# Théorie KAM

Mauricio Garay

KAM THEORY

M. Garay and D. van Straten

I

GROUP ACTIONS AND THE KAM PROBLEM

This book originated from courses taught by the first author in 2011 at the
Dijon University (France) and in 2013 at the University of Ouargla (Algeria)
on the Herman invariant tori conjecture.

The first author thanks M. Bahayou and A. Zeglaoui for the invitation in
Ouargla and R. Uribe and A. Dubouloz for the invitation in Dijon. Thanks also
to the audience and in particular to N. A’Campo, P. Cartier, Z. Fernane, A.
Kessi, F. Laudenbach, D. Smai and N. Yousfi.

To the memory of V.I. Arnold (1937-2010).

###### Contents

  1. 1  Darboux theorems and action-angle variables
    1. 1.1 Hamiltonian vector fields
    2. 1.2 Cotangent spaces
    3. 1.3 The Darboux theorem
    4. 1.4 The classical Darboux-Weinstein theorem
    5. 1.5 The relative Darboux-Weinstein theorem
    6. 1.6 Liouville integrability
    7. 1.7 Action-angle variables
    8. 1.8 Integrable systems and the Gauss-Manin connection
    9. 1.9 Affine structures and integrable systems
    10. 1.10 Bibliographical notes
  2. 2 The KAM problem
    1. 2.1 Quasi-periodic motions
    2. 2.2 Algebraic model
    3. 2.3 Perturbation theory
    4. 2.4 The formal model
    5. 2.5 Formal stability
    6. 2.6 Analytic model and holomorphic Fourier series
    7. 2.7 First order analytic obstruction
    8. 2.8 Bibliographical notes
  3. 3 The Kolmogorov invariant torus theorem
    1. 3.1 Kolmogorov’s Diophantine condition
    2. 3.2 Diophantine approximation
    3. 3.3 Vectors satisfying the Kolmogorov Diophantine condition
    4. 3.4 Kolmogorov’s invariant torus theorem: Statement
    5. 3.5 The one dimensional case
    6. 3.6 Kolmogorov’s theorem and group actions
    7. 3.7 Bibliographical notes
  4. 4 The generalised KAM problem
    1. 4.1 Invariant ideals
    2. 4.2 Poisson cohomology and Poisson vector fields
    3. 4.3 Pairs and homotopic stability
    4. 4.4 The formal Kolmogorov theorem
    5. 4.5 Bibliographical notes
  5. 5 The Lie iteration
    1. 5.1 Local transversals to a group action
    2. 5.2 Adjoint orbits for the linear group
    3. 5.3 The Lie iteration in the homogeneous case
    4. 5.4 Quadratic convergence of the Lie iteration
    5. 5.5 The Lie iteration in the general case
    6. 5.6 Bibliographical Notes

##  Chapter 1  Darboux theorems and action-angle variables

We review classical mechanics in its symplectic formulation and sketch proofs
of the basic results like the existence of action-angle coordinates. Some
familiarity with the language of global analysis is assumed.

###  1.1. Hamiltonian vector fields

A real $C^{\infty}$ manifold $M$ is called symplectic if it is endowed with a
closed $2$-form $\omega$ which induces an isomorphism between the tangent and
cotangent bundles:

$TM\longrightarrow T^{*}M,\ X\mapsto i_{X}\omega.$

Here $i_{X}$ denotes the interior product of a differential form with a vector
field $X$, that is:

$i_{X}\omega(Y)=\omega(X,Y).$

This isomorphism associates to each differential $1$-form a unique vector
field, which we call symplectically associated to it. Given a function

$H:M\longrightarrow\mathbb{R},$

the vector field $X_{H}$ associated to the $1$-form $dH$ is called the
Hamiltonian vector field of $H$.

The space $\mathbb{R}^{2n}$, equipped with the $2$-form

$\omega:=\sum_{i=1}^{n}dq_{i}\wedge dp_{i},$

gives a basic example of a symplectic manifold. The isomorphism between
tangent and cotangent bundles is given by

$\partial_{q_{i}}\mapsto dp_{i},\ \partial_{p_{i}}\mapsto-dq_{i}.$

In this way we recover the classical definition of the Hamiltonian vector
field

$X_{H}:=\sum_{i=1}^{n}(\partial_{p_{i}}H\partial_{q_{i}}-\partial_{q_{i}}H\partial_{p_{i}}),$

as it is symplectically associated to $dH$. Its integral curves are the
solutions to Hamilton’s canonical equations of motion:

