Complex Germen on invariant isotropic tori under the Hamiltonian phases flow with in involution Hamilton functions

TL;DR

This paper fully solves the problem of determining the existence and uniqueness of the complex germ on invariant isotropic tori under Hamiltonian flows with involutive functions, removing previous spectral restrictions.

Contribution

It provides necessary and sufficient conditions for the complex germ's existence and uniqueness without requiring the simple spectrum condition.

Findings

Established conditions for complex germ existence and uniqueness.

Extended previous results to cases without simple spectrum.

Analyzed Hamiltonian systems with cyclic variables.

Abstract

M. M. Nekhoroshev put forward the problem of to find the Complex Germ on a isotropic invariant torus with respect to Hamiltonian phases flows which come from k-functions in involution. This statement was partially solved in [9] establishing that if certain simplectic operator has a simple spectrum then the complex germ exist. In this work we solve this problem, providing a full solution, i.e. we present conditions for the existence and uniqueness of complex germ through the monodromy operator constructed in [9], but without the simple spectrum condition. We study also the Hamiltonian system with cyclic variables.

Equations

$$dq\wedge dp(x,y)=\sum _{i=1}^{n}d{q}_i(x)d{p}_i(y)-d{q}_i(y)d{p}_i(x),\forall x,y\in T_m{\Re }^{2n}$$

$$d{q}_i(x)=q_i(x)\text{and}d{p}_i(x)=p_i(x),$$

$${\omega }^2(x,Ja)=a(x),\text{for}x\in T_m{M}^{2n};a\in T_m^{\ast }{M}^{2n}.$$

$$[F,G]={\omega }^2(JdF,JdG),$$

$$[F,G]=-dG(JdF)=dF(JdG).$$

$$\displaystyle\left(\begin{array}[]{cc}0&-I_{n}\\ I_{n}&0\end{array}\right)$$

$$\dot{p}=H_q,\dot{q}=-H_p,$$

$$(g_H^t{)}^{\ast }{\omega }^2(x,y)={\omega }^2((g_H^t{)}_{\ast ,m}x,(g_H^t{)}_{\ast ,m}y),$$

$$(\alpha +i\beta )\cdot (V+iW)=(\alpha V-\beta W)+i(\alpha W+\beta V),$$

$$(V_1+i{W}_1)\cdot (V_2+i{W}_2)=(V_1{V}_2-W_1{W}_2)+i(V_1{W}_2+W_2{V}_2),$$

$$({\omega }^2{)}^{\mathbb{C}}(u+iv,x+iy)=({\omega }^2(u,x)-{\omega }^2(v,y))+i({\omega }^2(u,y)+{\omega }^2(v,x)),$$

$$\forall x\in r^n(m)\setminus T_m({\Lambda }^k{)}^{\mathbb{C}}\text{holds}(1/2i)[x,\bar{x}]>0,$$

$$[(g_H^t{)}_{\ast ,m}{]}^{\mathbb{C}}(r^n(m))=r(g_H^{t}m),$$

$$\Sigma =\{m\in M^{2n}:F_i(m)=f_i,i=1,\dots ,k,f=(f_1,\dots ,f_k)\in {\Re }^k\},$$

$$g^{\overrightarrow{t}}=g_1^{t_1}o\dots o{g}_k^{t_k},$$

$$G=\{\overrightarrow{t}\in {\Re }^k:g^{\overrightarrow{t}}m=m\},$$

$$G=\{l_1{T}_1+\dots ,l_k{T}_k;l_i\in \Re ;T_i\in G\text{are linearly independent},i=1,\dots ,k\}.$$

$$g_{\ast ,m}^T=g_{\ast ,m}^{l_1{T}_1+\dots +l_k{T}_k},$$

$$g_{\ast ,m}^{l_1{T}_1+\dots +l_k{T}_k}=g_{1\ast ,m}^{l_1{t}_{11}+\dots +l_k{t}_{k1}}o\dots o{g}_{k\ast ,m}^{l_1{t}_{1k}+\dots +l_k{T}_{kk}}.$$

$$g_{\ast ,m}^T=(g_{1\ast ,m}^{l_1{t}_{11}}+\dots +g_{k\ast ,m}^{l_1{t}_{1k}})o(g_{1\ast ,m}^{l_2{t}_{21}}+\dots +g_{k\ast ,m}^{l_2{t}_{2k}})o\dots o(g_{1\ast ,m}^{l_k{t}_{k1}}+\dots +g_{1\ast ,m}^{l_k{t}_{kk}}),$$

$$G_m=G_1^{l_1}o\dots o{G}_k^{l_k},$$

$$G_j(Jd{F}_i)=\sum _{n=1}^k{\beta }_{n}Jd{F}_n,$$

$$G_m(Jd{F}_i)=\sum _{n=1}^k{\mu }_{n}Jd{F}_n,$$

$$G_j({\theta }^{{}^{\prime }})=G_j(\theta )+\sum _{i=1}^k{\mu }_{i}Jd{F}_i,$$

$$G_m({\theta }^{{}^{\prime }})=G_m(\theta )+\sum _{i=1}^k{\tau }_{i}Jd{F}_i,$$

$${\Xi }_m:{\Gamma }_m\to {\Gamma }_m,\text{such that}[\theta ]\to [G_m(\theta )],$$

$${\Xi }_j:{\Gamma }_j\to {\Gamma }_j,\text{such that}[\theta ]\to [G_j(\theta )],$$

$$\theta =\sigma +\sum _{i=1}^k{\alpha }_{i}Jd{F}_i,{\theta }^{{}^{\prime }}={\sigma }^{{}^{\prime }}+\sum _{i=1}^k{\delta }_{i}Jd{F}_i.$$

$$[\theta ,{\theta }^{{}^{\prime }}]=[\sum _{i=1}^k{\alpha }_{i}Jd{F}_i,\sum _{i=1}^k{\delta }_{i}Jd{F}_i],$$

$$[\theta ,{\theta }^{{}^{\prime }}]=[\sigma ,{\sigma }^{{}^{\prime }}]+[\sum _{i=1}^k{\alpha }_{i}Jd{F}_i,\sum _{i=1}^k{\delta }_{i}Jd{F}_i],$$

$$+[\sigma ,\sum _{i=1}^k{\delta }_{i}Jd{F}_i]+[\sum _{i=1}^k{\alpha }_{i}Jd{F}_i,{\sigma }^{{}^{\prime }}].$$

Full text

1 Introduction

In several problems of the quantum and theoretical physics approximated solution to the partial differential equations which contain a small parameter in the higher derivative order are obtained, as well as to approximated eigenvalues and eigenvector of self-adjoint differential operator which depend on a small parameter. In such problems have been used the asymptotic methods [8, 7, 6], which nowadays are developed widely in several branches of the physics-mathematics. It is well known the success of asymptotic methods, e.g. with the quantification method was solved the older and sharp problem of the mechanic classic: calculation of the energetic level of the hydrogen atom [3].

In [7] over a 2n-dimensional phases space to obtain an asymptotic quasiclassical solution with respect to a small parameter on an isotropic tori k-dimensional ($\displaystyle k<n$) is obtained. This asymptotic on a torus is accomplished with a new geometric object which was called the Complex Germen .i.e. a family of complex planes with certain properties. Such object does not exist over any isotropic manifold. In such sense, V. P. Maslov put forward and solved the problem about its existence and construction techniques. At the same time, the uniqueness of Complex Germ has a great signification such that asymptotic be well defined. In [4] were completed the results obtained in [7], as well as have solved the uniqueness problem for the singular point and a closed trajectory of the Hamiltonian system. Further, was solved of existence and uniqueness of Germen on isotropic torus to Hamiltonian with cyclic variable.

Some more later, M. M. Nekhoroshev put forward this problem but on isotropic invariant torus with respect to Hamiltonian phases flows which come from k-functions in involution. This statement was partially solved in [9] establishing that if certain simplectic operator has a simple spectrum then the complex germ exist. In this work we solve this problem, providing a full solution, i.e. we present conditions for the existence and uniqueness of complex germ through the monodromy operator constructed in [9], but without the simple spectrum condition. We study also the Hamiltonian system with cyclic variables.

2 Preliminaries

Let $\displaystyle M^{2n}$ be a $\displaystyle 2n$-dimensional differentiable manifold with local coordinates $\displaystyle(p,q)$. We assume $\displaystyle C^{\infty}$ manifolds and scalar functions and vectorial fields as well. Let us introduce basic results, details can be found in [1]. Let $\displaystyle T_{m}M^{2n}$ be the tangent space at point $\displaystyle m$ of manifold $\displaystyle M^{2n}$. We introduce natural manifold on $\displaystyle TM^{2n}=\displaystyle\cup_{m\in M^{2n}}T_{m}M^{2n}$.

Definition 1**.**

It is called exterior form of degree two or 2-form at point $\displaystyle m$ on the manifold $\displaystyle M^{2n}$ to the bilinear and antisymmetric application $\displaystyle\omega^{2}:T_{m}M^{2n}\times T_{m}M^{2n}\rightarrow\Re$ , i.e.

• •

$\displaystyle\omega^{2}(\alpha x+\beta y,z)=\alpha\omega^{2}(x,z)+\beta\omega^{2}(y,z)$,

• •

$\displaystyle\omega^{2}(x,y)=-\omega^{2}(y,x)$,

for all $\displaystyle x,y,z\in T_{m}M^{2n}$ and $\displaystyle\alpha,\beta\in\Re$ (see [2]).

A 2-form $\displaystyle\omega^{2}$ is closed if $\displaystyle d\omega^{2}\equiv 0$, where $\displaystyle d:\Omega^{2}(M^{2n})\rightarrow\Omega^{3}(M^{2n})$ is an operator of exterior differentiation on the space of the 2-form $\displaystyle\Omega^{2}(M^{2n})$. Besides, it is called non-degenerate if $\displaystyle\omega^{2}(x,y)=0$ for all $\displaystyle x\in T_{m}M^{2n}$ then $\displaystyle y=0$.

