Lie transformations and the Birkhoff-Gustavson normal form

Consider an autonomous Hamiltonian system with $n$ degrees of freedom and a fixed point in the origin. We can always write the Hamiltonian $H$ as a formal power series in the phase space coordinates $q_\nu$, $p_\nu$, $\nu=1,2,\dots,n$. With ${\mbox{\protect\boldmath$z$}} = \left( q_1,\dots,q_n,p_1,\dots,p_n \right) \in {\bf R}^{2n}$ we have

Here $\L _l$ is the ${2n+l-1 \choose 2n-1}$-dimensional vector space of homogeneous polynomials of degree $l$ in $2n$ variables, and we employ the multiindex notation

$$\begin{eqnarray*} {\mbox{\protect\boldmath$m$}} & \in & {\bf N}_0^{2n} \\ \ve... ...tect\boldmath$m$}} & = & h_{m_1,\dots,m_{2n}} \in {\bf R} \quad. \end{eqnarray*}$$

Note that the dimension of $\L _l$ grows rapidly with $l$ (e.g. for $n=2$ we have $\dim\left(\L _{14}\right)=680$), such that any manipulation of $H_l$ for larger values of $l$ will have to be done by computer algebra rather than ``by hand''. We denote the space of all formal power series beginning with degree 2 by $\L =\bigoplus\limits_{l=2}^\infty\L _l$.

The Lie operator $\mbox{\rm ad}_F$ adjoint to a power series $F\in\L$ is the Poisson bracket of $F$ with some $G\in\L$:

$$\quad \mbox{\rm ad}_F(G) := \left\{ G,F \right\} = \sum_{... ...\partial p_i} \frac{\partial F}{\partial q_i} \right) \quad.$$ (-1)

$\mbox{\rm ad}_F$ is a linear operator on $\L$ for all $F$. The Lie operator adjoint to the quadratic part of the Hamiltonian and restricted to the subspace $\L _m$ is of central importance:

$$\quad {\cal A}_m := \mbox{\rm ad}_{H_2} \Big\vert _{\L _m} \quad, \quad m \geq 2 \quad,$$

Note that ${\cal A}_m$ maps monomials of degree $m$ to monomials of degree $m$.

The Lie transformation associated with $F\in\L$ is the exponential of $\mbox{\rm ad}_F$:

$$\quad \exp(\mbox{\rm ad}_F) = \sum_{i=0}^\infty \frac{1}{i!} {\mbox{\rm ad}_F}^i \quad.$$ (1)

Lie transformations are an adequate tool for Hamiltonian normal form theory because they are canonical [20].

Let $H$ be a Hamiltonian of type (1). $H$ is in Birkhoff-Gustavson normal form up to order $m$ if

$${\cal A}_l(H_l) = 0 \quad \mbox{for} \quad l=2,3,\ldots,m.$$ (2)

$H$ is in Birkhoff-Gustavson normal form if (6) holds for all $l\geq 2$. This definition is motivated by the fact that $H_2$ is an integral of motion if $H$ is in BGNF:

$$\quad {\cal A}_l(H_l)=0 \;\; \forall l \quad \Leftrightarrow \quad \{H,H_2\} = 0 \quad.$$

For any given Hamiltonian $H$ of type (1) we can proceed to the BGNF of $H$ in the following way: With some $F_m\in\L _m$ determine a new Hamiltonian $G=\sum_{l\geq 2}G_l$ by

$$\quad G = \exp\left( \mbox{\rm ad}_{F_m} \right) (H) \quad.$$ (3)

More explicitely we have

$$\quad G_2+G_3+\cdots = H_2+\cdots+H_m + \mbox{\rm ad}_{F_m}... ...t(\vert{\mbox{\protect\boldmath$z$}}\vert^{m+1}\right)} \quad,$$

where ${{\cal O}\left(\vert{\mbox{\protect\boldmath$z$}}\vert^{m+1}\right)}$ stands for terms of order $m+1$ and higher. Collecting terms of equal order we get, using ${\mbox{\rm ad}_{F_m}}^i(H_l) \in \L _{l+i(m-2)}$:

Assuming that $H$ is already in BGNF up to order $m-1$, equation (8) allows several important conclusions: First of all, according to (8a) the contributions of order less than $m$ remain unchanged under the Lie transformation associated to $F_m$. This shows that the transformed Hamiltonian $G$ is at least in normal form up to order $m-1$, too. Secondly, (8b) indicates how to obtain a Hamiltonian $G$ which is in BGNF even up to order $m$: From (8b) we get

$$\quad H_m = G_m+{\cal A}_m(F_m) \quad.$$ (3)

This homological equation [23] must be solved for $F_m$ and $G_m$ under the additional condition

$$\quad {\cal A}_m(G_m) = 0 \quad.$$ (4)

In other words, $G_m$ must be in the kernel (or null space) of ${\cal A}_m$:

$$\quad G_m \in \mbox{Ker}\left({\cal A}_m\right) = \left\{ P\in\L _m \;\big\vert\; {\cal A}_m(P)=0 \right\} \quad.$$

Thus we have the following iterative process for the normalization of $H$: For all $m\geq 3$ we first solve the homological equation for the polynomials $G_m$ and $F_m$ and then obtain the remaining terms $G_{l>m}$ of the new Hamiltonian by evaluating (7). The calculation of the $G_l$ is a tedious but straight-forward task that can be left to computer algebra. The non-trivial key point is solving the homological equation.

Let us assume that the vector space $\L _m$ can be decomposed into the direct sum of the kernel and range spaces of ${\cal A}_m$,

$$\quad \L _m = \mbox{Ker}\left({\cal A}_m\right) \oplus \mbox{Im}\left({\cal A}_m\right) \quad,$$ (5)

with $\mbox{Im}\left({\cal A}_m\right) = {\cal A}_m(\L _m)$. Then $H_m\in\L _m$ can uniquely be split into its kernel and range components

Hence, $G_m$ is uniquely determined by the kernel component of $H_m$:

$$\quad G_m = H_m' \quad.$$ (5)

Finally $F_m$ can be obtained by inverting

$$\quad {\cal A}_m(F_m) = H_m'' \quad.$$ (6)

Since there may be several preimages of $H_m''$ under ${\cal A}_m$ ($F_m$ is uniquely determined by (14) up to any element of the null space of ${\cal A}_m$) the normalization procedure is not unambiguous. However, we can always achieve unambiguity by additionally requiring $F_m$ to lie in the range space of ${\cal A}_m$.

The key point of the above procedure is the splitting (11). By means of the canonical transformation $({\mbox{\protect\boldmath$q$}},{\mbox{\protect\boldmath$p$}}) \mapsto (\tilde{{\mbox{\protect\boldmath$q$}}},\tilde{{\mbox{\protect\boldmath$p$}}})$ with

$$\begin{array}{rcl} \quad \tilde{{\mbox{\protect\boldmath$... ...p$}}-i{\mbox{\protect\boldmath$q$}}\right) \quad, \end{array}$$ (7)

Gustavson showed that for a Hamiltonian of type (1) equation (11) holds, since in the new coordinates $\tilde{{\mbox{\protect\boldmath$q$}}},\tilde{{\mbox{\protect\boldmath$p$}}}$ the corresponding transformed operator $\tilde{{\cal A}}_m$ is diagonal. Since $\tilde{{\cal A}}_m$ yields the splitting (11), ${\cal A}_m$ does as well. This proves the applicability of the BGNF theory to Hamiltonians of the Gustavson type (1): Every such Hamiltonian $H$ can, by means of a formal canonical transformation, be transformed into the equivalent Hamiltonian

$$\quad G = \Big[ \cdots \circ \exp\left(\mbox{\rm ad}_{F_4}... ...\circ \exp\left(\mbox{\rm ad}_{F_3}\right) \Big] \; (H) \quad,$$ (8)

where $F_m\in\L _m$ and $G$ is in BGNF. The term ``formal'' indicates that we do not yet consider the convergence properties of the power series $H$, $F$ and $G$.