$\left\\{\begin{matrix}\dot{q}_{i}&=&\partial_{p_{i}}H\\\
\dot{p}_{i}&=&-\partial_{q_{i}}H\;\;.\end{matrix}\right.$

A smooth map $\varphi:M\longrightarrow M^{\prime}$ between two symplectic
manifolds $(M,\omega)$ and $(M^{\prime},\omega^{\prime})$ is called
symplectic, if it preserves the symplectic forms:

$\varphi^{*}(\omega^{\prime})=\omega.$

By a symplectomorphism we will mean a symplectic diffeomorphism. A
symplectomorphism

$\varphi:M\longrightarrow M$

maps the Hamiltonian vector field of $H\circ\varphi$ to that of $H$. Therefore
the qualitative behaviour of a dynamical system only depends on the orbit of
Hamiltonian functions under the group of symplectomorphism.

If one can integrate the Hamiltonian vector field $X_{H}$ up to time $t$, we
obtain a flow

$\varphi_{t}:=\varphi^{t}_{X_{H}}:M\longrightarrow
M,\;\;\;\varphi_{0}=Id_{M},\;\;\;\varphi_{t}\circ\varphi_{s}=\varphi_{t+s},$

which is a one parameter family of symplectomorphism depending on $t$. To show
this, recall the formula

$\frac{d}{dt}(\varphi_{t}^{*}\omega)=\varphi_{t}^{*}({\mathcal{L}}_{X_{H}}\omega)$

where ${\mathcal{L}}_{\cdot}$ stands for the Lie derivative. Applying Cartan’s
formula

${\mathcal{L}}_{X}=di_{X}+i_{X}d$

we find:

${\mathcal{L}}_{X_{H}}\omega=di_{X_{H}}\omega+i_{X_{H}}d\omega=ddH+0=0.$

Hence $\varphi_{t}^{*}\omega$ is constant in $t$, and as $\varphi_{0}=Id_{M}$
we find $\varphi_{t}^{*}\omega=\omega$.

So any function $H$ gives rise to a differential $dH$ and therefore to a
vector field which preserves the symplectic form, but the converse is not
always true. Vector fields preserving the symplectic form will be called
symplectic vector fields. These are usually called Hamiltonian vector fields,
but we wish to distinguish them from the vector fields defined by Hamiltonian
functions (which are usually called exact Hamiltonian vector fields). So we
use the name Hamiltonian vector fields for the vector fields defined by a
Hamiltonian function.

######  Proposition 1.1.

In a symplectic manifold the closed one-forms are symplectically associated to
symplectic vector fields.

###### Proof.

A vector field $X$ is symplectic if and only if

${\mathcal{L}}_{X}\omega=0.$

By Cartan’s formula

${\mathcal{L}}_{X}\omega=di_{X}\omega+i_{X}d\omega=di_{X}\omega.$

Therefore $i_{X}\omega$ is a closed 1-form if and only if $X$ is symplectic. ∎

We denote by $\Omega^{1}(M)$ the vector space of $C^{\infty}$ real valued
$1$-forms on $M$ and let

$\Omega^{1}_{closed}(M):=\\{\eta\;|\;d\eta=0\\}\supset\Omega^{1}_{exact}(M):=\\{dH\;|\;H\in
C^{\infty}(M)\\}$

be the sub-spaces of closed and exact $1$-forms. The exterior derivative is a
map

$C^{\infty}(M)\longrightarrow\Omega_{closed}^{1}(M)$

which has the space of locally constant functions $H^{0}(M)$ as kernel, and
the de Rham cohomology space $H^{1}(M)$ as cokernel, so we obtain an exact
sequence of vector spaces

$0\longrightarrow\Omega^{1}_{exact}(M)\longrightarrow\Omega^{1}_{closed}(M)\longrightarrow
H^{1}(M)\longrightarrow 0.$

Looking at the symplectically associated spaces of vector fields, the above
exact sequence is converted into the exact sequence

$0\longrightarrow{\rm Ham\,}(M)\longrightarrow{\rm Symp\,}(M)\longrightarrow
H^{1}(M)\longrightarrow 0,$

where ${\rm Ham\,}(M)$ denotes the vector space of Hamiltonian vector fields
and ${\rm Symp\,}(M)$ the space of vector fields which preserve the symplectic
form. So we see that the difference between these two types of vector fields
has a simple cohomological interpretation.