Definition 2**.**

A closed, non-degenerate and differential 2-form $\displaystyle\omega^{2}$ on the manifold $\displaystyle M^{2n}$ is called symplectic structure. The couple $\displaystyle(M^{2n},\omega^{2})$ is called symplectic manifold and the tangent space in each point $\displaystyle m$ of the manifold is called symplectic vectorial field whose symplectic structure is the restriction of $\displaystyle\omega^{2}$ to $\displaystyle T_{m}M^{2n}\times T_{m}M^{2n}$.

One example of symplectic structure consist of $\displaystyle M^{2n}=\Re^{2n}$ with $\displaystyle\omega^{2}=dq\wedge dp$, where

[TABLE]

and for all $\displaystyle m\in\Re^{2n}$. In this case $\displaystyle T_{m}\Re^{2n}$ is identified with $\displaystyle\Re^{2n}$, where the 1-forms $\displaystyle dq_{i}$ and $\displaystyle dp_{i}$ are defined as

[TABLE]

where $\displaystyle q_{i}$ and $\displaystyle p_{i}$: $\displaystyle\Re^{2n}\rightarrow\Re$ are the coordinates system. i.e. $\displaystyle q_{i}(x)$ ($\displaystyle p_{i}(x)$) are $\displaystyle i$-th ($\displaystyle n+i$-th) coordinates of the vector $\displaystyle x$ in a prefixed basis of the real linear space $\displaystyle\Re^{2n}$. The symplectic structure defined in this way is called standard.

Definition 3**.**

The coordinates of the local chart $\displaystyle(q_{1},\ldots,q_{n},p_{1},\ldots,p_{n})$ of a symplectic manifold are called canonical if the expression of the symplectic structure $\displaystyle\omega^{2}$ in this coordinate system coincides with the standard.

As consequence of the Darboux Theorem, in each point of a symplectic manifold, there is a neighbour with canonical coordinates [1].

Definition 4**.**

Let $\displaystyle M^{2n}$ be a symplectic manifold and let $\displaystyle T_{m}M^{2n}$ and $\displaystyle T_{m}^{*}M^{2n}$ be tangent and cotangent spaces at point $\displaystyle m\in M^{2n}$, we define the operator $\displaystyle J:T_{m}^{*}M^{2n}\rightarrow T_{m}M^{2n}$ as

[TABLE]

The operator $\displaystyle J$ defined in this way constitutes an isomorphism between vectorial spaces.

Definition 5**.**

The Poisson bracket of two functions $\displaystyle F$, $\displaystyle G$ over the manifold $\displaystyle M^{2n}$ is defined as

[TABLE]

where $\displaystyle dF$ and $\displaystyle dG$ are the differentials 1-forms of $\displaystyle F$ and $\displaystyle G$ on $\displaystyle M^{2n}$.

If $\displaystyle[F,G]=0$, we say that functions $\displaystyle F$ and $\displaystyle G$ are in involution.

From Definition 4, we have

[TABLE]

Definition 6**.**

A Hamiltonian system is the triple $\displaystyle(M^{2n},\omega^{2},H)$, where $\displaystyle(M^{2n},\omega^{2})$ is a symplectic manifold and functions $\displaystyle H$ is defined on it. The field $\displaystyle JdH$ is called the Hamiltonian vectorial field.

The matrix of the Hamiltonian operator $\displaystyle H$ in canonical coordinates is

[TABLE]

where $\displaystyle 0$ and $\displaystyle I_{n}$ denotes the zero and identity n-dimensional matrices. Thus, in canonical coordinates the Hamiltonian system takes the form

[TABLE]

where $\displaystyle H_{q}=(\frac{\partial H}{\partial q_{1}},\ldots,\frac{\partial H}{\partial q_{n}})$ and $\displaystyle H_{p}=(\frac{\partial H}{\partial p_{1}},\ldots,\frac{\partial H}{\partial p_{n}})$.

Let us assume that the solution of the Hamiltonian system $\displaystyle(M^{2n},\omega^{2},H)$ can be extended to $\displaystyle\Re$, i.e. for $\displaystyle-\infty<t<+\infty$. In this case, $\displaystyle g^{t}_{H}m$ denotes the value of the solution $\displaystyle\theta$ with initial condition $\displaystyle\theta(0)=m$, we obtain the application $\displaystyle g^{t}_{H}M^{2n}\rightarrow M^{2n}$ for $\displaystyle t$ fixed. This application constitute a one parametric group of diffeomorphism, i.e. $\displaystyle g^{0}_{H}m=m$ and $\displaystyle g^{t+s}_{H}m=g^{t}_{H}m\circ g^{s}_{H}m$, for all $\displaystyle m\in M^{2n}$. This group is called flow of phases of the Hamiltonian system. Moreover, the application $\displaystyle g^{t}_{M}$ is symplectic for each $\displaystyle t$ fixed, i.e. $\displaystyle(g^{t}_{H})^{*}\omega^{2}=\omega^{2}$, where

[TABLE]

for all $\displaystyle m\in M^{2n}$. Here $\displaystyle(g^{t}_{H})_{*,m}$ denotes the derivatives of $\displaystyle(g^{t}_{H})$ for each $\displaystyle t\in\Re$ (see [2]).

Definition 7**.**

Let $\displaystyle\varLambda$ a submanifold of $\displaystyle M^{2n}$ and $\displaystyle T_{m}\varLambda$ the tangent space of $\displaystyle\varLambda$ at point $\displaystyle m$. The submanifold $\displaystyle\varLambda$ is called isotropic if the symplectic structure $\displaystyle\omega^{2}$ is null on it, i.e., $\displaystyle\omega^{2}(x,y)=0,\forall x,y\in T_{m}\varLambda$ and $\displaystyle m\in M^{2n}$.

The submanifold is invariant respect to the Hamiltonian system $\displaystyle(M^{2n},\omega^{2},H)$ if $\displaystyle IdH(m)\in T_{m}\varLambda,\forall m\in\varLambda$, which in terms of the phases flow is rewritten as $\displaystyle g^{t}_{H}(\varLambda)=\varLambda$, for all $\displaystyle t\in\Re$, hence $\displaystyle(g^{t}_{H})_{*,m}(T_{m}\varLambda)=T_{m}\varLambda$.

Let us consider the complexification of the linear space $\displaystyle\Re^{2}$ and a linear operator for a positive integer $\displaystyle n$. The complexification of $\displaystyle\Re^{n}$ is a n-dimensional linear space $\displaystyle(\Re^{n})^{\mathbb{C}}$ constructed as follow: the point in $\displaystyle(\Re^{n})^{\mathbb{C}}$ is denotes either by $\displaystyle(x,y)$ or $\displaystyle x+iy$ , where $\displaystyle x,y\in\Re^{n}$.

If $\displaystyle\alpha\cdot V$ denotes the multiplication of a scalar by $\displaystyle V\in\Re^{n}$ and $\displaystyle V+W$ the sum of two any vectors in $\displaystyle\Re^{n}$, the the multiplication of complex scalar $\displaystyle\alpha+i\beta$ by a vector $\displaystyle V+iW\in(\Re^{n})^{\mathbb{C}}$ and the sum of two vectors $\displaystyle V_{1}+iW_{1}$, $\displaystyle V_{2}+iW_{2}$ are defined as:

[TABLE]

thus $\displaystyle(\Re^{n})^{\mathbb{C}}$ constitutes an complex lineal space $\displaystyle((\Re^{n})^{\mathbb{C}}=\mathbb{C}^{n})$.

Remark 8**.**

Similarly, for any real vectorial subspace $\displaystyle P$ is possible to define its complexification $\displaystyle P^{\mathbb{C}}$.

It is well known, that between the tangent space $\displaystyle T_{m}(M^{2n})$ to submanifold 2n-dimensional $\displaystyle M^{2n}$ and $\displaystyle\Re^{2n}$ there is an isomorphism $\displaystyle X:T_{m}(M^{2n})\rightarrow\Re^{2n}$. Analogously, we can establish an isomorphism between the complexifications $\displaystyle T_{m}(M^{2n})^{\mathbb{C}}$ and $\displaystyle\Re^{\mathbb{C}}$.

Let us denote by $\displaystyle A:\Re^{m}\rightarrow\Re^{d}$ a $\displaystyle\Re$-linear operator. A complexification of the operator $\displaystyle A$ is a $\displaystyle\mathbb{C}$-linear operator $\displaystyle A^{\mathbb{C}}:(\Re^{m})^{\mathbb{C}}\rightarrow(\Re^{d})^{\mathbb{C}}$ defined by the relation $\displaystyle A^{\mathbb{C}}(x+iy)=Ax+iAy$. The following relations are valid: $\displaystyle(A+B)^{\mathbb{C}}=A^{\mathbb{C}}+B^{\mathbb{C}}$, where $\displaystyle A$ and $\displaystyle B$ are $\displaystyle\Re$-linear operators.

Let us introduce the following concepts: let $\displaystyle\omega^{2}$ be a real simplectic structure on the manifold $\displaystyle M^{2n}$. We call complexification of $\displaystyle\omega^{2}$ defined on $\displaystyle T_{m}(M^{2n})^{\mathbb{C}}$, for all $\displaystyle m\in M^{2n}$ to the form $\displaystyle\omega^{\mathbb{C}}:T_{m}(M^{2n})^{\mathbb{C}}\times T_{m}(M^{2n})^{\mathbb{C}}\rightarrow\mathbb{C}$, given by

[TABLE]

$\displaystyle\forall u+iv,x+iy\in T_{m}(M^{2n})^{\mathbb{C}}$.

