\subset V_{-1}\subset V_{0}\subset V_{1}\subset V_{2}\dots$,
$\dots \widetilde{V}_{-2}\subset\widetilde{V}_{-1}\subset
\widetilde{V}_{0}\subset\widetilde{V}_{1}\subset\widetilde{V}_{2}\dots$,
%\end{eqnarray*}
and as usually,
$W_0$ is complement to $V_0$ in $V_1$, but now not necessarily orthogonal
complement.
New orthogonality conditions have now the following form:
$\widetilde {W}_{0}\perp V_0$, $ W_{0}\perp\widetilde{V}_{0}$,
$V_j\perp\widetilde{W}_j$, $\widetilde{V}_j\perp W_j$,
translates of $\psi$  span $W_0$,
translates of $\tilde\psi$ span $\widetilde{W}_0$.
Biorthogonality conditions are
$<\psi_{jk},\tilde{\psi}_{j'k'}>=$
$\int^\infty_{-\infty}\psi_{jk}(x)\tilde\psi_{j'k'}(x)dx=$
$\delta_{kk'}\delta_{jj'}$, where
$\psi_{jk}(x)=2^{j/2}\psi(2^jx-k)$.
Functions $\varphi(x), \tilde\varphi(x-k)$ form  dual pair:
$<\varphi(x-k), \tilde\varphi(x-\ell)>=\delta_{kl}$,
$ <\varphi(x-k)$, $\tilde\psi(x-\ell)>=0$  for $ \forall k$,
$\forall\ell$.
Functions $\varphi, \tilde\varphi$ generate a multiresolution analysis.
$\varphi(x-k)$, $\psi(x-k)$ are synthesis functions,
$\tilde\varphi(x-\ell)$, $\tilde\psi(x-\ell)$ are analysis functions.
Synthesis functions are biorthogonal to analysis functions. Scaling spaces
are orthogonal to dual wavelet spaces.
Two multiresolutions are intertwining
$
V_j+W_j=V_{j+1}, \widetilde V_j+ \widetilde W_j = \widetilde V_{j+1}
$.
These are direct sums but not orthogonal sums.

So, our representation for solution has now the form
$f(t)=\sum_{j,k}\tilde b_{jk}\psi_{jk}(t)$,
where synthesis wavelets are used to synthesize the function. But
$\tilde b_{jk}$ come from inner products with analysis wavelets.
Biorthogonality yields
$\tilde b_{\ell m}=\int f(t)\tilde{\psi}_{\ell m}(t) dt$.
So, now we can introduce this more complicated construction into
our variational approach. We have modification only on the level of
computing coefficients of reduced nonlinear algebraical system.
This new construction is more flexible.
Biorthogonal point of view is more stable under the action of large
class of operators while orthogonal (one scale for multiresolution)
is fragile, all computations are much more simpler and we accelerate
the rate of convergence. In all type of Hamiltonian calculation,
which is based on some bilinear structures (symplectic or
Poissonian structures, bilinear form of integrand in variational
integral) this framework leads to much success.



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\end{thebibliography}

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