######  Example 1.2.

Consider the $n$-dimensional torus $(\mathbb{R}/\mathbb{Z})^{n}$; its points
are described by $n$ angular coordinates $\theta_{1},\dots,\theta_{n}$. These
$\theta_{i}$ are multivalued functions, but their differentials
$\alpha_{i}:=d\theta_{i}$ are well-defined closed $1$-forms on the torus. We
consider on the space $M:=(\mathbb{R}/\mathbb{Z})^{n}\times\mathbb{R}^{n}$ the
symplectic form

$\omega=\sum_{i=1}^{n}d\theta_{i}\wedge dp_{i}.$

where the $p_{i}$ are coordinates on the $\mathbb{R}^{n}$-factor. The
cohomology classes $[\alpha_{i}],i=1,2,\ldots,n$ define a basis of the de Rham
cohomology space

$H^{1}(M)=\oplus_{i=1}^{n}\mathbb{R}[\alpha_{i}]\approx\mathbb{R}^{n}.$

These forms are associated to the symplectic vector fields:

$X_{1}=\partial_{p_{1}},\dots,X_{n}=\partial_{p_{n}}.$

Therefore any symplectic vector field on $M$ is of the form

$\sum_{i=1}^{n}a_{i}\partial_{p_{i}}+X_{H}$

where

$H:M\longrightarrow\mathbb{R}$

is a $C^{\infty}$ function and $a_{1},\dots,a_{n}\in\mathbb{R}$.

###  1.2. Cotangent spaces

The cotangent space $M=T^{*}L$ of a manifold $L$ naturally inherits a
symplectic form. The points of this manifold are pairs $(q,p)$, where $p\in
T^{*}_{q}L$ is a covector at a point $q\in L$. The derivative of the bundle
projection

$\pi:M\longrightarrow L$

is a map

$d\pi:TM\longrightarrow TL.$

The action one-form $\alpha\in\Omega^{1}(M)$ is defined by the formula

$\alpha_{q,p}(\xi)=p(d\pi_{q,p}(\xi)),\ \ \xi\in T_{q,p}M.$

Then the exact two form:

$\omega:=d\alpha$

is non-degenerate and is called the canonical symplectic form on $T^{*}L$.

Let us analyse this construction for $L=\mathbb{R}$. In this case, the
cotangent bundle can be identified with the projection

$\mathbb{R}^{2}\longrightarrow\mathbb{R},(q,p)\mapsto q.$

If $\xi=(\xi_{1},\xi_{2})\in T_{(q,p)}\mathbb{R}^{2}$ is a tangent vector,
then

$d\pi_{q,p}(\xi)=\xi_{1}$

so the action form is defined by

$\alpha_{q,p}(\xi)=p\,\xi_{1}.$

This means that

$\alpha=pdq$

and therefore $\omega=dp\wedge dq$. Similarly, for the space
$\mathbb{R}^{2n}$, viewed as $T^{*}\mathbb{R}^{n}$, we recover the symplectic
form defined previously, up to a sign and in Example 1.2 we were in fact
dealing with the cotangent space to the $n$-torus
$(\mathbb{R}/\mathbb{Z})^{n}$.

Note that the canonical symplectic form vanishes on the zero-section of
$T^{*}L$. More generally, an $n$-dimensional submanifold $L$ of a $2n$
dimensional symplectic manifold $(M,\omega)$ is called Lagrangian if
$\omega_{|L}=0$. In such a case the class of $\omega$ vanishes inside the de
Rham cohomology group $H^{2}(L)$. In any neighbourhood of $L$ which retracts
onto $L$, the symplectic form is exact. A form $\alpha$ such that

$d\alpha=\omega$

is called an action form .

######  Proposition 1.3.

Consider the symplectic space $\mathbb{R}^{2n}$ with coordinates $q,p$ and
symplectic form

$\omega=\sum_{i=1}^{n}dq_{i}\wedge dp_{i}.$

Let $L\subset\mathbb{R}^{2n}$ be a Lagrangian manifold given as the graph of a
map

$f=(f_{1},\dots,f_{n}):\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$

over the $q$-space. Then there exists a function

$S:\mathbb{R}^{n}\longrightarrow\mathbb{R}$

such that

$f=\nabla S=(\partial_{q_{1}}S,\partial_{q_{2}}S,\dots,\partial_{q_{n}}S).$

Conversely, any such graph is Lagrangian.

###### Proof.

The action form

$\sum_{i=1}^{n}p_{i}dq_{i}$

is closed on $L$, thus the Poincaré lemma implies that it is the differential
of a function $S$:

$dS=\sum_{i=1}^{n}p_{i}dq_{i},$

hence $p_{i}=\partial_{i}S$. ∎

A function $S$ as in the proposition is called a generating function of $L$.
It is unique up to an additive constant.