Remark 9**.**

Since $\displaystyle\omega^{2}$ is antisimetric then $\displaystyle(1/2i)(\omega^{2})^{\mathbb{C}}(x,\bar{x})$ is real for all $\displaystyle x\neq 0$ in $\displaystyle T_{m}(M^{2n})$. We denote $\displaystyle(\omega^{2})^{\mathbb{C}}(x,y)=[x,y]$. It is possible to verify that the complexification of the standard simplectic structure on $\displaystyle T_{m}(M^{2n})$ is the standard simplectic structure on $\displaystyle T_{m}(M^{2n})^{\mathbb{C}}$ at any point $\displaystyle m$ of the manifold $\displaystyle M^{2n}$ ([5]).

3 Statement of the problem

Let $\displaystyle M^{2n}$ be a $\displaystyle 2n$-dimensional simplectic manifold and $\displaystyle F_{1},\ldots,F_{k}$ a family of functions defined on $\displaystyle M^{2n}$: $\displaystyle F_{j}:M^{2n}\rightarrow\Re$, $\displaystyle j=1,\ldots,k$ such that $\displaystyle k<n$, which stay in involution on $\displaystyle M^{2n}$. Let the $\displaystyle k$-Hamiltonian systems $\displaystyle(M^{2n},\omega^{2},F_{j})$ with the correspond Hamiltonian flux of phases $\displaystyle{g^{t}_{F_{j}}},t\in\Re;j=1,\ldots,k$. Since the functions $\displaystyle F_{j},j=1,\ldots,k$ are in involution, the flux of phases commute, i.e. $\displaystyle g^{t}_{F_{i}}og^{t}_{F_{j}}=g^{t}_{F_{j}}og^{t}_{F_{i}}$, for all $\displaystyle t\in(-\infty,+\infty)$ and for all $\displaystyle i,j=1,\ldots,k$.

We assume that

• •

The Hamiltonian flux phases $\displaystyle{g^{t}_{F_{j}}}$, $\displaystyle j=1,\ldots,k$ are global, i.e. they are defined for all $\displaystyle t\in(-\infty,+\infty)$. It is valid for example if we assume that the manifold $\displaystyle M^{2n}$ is compact.

• •

The torus $\displaystyle\Lambda^{k}=S^{1}\times,\ldots,\times S^{1}$ (where $\displaystyle S^{1}$ denotes the unit circle), $\displaystyle k<n$ is a $\displaystyle k$-dimensional isotropic submanifold of $\displaystyle M^{2n}$ ($\displaystyle\Lambda^{k}\subset M^{2n}$) and is invariant respect to the Hamiltonian system $\displaystyle(M^{2n},\omega^{2},F_{j}),j=1,\ldots,k$.

• •

The differential $\displaystyle dF_{1},\ldots,dF_{k}$ are lineal independent at each point of the manifold $\displaystyle M^{2n}$.

For the construction of asymptotic solution of several partial differential equation in [7] the concept of Complex Germ on isotropic manifold is introduced. The issues of existence and uniqueness of such object is treated in this work. The main difficulty is that not always exist the Germ over any isotropic manifold.

Definition 10**.**

A Complex Germ over the isotropic tori $\displaystyle\Lambda^{k}(k<n)$ is a smooth map on $\displaystyle\Lambda^{k}$, $\displaystyle r^{n}:m\rightarrow r^{n}(m),\forall m\in\Lambda^{k}$, such that to each point $\displaystyle m\in\Lambda^{k}$ correspond a $\displaystyle n$-dimensional complex subspace $\displaystyle r^{r}(m)$ of the complexification of the tangent space to $\displaystyle M^{2n}$ at the point $\displaystyle m$ ($\displaystyle T_{m}(M^{2n})^{\mathbb{C}}$) with following properties:

i) $\displaystyle r^{n}(m)$ is a lagrangian subspace, i.e. $\displaystyle dim(r^{n}(m))=n$ and isotropic ($\displaystyle[x,y]=0,\forall x,y\in r^{n}(m)$),

ii) $\displaystyle r^{n}(m)\supset T_{m}(\Lambda^{k})^{\mathbb{C}}$,

iii) $\displaystyle r^{n}(m)$ is disipative respect of $\displaystyle T_{m}(\Lambda^{k})$, i.e.

[TABLE]

iv) $\displaystyle r^{n}(m)$ is invariant respect the Hamiltonian flux $\displaystyle{g^{t}_{H}},t\in\Re$ of a given function $\displaystyle H$,i.e. $\displaystyle\forall\in\Lambda^{k},\forall t\in(-\infty,+\infty)$ holds

[TABLE]

where $\displaystyle[(g^{t}_{H})_{*,m}]^{\mathbb{C}}$ is the complexification of the derivative at point $\displaystyle m$ of an element $\displaystyle g^{t}_{H}$ of Hamiltonian flux phases $\displaystyle H$ associated to Hamilton function $\displaystyle H$.

Remark 11**.**

From now we omit the supraindex $\displaystyle\mathbb{C}$ that indicated the complexification, in the space and operator. But implicitly we use the properties enunciated previously.

The aim of this paper is to seek conditions for the existence and uniqueness of a Complex Germ on the a invariant torus to Hamiltonian system $\displaystyle(M^{2n},\omega^{2},F_{j}),~{}j=1,\ldots,k;(k<n)$.

4 Condition for the existence of the Complex Germ on the torus

Let

[TABLE]

be the intersection of the level surfaces defined by the functions $\displaystyle F_{j},j=1,\ldots,k$ which contain the trajectories of the Hamiltonian system $\displaystyle\displaystyle x^{\textbullet}=JdF_{j}(x),j=1,\ldots,k$. Since $\displaystyle dF_{1},\ldots,dF_{k}$ are linearly independent then $\displaystyle\Sigma$ is a submanifold submerge in $\displaystyle M^{2n}$ of dimension $\displaystyle 2n-k$.

Let us define the action of additive group $\displaystyle\Re^{k}$ over $\displaystyle M^{2n}$, as follow: to each $\displaystyle\overrightarrow{t}=(t_{1},\ldots,t_{k})\in\Re^{k}$ correspond the difemeorphism of $\displaystyle M^{2n}$ as:

[TABLE]

where $\displaystyle g_{j}^{t}=g_{F_{j}}^{t},\forall j=1,\dots,k$. Since $\displaystyle g_{j}^{t}$ are simplectic diffemeorphism the $\displaystyle g^{\overrightarrow{t}}$ as well, i.e. $\displaystyle(g^{\overrightarrow{t}})^{*}\omega^{2}=\omega^{2},\forall\overrightarrow{t}\in\Re^{k}$.

4.1 Monodromy operator

The condition for existence of the Complex Germ are given in term of the monodromy operator which we defined in the following paragraph

Let us fix the point $\displaystyle m\in\Lambda^{k}$. Denoting by $\displaystyle G$ the discrete subgroup of $\displaystyle\Re^{k}$ defined as follow

[TABLE]

which does not depend on the choice of the point $\displaystyle m$. The subgroup $\displaystyle G$ can be generated by a set of $\displaystyle k$ elements linearly independent(see [1]), i.e.

[TABLE]

Since $\displaystyle\Lambda^{k}$ is invariant respect to Hamiltonian system $\displaystyle x^{\prime}=JdF_{j}(x);j=1,\ldots,k$ we have $\displaystyle g_{j}^{t}\Lambda^{k}=\Lambda^{k}$ therefore $\displaystyle g^{\overrightarrow{t}}\Lambda^{k}=\Lambda^{k}$. We have also that the map $\displaystyle f:\Re^{k}\rightarrow M^{2n}$ given by $\displaystyle f(t)=g^{\overrightarrow{t}}m$ which to any $\displaystyle\overrightarrow{t}$ correspond a point in $\displaystyle\Lambda^{k}$ is sobrejective, but is not injective because $\displaystyle\Lambda^{k}$ is compact and $\displaystyle\Re^{k}$ is not; therefore there exist $\displaystyle t_{1},t_{2}\in\Re^{k};t_{1}\neq t_{2}$, such that $\displaystyle g^{t_{1}}m=g^{t_{2}}m$ and $\displaystyle g^{t_{1}-t_{2}}m=m$ with $\displaystyle t_{1}-t_{2}\in G$; which mean that the subgroup $\displaystyle G$ is not trivial.

Definition 12**.**

Let $\displaystyle T\in G$. The operator $\displaystyle G_{m}=(g^{T}_{*,m}):T_{m}(M^{2n})\rightarrow T_{m}(M^{2n})$ is called monodromy operator of the subgroup $\displaystyle G$ at point $\displaystyle m\in\Lambda^{k}$.

Let us rewrite the monodromy operator in a way more amassing to describe the sufficient condition for the existence of the Complex Germ. We gave

[TABLE]

for certain $\displaystyle l=(l_{1},\ldots,l_{k})\in Z^{k}$, where $\displaystyle T_{1},\ldots,T_{k}$ are the generator of subgroup $\displaystyle G$. Let denote $\displaystyle T_{j}=(t_{j1},\ldots,t_{jk});t_{ij}\Re;\forall i,j=1,\ldots,k$. thus we obtain

[TABLE]

Using that $\displaystyle g_{j}^{t+s}=g_{j}^{t}og_{j}^{s}$ and $\displaystyle g_{j}^{t}og_{i}^{t}=g_{i}^{t}og_{j}^{t}$ and reordering (18) we obtain

[TABLE]

or

[TABLE]

where $\displaystyle G_{j}=g_{*,m}^{T_{j}}:T_{m}(M^{2n})\rightarrow T_{m}(M^{2n}),j=1,\ldots,k$ is called monodromy operator with period $\displaystyle T_{j}$. The following is valid

Lemma 13**.**

Over the torus $\displaystyle\Lambda_{k}$ is possible to defined $\displaystyle k$ vectorial fields linearly independent.

Proof.

Since $\displaystyle\Lambda^{k}$ is invariant respect to the $\displaystyle k$- Hamiltonian system $\displaystyle\dot{x}=JdfF_{j}(x)$,

$\displaystyle j=1,\ldots,k$, $\displaystyle JdF_{j}(m)\in T_{m}(\Lambda^{k}),\forall m\in\Lambda^{k}$. Also, since $\displaystyle dF_{1},\ldots,dF_{k}$ are linearly independent at each point $\displaystyle m\in\Lambda^{k}$ and the operator $\displaystyle J$ is regular we obtain that the vector $\displaystyle JdF_{j}(m),j=1,\ldots,k$ are linearly in each point of $\displaystyle m\in\Lambda^{k}$. ∎

Note that $\displaystyle JdF_{j},j=1,\dots,k$ in each point $\displaystyle m\in\Lambda^{k}$ constitute a basis of the tangent space $\displaystyle T_{m}(\Lambda^{k})$ of $\displaystyle\Lambda^{k}$. Using the $\displaystyle\Lambda^{k}$ is invariant we obtain

[TABLE]

[TABLE]

with $\displaystyle\beta_{n},\mu_{n}\in\mathbb{C};n=1,\ldots,k$. Let $\displaystyle\Gamma_{m}=T_{m}(\Sigma)\diagup T_{m}(\Lambda^{k})$; we have $\displaystyle dim(\Gamma_{m})=2(n-k)$.

It is valid the following

Lemma 14**.**

Let $\displaystyle[\theta]\in\Gamma_{m}$ then $\displaystyle G_{j}([\theta])=[G_{j}(\theta)]$ and $\displaystyle G_{m}([\theta])=[G_{m}(\theta)]$.

Proof.

Let $\displaystyle[\theta]=\{\theta:\theta^{{}^{\prime}}\equiv\theta mod(T_{m}(\Lambda^{k}))\}$, $\displaystyle\theta^{{}^{\prime}}\equiv\theta mod(T_{m}(\Lambda^{k}))$ if and only if $\displaystyle\theta^{{}^{\prime}}=\theta+\displaystyle\sum_{i=1}^{k}\beta_{i}JdF_{i}$, where $\displaystyle\beta_{i}\in\mathbb{C},i=1,\ldots,k$, Also the following equality is valid

[TABLE]

and by using equality (21) we have $\displaystyle G_{j}(\theta^{{}^{\prime}})=G_{j}(\theta)+\sum_{i=1}^{k}\mu_{i}JdF_{i}$ for certain $\displaystyle\mu_{i}\in\mathbb{C},i=1,\ldots,k$ and due to (20) also satisfy that

[TABLE]

[TABLE]

[TABLE]

with $\displaystyle j=1,\ldots,k$. From now, the structure $\displaystyle[,]$ must be defined between equivalence class modulo $\displaystyle T_{m}(\Lambda)$ on each point $\displaystyle m\ \in\Lambda^{k}$. To do so, we need to prove that $\displaystyle[,]$ is compatible with respect to equivalence class i.e. if $\displaystyle\theta\equiv\sigma mod(T_{m}(\Lambda^{k}))$ and $\displaystyle\theta^{{}^{\prime}}\equiv\sigma^{{}^{\prime}}mod(T_{m}(\Lambda^{k}))$, then $\displaystyle[\theta,\theta^{{}^{\prime}}]=[\sigma,\sigma^{{}^{\prime}}]$. We assume by definition that $\displaystyle[[\theta],[\theta]]=[\theta,\theta]$. We prove that $\displaystyle[\theta,\theta^{{}^{\prime}}]=[\sigma,\sigma^{{}^{\prime}}]$:

[TABLE]

By using bi-linearity property we have

[TABLE]

or

[TABLE]

We have

[TABLE]

because $\displaystyle JdF_{i}(\sigma)=0$ since $\displaystyle\sigma\in T_{m}(\Sigma)$. Analogously is prove that $\displaystyle[\displaystyle\sum_{i=1}^{k}\alpha_{i}JdF_{i},\sigma^{{}^{\prime}}]=0$. Also that $\displaystyle[\displaystyle\sum_{i=1}^{k}\alpha_{i}JdF_{i},\displaystyle\sum_{i=1}^{k}\delta_{i}JdF_{i}]=0$ holds , since the functions $\displaystyle F_{i},i=1,\ldots,k$ stay in involution. Finally we have the proof.

∎

Definition 15**.**

The operator $\displaystyle\Xi_{m}:\Gamma_{m}\rightarrow\Gamma_{m}$ is called reduced monodromy operator at point $\displaystyle m$. i.e.,

[TABLE]

where the operators $\displaystyle\Xi_{j}$, $\displaystyle j=1,\ldots,k$ are called reduced monodromy operator with period $\displaystyle T_{j}$.

Also the following proposition is valid

Proposition 16**.**

The quotient space $\displaystyle\Gamma_{m}=T_{m}(\Sigma)/T_{m}(\Lambda^{k})$ has a natural symplectic structure such that the operators $\displaystyle\Xi_{j}$, $\displaystyle j=1,\ldots,k$ are symplectic respect to the structure $\displaystyle[,]$ of the space $\displaystyle T_{m}(M^{2n})$ induce over this space.

Proof.

It is known that if the vectorial space $\displaystyle V$ is given a bilinear, antisymmetric, degenerate form then over the quotient vectorial space $\displaystyle V/V^{\perp}$ is induced a bilinear, antisymmetric, no degenerate form. Taking $\displaystyle V=T_{m}(\Sigma)$ and the bilinear form $\displaystyle[,]$ which is degenerate over $\displaystyle T_{m}(\Sigma)$ and we use that $\displaystyle T_{m}(\Sigma)^{\perp}=T_{m}(\Lambda^{k})$. Let $\displaystyle\sigma=\sum_{i=1}^{k}\sigma_{i}JdF_{i}$, which $\displaystyle\sigma_{i}\neq 0$ for some $\displaystyle i=1,\ldots,k$. Since $\displaystyle\sigma\in T_{m}(\Lambda^{k})$ holds that $\displaystyle[x,\sigma]=0$,  $\displaystyle\forall x\in T_{m}(\Sigma)$ because $\displaystyle[x,IJdF_{i}]=dF_{i}(x)=0$ ($\displaystyle F_{i}$ is constant on $\displaystyle\Sigma$). This proved that $\displaystyle[,]$ is degenerate on $\displaystyle T_{m}(\Sigma)$ and $\displaystyle JdF_{i}\in T_{m}(\Sigma)$. Since $\displaystyle\dim(T_{m}(\Sigma))=2n-k$ then $\displaystyle\dim(T_{m}(\Sigma)^{\bot})=k$ and $\displaystyle IJdF_{i}$ ($\displaystyle i=1,\ldots,k$) constitutes a basis of $\displaystyle T_{m}(\Lambda^{k})$. Since $\displaystyle T_{m}(\Sigma)^{\bot}=T_{m}(\Lambda^{k})$ we have that $\displaystyle\Gamma_{m}$ is a subspace lineal symplectic. Now, we verify that the operator $\displaystyle\Xi_{j}$ ($\displaystyle j=1,\ldots,k$) are symplectic. Let $\displaystyle[\theta],[\theta^{{}^{\prime}}]\in\Gamma_{m}$. we have

[TABLE]

where $\displaystyle G_{j}$, with $\displaystyle j=1,\ldots$ are the monodromy operators in Definition 12. From Lemma 14 we have

[TABLE]

By definition of the product of two class we obtain

[TABLE]

Since $\displaystyle G_{j}$ are symplectic we have $\displaystyle[\Xi_{j}([\theta]),\Xi_{j}([\theta^{{}^{\prime}}])]=[\theta,\theta^{{}^{\prime}}]$. Using again the definition of the product of two class we have $\displaystyle[\Xi_{j}([\theta]),\Xi_{j}([\theta^{{}^{\prime}}])]=[[\theta],[\theta^{{}^{\prime}}]]$; so $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\ldots,k$ are symplectic. ∎

Definition 17**.**

Two symplectic lineal operator $\displaystyle A_{i}:L^{1}\rightarrow L^{2}$, with $\displaystyle i=1,2$. are called equivalent if there exist a lineal symplectic $\displaystyle\tau:L^{1}\rightarrow L^{2}$ such that $\displaystyle A_{2}=\tau A_{2}\tau^{-1}$.

For the purposes of this work we need to verify that the reduced monodromy operator does not depends on the point $\displaystyle m\in T_{m}(\Lambda^{k})$. They are precisely these operators which will serve to establish sufficient conditions for the existence of the complex germ. Notice $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\ldots,k$ are symplectic operators then by Definition 15, operator $\displaystyle\Xi_{m}$ in (31) is symplectic as well.

It is valid the following

Proposition 18**.**

Let $\displaystyle m,m^{{}^{\prime}}\in\Lambda^{k}$ then operators $\displaystyle\Xi_{m}$ and $\displaystyle\Xi_{m^{{}^{\prime}}}$ defined in (31) are equivalent.

Proof.

There exist $\displaystyle\overrightarrow{t}\in\Re^{k}$ such that $\displaystyle m=g^{\overrightarrow{t}}m^{{}^{\prime}}$. Let the operator $\displaystyle G_{m,m^{{}^{\prime}}}=(g^{\overrightarrow{t}})_{*,m}:T_{m}(M^{2m}\rightarrow T_{m^{{}^{\prime}}}(M^{2m})$, which is symplectic (see Section 2 ). Also $\displaystyle G_{m,m^{{}^{\prime}}}(T_{m}(\Sigma))=T_{m^{{}^{\prime}}}(\Sigma)$ and $\displaystyle G_{m,m^{{}^{\prime}}}(T_{m}(\Lambda^{k}))=T_{m}(\Lambda^{k})$ hold. Then we define the operator $\displaystyle\tau_{m,m^{{}^{\prime}}}:\Gamma_{m}\rightarrow\Gamma_{m^{{}^{\prime}}}$ by $\displaystyle\tau_{m,m^{{}^{\prime}}}([\theta])=[G_{m,m^{{}^{\prime}}}(\theta)]$. We can check that $\displaystyle\Xi_{m^{{}^{\prime}}}\tau_{m,m^{{}^{\prime}}}=\tau_{m,m^{{}^{\prime}}}\Xi_{m}$. ∎

Remark 19**.**

Analogously it can prove that the operators $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\ldots,k$ are equivalents at different points in the torus $\displaystyle\Lambda^{k}$.