###  1.3. The Darboux theorem

Recall that two mappings define the same germ along a subset, if their
restriction to a common neighbourhood of the subset agree. We denote by

$f:(M,X)\longrightarrow(N,Y),\ f(X)=Y$

the germ of $f:M\longrightarrow N$ along $X$. The following fundamental result
is called the Darboux theorem.

######  Theorem 1.4.

The germ of a $2n$-dimensional symplectic manifold $(M,\omega)$ at an
arbitrary point is isomorphic to the germ of
$(\mathbb{R}^{2n},\sum_{i}dq_{i}\wedge dp_{i})$ at the origin.

###### Proof.

Take a local chart

$\varphi:(M,p)\longrightarrow(\mathbb{R}^{2n},0)$

at a point $p\in M$. We get two symplectic forms on a neighbourhood of the
origin in $\mathbb{R}^{2n}$:

$\omega_{0}=(\varphi^{-1})^{*}\omega,\;\;\textup{and}\;\;\;\omega_{1}=\sum_{i=1}^{n}dq_{i}\wedge
dp_{i}.$

The value of these forms at the origin are anti-symmetric bilinear forms which
are conjugate by a linear transformation. Thus, up to a linear change of
coordinates, we may and will assume that they are equal. It follows that

$\omega_{t}=(1-t)\omega_{0}+t\omega_{1},\;\;t\in[0,1]$

defines a $1$-parameter family of symplectic forms in a sufficiently small
contractible neighbourhood of the origin.

We now search for a $1$-parameter family $\varphi_{t}$ of symplectomorphisms
such that:

$\varphi_{t}^{*}\omega_{t}=\omega_{0}.$

We differentiate this equation with respect to $t$ and as the right hand side
is $t$-independent we obtain:

$0=\frac{d}{dt}\varphi_{t}^{*}\omega_{t}=\varphi_{t}^{*}({\mathcal{L}}_{X_{t}}\omega_{t}+\dot{\omega}_{t}),\;\;\dot{\omega}_{t}:=\frac{d}{dt}\omega_{t},$

where $X_{t}$ is the time-dependent Hamiltonian vector field associated to
$v_{t}$. Composing with the inverse of $v_{t}^{*}$, we get the equation:

${\mathcal{L}}_{X_{t}}\omega_{t}=-\dot{\omega}_{t}$

Cartan’s formula shows that:

${\mathcal{L}}_{X_{t}}\omega_{t}=di_{X_{t}}\omega_{t}+i_{X_{t}}d\omega_{t}=di_{X_{t}}\omega_{t}.$

As the neighbourhood was assumed to be contractible, the closed form
$\omega_{t}$ is in fact exact by the Poincaré lemma, so we have

$\omega_{t}=d\alpha_{t}.$

It is therefore sufficient to solve the equation

$i_{X_{t}}\omega_{t}=-\dot{\alpha}_{t}$

for the vector field $X_{t}$, which is possible, since the interior product
with a symplectic form is an isomorphism. As the forms $\omega_{1}$ and
$\omega_{2}$ are equal at the origin, the vector field $X_{t}$ vanishes at
$0$. Thus, the time $1$ flow of the vector field $X_{t}$ exists in a small
neighbourhood of the origin and yields the sought for symplectomorphism. This
proves the theorem. ∎

Coordinates such as in the above theorem are called Darboux coordinates,
sometimes canonical coordinates. The existence of Darboux coordinates shows
that there are no local symplectic invariants.

######  Corollary 1.5.

Let $L\subset M$ be a Lagrangian submanifold of a symplectic manifold
$(M,\omega)$. The germ of $L$ at a point is symplectomorphic to the germ at
the origin of the zero section in $T^{*}\mathbb{R}^{n}$ with its canonical
symplectic structure.

###### Proof.

Choosing Darboux coordinates at the point considered, we can reduce to the
case $M=T^{*}\mathbb{R}^{n}$ equipped with its canonical symplectic form. The
$\pi/2$-rotations in the planes $(q_{i},p_{i})$

$(q_{i},p_{i})\mapsto(-p_{i},q_{i})$

are symplectomorphisms and by applying these, we may always assume that $L$ is
the graph of a map. By Proposition 1.3, this map is itself the gradient of a
function

$S:\mathbb{R}^{n}\longrightarrow\mathbb{R}.$

The map

$(q,p)\mapsto(q,p-\nabla S)$

is a symplectomorphism which maps $L$ to the zero section. ∎

###  1.4. The classical Darboux-Weinstein theorem

The theorem of Darboux implies the fundamental fact that a Lagrangian manifold
has no local symplectic invariants. In some situations there are global
versions of this result.