Let $\displaystyle\Pi:T_{m}(\Sigma)\rightarrow\Gamma_{m}$ the canonical map that each $\displaystyle\theta\in T_{m}(\Sigma)$ correspond its equivalence class $\displaystyle[\theta]$ modulo $\displaystyle T_{m}(\Lambda^{k})$.

We have that is valid

Proposition 20**.**

It is valid that

[TABLE]

Proof.

Let $\displaystyle\theta\in T_{m}(\Sigma)$; $\displaystyle\Pi(G_{j}(\theta))=[G_{j}(\theta)]$. Besides that $\displaystyle\Xi_{j}(\Pi(\theta))=\Xi_{j}([\theta])=[G_{j}(\theta)]$, being proved the proposition. ∎

Definition 21**.**

A complex linear subspace $\displaystyle R$ $\displaystyle\subset$ $\displaystyle\mathbb{C}^{n}$ is called positive if   $\displaystyle\forall x\in R;x\neq 0$ $\displaystyle(1/2i)[x,\bar{x}]>0$ holds. It subspace is called negative if $\displaystyle(1/2i)[x,\bar{x}]<0$.

Lemma 22**.**

Let $\displaystyle R\subset\Gamma_{m}$ a lineal, positive, lagrangian and invariant respect to the operators $\displaystyle\Xi_{i}$, $\displaystyle i=1,\ldots,k$. then the subspace $\displaystyle r^{n}=\Pi^{-1}(R)$ is dissipative respect to $\displaystyle T_{m}(\Lambda^{k})$, lagrangian and invariant respect of the operator $\displaystyle G_{i}$, $\displaystyle i=1,\ldots,k$, where $\displaystyle\Pi$ is the canonical map.

Proof.

We have that $\displaystyle\Pi(x)=[x]modT_{m}(\Lambda^{k})=\{x+y:y\in T_{m}(\Lambda^{k})\}$;

a) Let us prove that $\displaystyle r^{n}=\Pi^{-1}(R)$ is isotropic. Let $\displaystyle x_{1},x_{2}\in\Pi^{-1}(R)$ then $\displaystyle\Pi(x_{1}),\Pi(x_{2})\in R;[\Pi(x_{1}),\Pi(x_{2})]=0$ for being $\displaystyle r^{n}$ isotropic; by definition we have $\displaystyle[x_{1},x_{2}]=[\Pi(x_{1}),\Pi(x_{2})]=0$, therefore $\displaystyle r^{n}$ is isotropic.

b) Let us prove that $\displaystyle r^{n}$ is lagrangian in $\displaystyle T_{m}(M^{2n}),i.e.dim(R)=(1/2)dim(\Gamma_{m})=n-k,\Pi^{-1}(0)=T_{m}(\Lambda^{k})$, then $\displaystyle dim(\Pi^{-1}(0))=k$, also since $\displaystyle\Pi^{-1}(0)\subset\Pi^{-1}(R)$, then and $\displaystyle dim(r^{n})=dim(Ker(\Pi))+dim(Im(\Pi^{-1}(R)))=k+n-k=n$, then $\displaystyle r^{n}$ is lagrangian.

c) Let us verify that $\displaystyle r^{n}$ is dissipative respect to $\displaystyle T_{m}(\Lambda^{k})$: we have that $\displaystyle T_{m}(\Lambda^{k})\subset r^{n}$. Let $\displaystyle x\in r^{n}\diagdown T_{m}(\Lambda^{k})$ then $\displaystyle\Pi(x)=[x]\neq 0$. Besides we have $\displaystyle[\bar{x}]=\bar{[x]}$, the dissipative condition of $\displaystyle r^{n}$ is a consequence of the positivity of $\displaystyle R$, i.e.

[TABLE]

due to $\displaystyle R$ is positive.

d) The invariance of $\displaystyle r^{n}$ respect of $\displaystyle G_{i}$ ($\displaystyle i=1,\ldots,k$) is a consequence of (32) and the invariance of $\displaystyle R$ respect to $\displaystyle\Xi_{j}$ ($\displaystyle i=1,\ldots,k$). ∎

It is valid the following

Theorem 23**.**

Let $\displaystyle m\in\Lambda^{k}$ fixed. If there exist a lineal subspace $\displaystyle N$ lagrangian, dissipative respect to $\displaystyle T_{m}(\Lambda^{k})$, invariant respect all the operators $\displaystyle G_{i}$ ($\displaystyle i=1,\ldots,k$) then the exist a complex germ respect to the Hamiltonian system $\displaystyle x^{{}^{\prime}}=JdF_{j}(x)$ ($\displaystyle j=1,\ldots,k$).

Proof.

The idea is to construct a smooth map $\displaystyle r^{n}:m\rightarrow r^{n}(m)$ such $\displaystyle r^{n}(m)\subset T_{m}(M^{2n})$, $\displaystyle\forall m\in\Lambda^{k}$ satisfying Definition 10. Let us define $\displaystyle r^{n}(m)=N$. Since that map $\displaystyle f:\vec{t}\rightarrow g^{\vec{t}}m$, $\displaystyle\vec{t}\in\Re^{k}$ is subjective, we have that $\displaystyle\forall p\in\Lambda^{k}$, there exist $\displaystyle m\in\Lambda^{k}$ and $\displaystyle\vec{s}\in\Re^{k}$ such that $\displaystyle g^{\vec{s}}m=p$. Let us put

[TABLE]

Since the operator $\displaystyle(g^{\vec{s}})_{*,m}$ is symplectic and carries the subspace $\displaystyle T_{m}(\Lambda^{k})$ in $\displaystyle T_{p}(\Lambda^{k})$ then $\displaystyle r^{n}(p)~{}~{}\forall p\in\Lambda^{k}$ is lagrangian, disipative with respect to $\displaystyle T_{p}(\Lambda^{k})$.

Now, we verify that $\displaystyle r^{n}(m)$ is invariant: By Definition of Germ in (33) we have

[TABLE]

Using $\displaystyle g^{\vec{t}+\vec{s}}=g^{\vec{t}}og^{\vec{s}}$ in (34) we obtain

[TABLE]

Using again (33) in (35), we obtain the invariance

[TABLE]

On the other hand, since $\displaystyle(g^{0})_{*,m}=E_{2n}$, where $\displaystyle E_{2n}$ is identity map, then by choosing appropriately the vector $\displaystyle\vec{t}$ we obtain $\displaystyle(g^{\vec{t}})_{*,m}=g_{j}^{t}m$, where $\displaystyle t\in\Re$ and $\displaystyle g^{t}_{j}$ is the Hamiltonian flux associated to the function $\displaystyle F_{j}$, with $\displaystyle j=1,\ldots,k$. As a consequence the Germ is invariant respect to $\displaystyle g^{t}_{j}$. The smoothness of the Germ we proof in the Appendix A. ∎

Remark 24**.**

From Lemma 22 we have that is possible to construct the complex germ if the operators $\displaystyle\Xi_{i}$, $\displaystyle i=1,\ldots,k$ have a common positive, lagragian and invariant (P.L.I) linear subspace.

Remark 25**.**

Proof of Theorem 23 consists in the construction of complex germ analogously as done in [10]. In this way, we proof the the map $\displaystyle r^{n}:m\rightarrow r^{n}(m)$ is smooth.

Now following Remark 24, we find that sufficient conditions on operators $\displaystyle\Xi_{i}$, $\displaystyle i=1,\ldots,k$ such that they has a common P. L. I subspace.

Definition 26**.**

Let $\displaystyle(M^{2n},\omega^{2})$ a symplectic manifold. The subspace $\displaystyle L\subset T_{m}(M^{2n})$ in a point $\displaystyle m\in M^{2n}$ is simplectic if the restriction of $\displaystyle\omega^{2}$ a $\displaystyle L$ is no-degenerate.

Definition 27**.**

A lineal transformation $\displaystyle SL_{1}\rightarrow L_{2}$ between two lineal spaces is called stable if $\displaystyle\forall\epsilon>0$ $\displaystyle\exists\delta>0$ such that $\displaystyle|x|<\delta$ then $\displaystyle|S^{n}(x)|<\epsilon$ $\displaystyle\forall n\mathbb{N},n>0$ (see [2]).

Proposition 28**.**

A symplectic map is stable if and only if all its eigenvalues belong to the unitary circle and $\displaystyle S$ is diagonalizable.

The proof can be found in [10], where also is proved that the stability condition of symplectic operator is equivalent to have a P. L. I subspace. With this idea, we present previous Lemmas used to prove sufficient condition the existence of a P.L.I subspace for the monodromy operators $\displaystyle\Xi_{i}$, $\displaystyle i=1,\ldots,k$.

Let $\displaystyle A$ y $\displaystyle B$ two stable operators. Let $\displaystyle K_{1}=\{\sigma_{1},\ldots,\sigma_{2n}\}$ the eigenvalues of $\displaystyle A$, where $\displaystyle 1$ and $\displaystyle-1$ may also be included. Therefore, we can do the partition $\displaystyle K_{1}=K_{a}\cup\{-1,1\}$, where $\displaystyle K_{a}=\{\sigma_{1},\ldots,\sigma_{r},\bar{\sigma_{1}},\ldots,\bar{\sigma_{r}}:Im\sigma_{i}\neq 0\}$ with $\displaystyle r<=n$ and $\displaystyle\sigma_{i}$ distinct. Analogously for the operator $\displaystyle B$ we have the partition $\displaystyle K_{2}=K_{b}\cup\{-1,1\}$, where $\displaystyle K_{b}=\{\mu_{1},\ldots,\mu_{s},\bar{\mu_{1}},\ldots,\bar{\mu_{s}}:Im\mu_{i}\neq 0\}$ with $\displaystyle s<=n$ and $\displaystyle\mu_{i}$ distinct.

Let us denote by $\displaystyle S_{\sigma}$ the subspace associated to the eigenvalue $\displaystyle\sigma$. In general the eigenvalues $\displaystyle 1$ and $\displaystyle-1$ they may not be included, these will be analyzed separately.

The following Lemma are valid

Lemma 29**.**

Let $\displaystyle A$ y $\displaystyle B$ two stable operators. Let us consider the restriction of the operator $\displaystyle A_{j}=A_{|S_{\sigma_{j}}\oplus S_{{S_{\bar{\sigma}_{j}}}}}$ and $\displaystyle B_{j}=B_{|S_{\sigma_{j}}\oplus S_{{S_{\bar{\sigma}_{j}}}}}$, where $\displaystyle\sigma_{j}\in K_{a}$. Then there exist a common P. L. I subspace for $\displaystyle A_{j}$ and $\displaystyle B_{j}$.

Proof.

Let us consider the subspace $\displaystyle L=S_{\sigma_{j}}\oplus S_{{S_{\bar{\sigma}_{j}}}}$ which is symplectic and invariant respect to the operator $\displaystyle A$ (see [10]), i.e. $\displaystyle AL=L$. We have that if $\displaystyle p$ the multiplicity of $\displaystyle\sigma_{j}\in K_{a}$ then $\displaystyle\dim(L)=2p$.

a)Now we prove that the subspace $\displaystyle L$ is also invariant for the operator $\displaystyle B$. Since $\displaystyle A_{j}B=BA_{j}$ we have $\displaystyle A_{j}BL=BL$ therefore $\displaystyle BL$ is invariant for the operator $\displaystyle A_{j}$ and since $\displaystyle B$ is a symplectic diffeomorphism then $\displaystyle\dim(BL)=2p$.

Let $\displaystyle S_{\sigma_{j}}=\{A_{j}x=\sigma_{j}x\}$, then for $\displaystyle x\in S_{\sigma_{j}}$ we have $\displaystyle B(A_{j}(x))=\sigma_{j}Bx$. Using $\displaystyle BA_{j}=A_{j}B$ we obtain $\displaystyle A_{j}(Bx)=\sigma_{j}B(x)$ therefore $\displaystyle Bx\in S_{\sigma_{j}}$ and as a consequence $\displaystyle BL\subset L$. Using that the operator $\displaystyle B$ is a diffeomorphism we have $\displaystyle BL=L$ and the operator $\displaystyle B_{j}=B_{|L}$ is well defined.

b) Now since $\displaystyle B$ is diagonalizable and symplectic operator in $\displaystyle L$, it is possible to obtain a descomposition through $\displaystyle K_{2}$ of L, i.e.

[TABLE]

where $\displaystyle S_{1}$ and $\displaystyle S_{-1}$ appear if $\displaystyle 1$ or $\displaystyle-1$ are eigenvalues of $\displaystyle B$ in $\displaystyle L$. Let us consider the restriction $\displaystyle{B_{j}}_{|L_{\mu_{j}}}$ of the operator $\displaystyle B_{j}$ to $\displaystyle L_{\mu_{j}}=S_{\mu_{j}}\oplus S_{\bar{\mu_{j}}}$ then the following affirmation are true

1- $\displaystyle L_{\mu_{j}},j=1,\ldots,d$ are symplectic operators and $\displaystyle B_{j}$ are stable, therefore there exist a subspace P.L.I for $\displaystyle{B_{j}}_{|L_{\mu_{j}}}$ in this subspace which we denote by $\displaystyle R_{\mu_{j}}$ (see [10] for the construction of this subpace).

2-The subspace $\displaystyle S_{1}$ is symplectic (see [10]). Also we have $\displaystyle{B_{j}}_{|S_{1}}=Id_{|S_{1}}$, therefore all vector is eigenvalues for $\displaystyle{B_{j}}_{|S_{1}}$. Then we can choose a collection of vector in $\displaystyle S_{1}$ such as they generated a P.L.I. subspace. Let us denote by $\displaystyle R_{o}$. Also satisfy $\displaystyle AR_{o}=R_{o}$. Analogously, for $\displaystyle{B_{j}}_{|S_{-1}}=Id_{|S_{-1}}$ is possible to construct a P.L.I subspace denoted by $\displaystyle R_{-1}$.

Finally the subspace

[TABLE]

is a common P.L.I subspace to $\displaystyle A_{j}$ and $\displaystyle B_{j}$ in $\displaystyle L$. Thus, the proof is a consequence of the way that such subspace are constructed, i.e. it are positive and Lagrangian in $\displaystyle L$. Also, they are invariant for $\displaystyle B_{j}$,i.e. $\displaystyle B_{j}R_{\mu}=R_{\mu}$. So, we need to prove that is invariant for $\displaystyle A_{j}$, i.e. $\displaystyle A_{j}R_{\mu}=R_{\mu}$ for any $\displaystyle j=1,\ldots,d$.

The subspace $\displaystyle R_{\mu_{j}}$ is constructed from $\displaystyle h$ eigenvector $\displaystyle e_{1},\ldots,e_{h}$ ($\displaystyle h$ is a multiplicity of $\displaystyle\mu_{j}$ as eigenvalues of $\displaystyle B_{j}$ in $\displaystyle L$), which are associated either $\displaystyle\mu_{j}$ or $\displaystyle\bar{\mu_{j}}$.

Let $\displaystyle e_{i}$ an arbitrary with $\displaystyle i=1,\ldots,h$. Since $\displaystyle R_{\mu_{1}}\subset L$, there $\displaystyle e_{i}$ is an eigenvalues of $\displaystyle A$ associated to either $\displaystyle\sigma_{j}$ or $\displaystyle\bar{\sigma_{j}}$. Assume that $\displaystyle A_{j}e_{i}=\sigma_{j}e_{i}$, therefore for $\displaystyle x\in R_{\mu_{1}}$, we have $\displaystyle A_{j}x\in R_{\mu_{1}}$.

∎

Lemma 30**.**

Let $\displaystyle A$ y $\displaystyle B$ two stable operators. Let us consider the restriction of the operator $\displaystyle A_{1}=A_{|S_{\beta}}$ and $\displaystyle B_{1}=B_{|S_{\beta}}$, where $\displaystyle\beta$ is either $\displaystyle 1$ or $\displaystyle-1$ which are eigenvalues of the operator $\displaystyle A$ and $\displaystyle B$. Then there exist a common P. L. I subspace for $\displaystyle A_{1}$ and $\displaystyle B_{1}$.

Proof.

The subspace $\displaystyle S_{1}$ is symplectic (see [10]). Also we have $\displaystyle{B}_{|S_{1}}=Id_{|S_{1}}$, therefore all vector is eigenvalues for $\displaystyle{B}_{|S_{1}}$. Then we can choose a collection of vector in $\displaystyle S_{1}$ such as they generated a P.L.I. subspace. Let us denote by $\displaystyle R_{o}$. Also satisfy $\displaystyle AR_{o}=R_{o}$. Analogously, for $\displaystyle{B}_{|S_{-1}}=Id_{|S_{-1}}$ is possible to construct a P.L.I subspace denoted by $\displaystyle R_{-1}$. ∎

Due to the condition impose on these operators, we have that Lemma 32 is a Corollary of

Lemma 31**.**

Let $\displaystyle A$ and $\displaystyle B$ two stable symplectic operators that commutate in the symplectic space $\displaystyle\mathbb{C}^{2n}$, then they has a common P.L.I subspace.

Proof.

By proposition 28 the operators $\displaystyle A$ and $\displaystyle B$ have its eigenvalues in the unitary circle and are diagonalizable. Then we have $\displaystyle\mathbb{C}^{2n}=\sum_{i=1}^{d}(S_{\sigma_{j}}\oplus S_{\bar{\sigma_{j}}})\oplus S_{1}\oplus S_{-1}$, where $\displaystyle\sigma_{j}$, $\displaystyle\bar{\sigma_{j}}$, $\displaystyle 1$ and $\displaystyle-1$ are eigenvalues of $\displaystyle A$ (in general $\displaystyle 1$ and $\displaystyle-1$ are not necessarily eigenvalues of $\displaystyle A$). Now from Lemmas 29 and 30 we obtain that there exist a common P.L.I subspace for the $\displaystyle A$ and $\displaystyle B$. ∎

Lemma 32**.**

The reduce monodromy operators $\displaystyle\Xi_{i}$, $\displaystyle i=1,\ldots,k$ are stable then they have a common P.L.I. subspace.

Proof.

The construction of the a common subspace for $\displaystyle k>2$ is similar. After we have for two operators we construct for the rest. ∎

4.2 Existence of the Germ

It is valid the following

Theorem 33**.**

If the reduce monodromy operators $\displaystyle\Xi_{i}$, $\displaystyle i=1,\ldots,k$ are stable then there exist a complex germ invariant respect Hamiltonian system $\displaystyle(M^{2n},\omega^{2},F_{j})$, with $\displaystyle j=1,\ldots,k$ on the torus $\displaystyle\Lambda^{k}$.

The proof is a consequence of Lemma 32 and Theorem 23.

Further we despited a necessary condition for the existence of the Germ.

Theorem 34**.**

Assume that there exist a complex germ on the isotropic torus $\displaystyle\Lambda^{k}$ invariant respect to the Hamiltonian flow $\displaystyle\dot{x}=IdF_{j}(x)$, with $\displaystyle j=1,\ldots,k$. Then the reduce monodromy operators $\displaystyle\Xi_{i}$, $\displaystyle i=1,\ldots,k$ are stable.

Proof.

We have $\displaystyle\forall m\in\Lambda^{k}$ there exist complex subspace lagrangian $\displaystyle r^{n}(m)\subset T_{n}(M^{2n})$ such that $\displaystyle r^{n}(m)\supset T_{m}(\Lambda^{k})$ which is dissipative respect of $\displaystyle T_{m}(\Lambda^{k})$, i.e. $\displaystyle\forall x\in r^{n}(m)\smallsetminus T_{m}(\Lambda^{k})$ we have $\displaystyle[x,\bar{x}]/2i>0$ (condition (11) of Definition 10). Also, for $\displaystyle t\in\Re$ we have condition (12) of Definition 10. Therefore, the flow $\displaystyle g^{\vec{t}}=g^{t_{1}}o\ldots og^{t_{k}}:M^{2n}\rightarrow M^{2n}$, where $\displaystyle t=(t_{1},\dots,t_{n})\Re^{k}$ satisfy $\displaystyle g^{\vec{t}}_{*,m}(r^{n}(m))=r^{n}(g^{\vec{t}}m)$. For $\displaystyle T\in G$ ($\displaystyle G$ discrete subgroup of $\displaystyle\Re^{k}$ defined in ) we have

[TABLE]

which mean that $\displaystyle G_{j}(r^{n}(m))=r^{n}(m)$, where $\displaystyle G_{j}$, $\displaystyle j=1,\ldots,k$ are the monodromy operators, which with period $\displaystyle T_{j}$, with $\displaystyle j=1,\ldots,k$ generating the subgroup $\displaystyle G$. Let us consider the canonical projection map $\displaystyle\Pi:T_{m}(\Sigma)\rightarrow\Gamma_{m}=T_{m}(\Sigma)/T_{m}(\Lambda^{k})$ and let $\displaystyle R=\Pi(r^{n}(m))$.

Now only rest to prove that the subspace $\displaystyle R$ is lagragian, positive in $\displaystyle\Gamma_{m}$ and invariant respect of the operators $\displaystyle\Xi_{j}$ with $\displaystyle j=1,\ldots,k$. Since $\displaystyle\Pi^{-1}(0)=T_{m}(\Lambda^{k})$ we have $\displaystyle\dim(R)=\dim(r^{n}(m))-k=n-k=(1/2)\dim(\Gamma_{m})$. From the definition of symplectic structure in the quotient space $\displaystyle\Gamma_{m}$ we obtain that $\displaystyle R$ is isotropic and a consequence lagragian en $\displaystyle\Gamma_{m}$.

We have that if $\displaystyle\Pi(x)\neq 0$ then $\displaystyle x\not\in T_{m}(\Lambda^{k})$. Using that $\displaystyle R$ is disipative respect of $\displaystyle T_{m}(\Lambda^{k})$ and that $\displaystyle\overline{\Pi(x)}=\Pi(\bar{x})$ we have $\displaystyle(1/2i)[x,\bar{x}]=[\Pi(x),\Pi(\bar{x})]=[\Pi(x),\overline{\Pi(x)}]>0$ and as consequence $\displaystyle R$ is positive.

Since $\displaystyle\Xi_{j}o\Pi=\Pi oG_{j}$, with $\displaystyle j=1,\ldots,k$ follows that $\displaystyle R$ is invariant respect of $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\dots,k$. Thus we obtain a P.L.I subspace common for the operators $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\dots,k$ therefore this operators are diagonalizables and its eigenvalues belong to unitary circle then by Proposition 28 are stables.

∎

5 Uniqueness of the Germ

In this section we discuss about sufficient and necessary conditions of complex Germ. This issue is crucial since the germ is used in the construction of an asymptotic quase-classic it necessary to verify if obtained in this way is unique, which depend on the uniqueness of the germ. To display here a full characterization we summarized some basic concept.

Definition 35**.**

An stable map $\displaystyle S:L_{1}\rightarrow L_{2}$ is called strong stable if all close map is stable.

What means close map in above Definition?. Let consider the group of the symplectic map which constitute a submanifold of the lineal map of $\displaystyle\Re^{n}$. We consider any distance between two lineal map on $\displaystyle\Re^{n}$ as a distance between the respective matrix in a prefixed basis. i.e. Let $\displaystyle[s_{ij}]$ and $\displaystyle[s^{{}^{\prime}}_{ij}]$ the matricial representantin of two lineal map $\displaystyle L_{1},L_{2}:L^{1}\rightarrow L^{2}$ then they are close if $\displaystyle\max|s_{ij}-s^{{}^{\prime}}_{ij}|\leq\epsilon$, $\displaystyle\forall\epsilon>0$ and $\displaystyle i,j=1,\ldots,k$.

Let $\displaystyle A:\mathbb{C}^{2n}\rightarrow\mathbb{C}^{2n}$ an symplectic operator and $\displaystyle\sigma$ an eigenvalues of $\displaystyle A$. We denote by $\displaystyle L_{\sigma}$ the maximal invariant subspace respect to $\displaystyle A$ associated to $\displaystyle\sigma$.

Definition 36**.**

The eigenvalues $\displaystyle\sigma$ is called elliptic positive (negative) if the subspace $\displaystyle L_{\sigma}$ is positive (negative).

In [10] a collection of results of the elliptic eigenvalues are obtained. Here we summarize those we will use

Proposition 37**.**

An symplectic map $\displaystyle L:\mathbb{C}^{2n}\rightarrow\mathbb{C}^{2n}$ is strong stable if all its eigenvalues are elliptic belong to the unitary circle.

Also the following proposition is valid

Proposition 38**.**

An symplectic map $\displaystyle L:\mathbb{C}^{2n}\rightarrow\mathbb{C}^{2n}$ is strong stable if has a unique P.L.I subspace.

The following Lemma holds

Lemma 39**.**

If there exist a unique common subspace P.L.I to the reduce monodromy operator $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\dots,k$ then there exist a unique complex germ on the torus $\displaystyle\Lambda^{k}$.

Proof.

Assume that there exist two distinct complex germ $\displaystyle r_{1}^{n}$ and $\displaystyle r_{2}^{n}$, i.e. there exist a point $\displaystyle m_{o}$ such that $\displaystyle r_{1}^{n}(m_{o})\neq r_{2}^{n}(m_{o})$. Let $\displaystyle\Pi$ the canonical subjective and let the subspace $\displaystyle R_{j}=\Pi(r_{j}(m_{o}))$, $\displaystyle j=1,2$ which are by definition lagragian and positive in $\displaystyle\Gamma_{m_{o}}$ and invariant respect the operators $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\dots,k$. Using analogous process to describe in Lemma 29 is possible to construct more that one common P.L.I subspace to the operators $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\dots,k$. ∎

Now we give sufficient condition for the existence of a unique complex germ.

Theorem 40**.**

In the reduce operator of monodromy are stable and there exist at least one strong stable then there exist a unique complex Germ on the torus $\displaystyle\Lambda^{k}$ invariant respect to the Hamiltonian system $\displaystyle(M^{2n},\omega^{2},F_{j})$ with $\displaystyle j=1,\ldots,k$.

Proof.

Let $\displaystyle\Xi_{j}$ for some $\displaystyle j\in J={1,\ldots,k}$ a strong stable operators. By Proposition 38 there exist a unique P.L.I subspace for $\displaystyle\Xi_{j}$ which we denote by $\displaystyle R$. As the operator $\displaystyle\Xi_{i}$, with $\displaystyle i\in I\diagdown{j}$ commute with $\displaystyle\Xi_{j}$ then by Lemma 32 is possible to construct a common P.L.I subspace for these operators, which is a unique. Then by Lemma 39 we have a Theorem. ∎

A necessary condition for the existence of germ is given the following

Lemma 41**.**

If there exist a unique complex germ on the torus $\displaystyle\Lambda^{k}$ invariant respect to the Hamiltonian system $\displaystyle(M^{2n},\omega^{2},F_{j})$ with $\displaystyle j=1,\ldots,k$. Then there exist a unique common P.L.I subspace for the reduce monodromy operators $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\dots,k$.

Proof.

Let the stable reduce monodromy operators $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\dots,k$. Then by Lemma 32 these operators have for a common P.L.I subspace $\displaystyle R\subset\Gamma_{m}$ for each $\displaystyle m\in\Lambda^{k}$.

From Theorem 23 (with this subspace), we can construct the complex germ $\displaystyle r^{n}=r^{n}(R)$. Besides, to distinct subspace correspond distinct Germ, i.e. if $\displaystyle R_{1}\neq R_{2}$ then for $\displaystyle r^{n}_{1}=r^{n}(R_{1})$ and $\displaystyle r^{n}_{2}=r^{n}(R_{2})$ we have $\displaystyle\forall m\in\Lambda^{k}$ that $\displaystyle r^{n}_{1}(m)\neq r^{n}_{2}(m)$. Assume that there exist $\displaystyle m\in\Lambda^{k}$ such $\displaystyle r^{n}_{1}(m)=r^{n}_{2}(m)$; let $\displaystyle m^{{}^{\prime}}\in\Lambda^{k}$ which serves to construct the reduce operator $\displaystyle\Xi_{m^{{}^{\prime}}}$ and the subspace $\displaystyle R_{1}$ and $\displaystyle R_{2}$. Since there exist $\displaystyle\vec{s}\in\Re^{k}$ such that $\displaystyle g^{\vec{s}}(m^{{}^{\prime}})=m$, and if we define $\displaystyle r_{1}(m)=g^{\vec{s}}_{*,m^{{}^{\prime}}}(r^{n}(m^{{}^{\prime}}))$; $\displaystyle r^{n}_{1}(m^{{}^{\prime}})=R_{1}$ and $\displaystyle r^{n}_{2}(m)=g^{\vec{s}}_{*,m^{{}^{\prime}}}(r^{n}(m^{{}^{\prime}}))$; $\displaystyle r^{n}(m^{{}^{\prime}})=R_{2}$ we obtain a contradiction $\displaystyle R_{1}=R_{2}$. ∎

We have the following

Lemma 42**.**

In there is a unique P.L.I subspace common to the reduce monodromy operators $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\dots,k$ then these operators are stable and at least one is strong stable.

Proof.

We assume that all the operators $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\dots,k$ are stable and none is strong stable then from Lemmas 29, 30 it is possible to construct more of one P.L.I subspace common for two operators, i.e. $\displaystyle R_{1}$ and $\displaystyle R_{2}$. With the procedure of Lemma 32 is possible to construct more of one common P.L.I for the operators $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\dots,k$ . ∎

We present a necessary condition for the complex germ

Theorem 43**.**

If there is a unique complex germ invariant respect to the Hamiltonian system $\displaystyle(M^{2n},\omega^{2},F_{j})$ with $\displaystyle j=1,\ldots,k$. Then the reduce monodromy operators $\displaystyle\Xi_{j}$, with $\displaystyle j=1,\dots,k$ are stable and at least one is strong stable.

The proof follow from the Lemmas 39 and 41.

6 Smoothness of the complex Germ

A p-distribution on a manifold $\displaystyle M$, with $\displaystyle p\leq dim(M)$ is a map such that to each point $\displaystyle m\in M$ correspond a subspace p-dimensional $\displaystyle\theta(m)$ of the $\displaystyle T_{m}(M)$. We say that such distribution is smooth if there are p smooth vectorial fields $\displaystyle X_{1},\ldots,X_{k}$ defined on a neighbor $\displaystyle U$ of the point $\displaystyle m$ such that $\displaystyle X_{1}(m),\ldots,X_{k}(m)$ generate the subspace $\displaystyle\theta(m)$.

Since the complex Germ is a n-dimensional distribution with certain additional properties (see Definition ) the smoothness property is a consequence of the own construction of the germ. From remark 24 the germ is generate by a P.L.I subspace $\displaystyle R$. We assume that this subspace is generated by the vector $\displaystyle r_{1},\ldots,r_{n}$. Since the map $\displaystyle(g^{t}_{F_{j}})_{*,m}$ are smooths the the vectorial fields $\displaystyle X_{j}$, $\displaystyle j=1,\ldots,k$ defined on each point $\displaystyle m\in\Lambda^{k}$ by

[TABLE]

where $\displaystyle\vec{t}\in\Re^{k}$ satisfy $\displaystyle g^{\vec{t}}m_{o}=m_{o}$, with $\displaystyle g^{\vec{t}}=g^{t_{1}}_{1}o\ldots,g^{t_{k}}_{k}$ and $\displaystyle g^{t_{1}}_{j}$ are Hamiltonian flows associated to the Hamilton functions $\displaystyle F_{j}$. It is possible to check that $\displaystyle X_{j}$, $\displaystyle j=1,\dots,k$ are smooth and generated the germ $\displaystyle r^{n}(m)$.

7 Application to the Hamiltonian system with cyclic variables

In this section we study a $\displaystyle 2n$-dimensional symplectic manifold that contains a 2k-dimensional manifold ($\displaystyle k<n$) which consist of $\displaystyle k$ invariant isotropic torus respect to certain Hamiltonian system. These system in the local coordinate $\displaystyle(I,p,\theta,q)$ has the form $\displaystyle H=H(I,p,q)$ and the rest are $\displaystyle k-1$ coordinates. These are called system with cyclic variables. The basis and the problem are described following. Let $\displaystyle(M^{2n},\omega^{2})$ a symplectic manifold with canonical variables $\displaystyle{p,q}.$.

Definition 44**.**

An atlas on the manifold $\displaystyle(M^{2n},\omega^{2})$ is called symplectic if the the coordinate space $\displaystyle(p,q)\in\Re^{2n}$ the symplectic structure take the form $\displaystyle\omega^{2}=\sum_{i=1}^{n}dp_{i}\wedge dq_{i}$ and the cart $\displaystyle\Phi o\Theta^{-1}:\Theta(U_{1}\cap U_{2})\rightarrow\Phi(W_{1}\cap W_{2})$ are symplectic transformations, where $\displaystyle\Phi:U_{1}\rightarrow W_{1}$, $\displaystyle\Theta:U_{2}\rightarrow W_{2}$, with $\displaystyle U_{1},U_{2}\subset M^{2n}$ and $\displaystyle W_{1},W_{2}\subset\Re^{2n}$,

Definition 45**.**

A Hamiltonian system $\displaystyle(M^{2n},w^{2},H)$ with Hamilton function $\displaystyle H$ is called system with $\displaystyle k$ cyclic variables if there are the system of symplectic coordinates $\displaystyle{I,p,\theta,q}$ such that $\displaystyle H=H(I,p,q)$. The cyclic variables do not appear in the expression of the Hamilton function which we denote by $\displaystyle\theta=(\theta_{1},\ldots,\theta_{k})$.

Let the variables $\displaystyle I=(I_{1},\ldots,I_{k})$, $\displaystyle p=(p_{1},\ldots,k)$ canonical conjugate of the variables $\displaystyle\theta=(\theta_{1},\ldots,\theta_{k})mod2\pi$, $\displaystyle q=(q_{1},\ldots,q_{k})$ with $\displaystyle 1<=k<=n$. i.e. the phase space $\displaystyle M^{2n}$ have the coordinate system $\displaystyle(I,p,\theta,q)$ where the symplectic structure $\displaystyle w^{2}$ can be written in the form

[TABLE]

We assume that coordinate system define a diffeormorphism from $\displaystyle M^{2n}$ under the space consist of the direct product of the torus $\displaystyle T^{k}$ with open region of $\displaystyle\Re^{2n-k}$. In this system the canonical equation of the Hamilton function take the form

[TABLE]

where $\displaystyle H_{q}=(\partial H/\partial q_{1},\ldots,\partial H/\partial q_{k}$ and $\displaystyle H_{p}=(\partial H/\partial p_{1},\ldots,\partial H/\partial p_{k}$.

We consider the subset

[TABLE]

and we assume that is connected. Let us denote by $\displaystyle\Lambda^{k}=\Lambda^{k}(I_{o},p_{o},q_{o})$ the k-dimensional torus defined by the the equalities $\displaystyle\{(I,p,\theta,q):I=I_{o},p=p_{o},q=q_{o}\}$. It is possible to check that N represent the union of such torus.

We consider the restriction of the vectorial Hamiltonian field to the torus $\displaystyle\Lambda^{k}$ where a trajectory on such field take the form $\displaystyle g_{H}^{t}=(I_{o},p_{o},H_{I}t+\theta,q_{o})$ defined for all $\displaystyle t\in(-\infty,+\infty)$ because $\displaystyle\Lambda^{k}$ is compact. It is possible to check that any torus $\displaystyle\Lambda\subset N$ is invariant respect to this system. Also, from the symplectic structure in (40) we deduce that $\displaystyle\omega^{2}|_{\Lambda^{k}}=0$, i.e. the torus are isotropic. Therefore $\displaystyle H$ is the union of isotropic, invariant respect to the Hamiltonian system $\displaystyle(M^{2n},\omega^{2},H)$.

We have the following Definition

Definition 46**.**

We say that F is first integral of the Hamiltonian system with function $\displaystyle H$ if the Poisson braked satisfy $\displaystyle[F,H]$=0.

It is possible to check that described above the coordinate function $\displaystyle I_{1},\ldots,I_{k}$ are first integral of the system $\displaystyle(M^{2n},w^{2},H)$.

We have

Proposition 47**.**

Assume that for any point in $\displaystyle N$ given in (42) the determinant of $\displaystyle Hess(H)$ is distinct of zero , i.e.

[TABLE]

Then $\displaystyle N$ is sub-manifold symplectic 2k-dimensional that has locally the form of a map $\displaystyle(I,\theta)\rightarrow(p,q)$ that does not depend on the coordinate $\displaystyle\theta$, i.e. $\displaystyle p=P(I),q=q(Q)$.

The proof of this result can be found in [10].

From now, we assume that $\displaystyle\Lambda^{k}$ consist of regular points, i.e. $\displaystyle H_{I}\neq 0$, as a consequence the functions $\displaystyle H,I2,\ldots,I_{k}$ are k first integral involution. The problem is under what conditions there is a complex Germ on $\displaystyle\Lambda^{k}$ with respect to the Hamiltonian system defined by Hamilton functions $\displaystyle H,I2,\ldots,I_{k}$.

The reduce monodromy operators for this case are the following: $\displaystyle G1=(g^{t}_{H})_{*,m},G_{j}=(g^{t}_{I_{j}})_{*,m}=E_{2n}$, with $\displaystyle j=2,\ldots,k$ and $\displaystyle E_{2n}$ is identity operator of order $\displaystyle 2n$. This mean that the invariance condition if only necessary to check for the operator $\displaystyle G_{1}$. Applying the result of this paper we obtain the sufficient and necessary for the existence of Germ different from those obtained in those obtained in [2].

8 Conclusion

In this work we give answer to the question about the existence and uniqueness of a complex germ on a isotropic torus invariant respect to the Hamiltonian flows defined by k function $\displaystyle F_{1},\ldots,F_{k}$ that stay in involution in the phase state $\displaystyle M^{2n}$.

We proof that there exist such germ if and only if the reduce monodromy operator with period $\displaystyle T_{j}$, $\displaystyle j=1,\dots,k$ are stable. This germ is unique if at least one operator is strong stable.

The result obtained here were applied to the case of an Hamiltonian with cyclic variables resulting in new condition for the existence and uniqueness of complex germ.