Our proof of the Darboux theorem consisted of two parts: starting with two
different symplectic forms in a neighbourhood of the origin in
$\mathbb{R}^{2n}$, we first chose linear coordinates so that both symplectic
forms agree at the origin. Then knowing that the linear path between the
symplectic forms remains inside the space of symplectic forms, we applied the
path homotopy method.

######  Theorem 1.6.

Let $L$ be compact submanifold of a symplectic manifold $M$ and
$\omega_{t},t\in[0,1]$ a one parameter family of symplectic forms on $M$ with
$C^{\infty}$ dependence on $t$. Assume that the restrictions of de Rham
classes $[\omega_{t}]$ to $H^{2}(L,\mathbb{R})$ are independent of $t$. Then
there exists a neighbourhood $T\subset M$ of $L$ such that the $t$-dependent
symplectic manifolds $(T,\omega_{t})$ are all symplectomorphic.

###### Proof.

By compactness of the interval $[0,1]$ it suffices to prove the theorem for
sufficiently small values of $t$. Choose a tubular neighbourhood $T$ of $L$.
As $T$ retracts to $L$, the assumptions imply that the class of
$\partial_{t}\omega_{t}$ vanishes in $H^{2}(T)$. This means that
$\partial_{t}\omega_{t}$ is exact in $T$, that is, we find a family of
$1$-forms $\alpha_{t}$ such that

$\partial_{t}\omega_{t}=d\alpha_{t}.$

As before one finds a time dependent vector field $X_{t}$ such that

$i_{X_{t}}\omega_{t}=-\alpha_{t}$

By compactness of $L$, the vector field can be integrated for sufficiently
small times and its time $t$-flow sends $\omega_{0}$ to $\omega_{t}$. This
concludes the proof of the theorem. ∎

From this theorem we will deduce the following celebrated result:

######  Theorem 1.7.

Any compact Lagrangian submanifold $L$ of a symplectic manifold $(M,\omega)$
admits a neighbourhood symplectomorphic to neighbourhood of $L$ in $T^{*}L$
with its standard symplectic structure.

In the proof we will make use of an important idea of Gromov, namely the
existence of an almost complex structure adapted to a given symplectic form.
Recall that that a complex structure on a real vector space $E$ is a linear
map $J\in GL(E)$ with the property that $J^{2}=-{\rm Id\,}$. It is said to be
adapted to a linear symplectic form $\omega$ on $E$ if

$\omega(-,J-)$

is an Euclidean scalar product on $E$.

######  Lemma 1.8.

Let $(E,\omega)$ be a symplectic vector space, $L\subset E$ a linear
Lagrangian subspace. A complex structure $J\in GL(E)$ is adapted if and only
if the following three conditions are satisfied:

  1. (i)

The quadratic form $L\longrightarrow\mathbb{R},x\mapsto\omega(x,Jx)$ is
positive definite,

  2. (ii)

$L$ and $JL$ are complementary,

  3. (iii)

$\omega(x,y)=\omega(Jx,Jy)$ for any $x,y\in E$.

###### Proof.

We first check that the conditions are necessary. The restriction of a
positive definite quadratic form to a vector sub-space remains positive
definite, so clearly we have $(i)$. If $x\in L\cap JL$, then $Jx$ lies also in
$L\cap JL$. As $L$ is Lagrangian, we deduce that

$\omega(x,Jx)=0$

and this implies that $x=0$ since $\omega(-,J-)$ is positive definite, so we
have $(ii)$. To verify check $(iii)$, let $z=Jy$, using that $\omega(-,J-)$
and $\omega$ are respectively symmetric and antisymmetric, we get that:

$\omega(x,y)=\omega(x,-Jz)=\omega(z,-Jx)=\omega(Jy,-Jx)=\omega(Jx,Jy).$

Let us now prove that the conditions are sufficient. By $(iii)$ the form
$\omega(-,J-)$ is symmetric and by $(ii)$ any $x\in E$ can be written as

$x=x_{1}+Jx_{2},\ x_{1},x_{2}\in L.$

Then

$Jx=Jx_{1}-x_{2},$

and so

$\omega(x,Jx)=\omega(x_{1},Jx_{1})+\omega(x_{1},-x_{2})+\omega(Jx_{2},Jx_{1})+\omega(Jx_{2},-x_{2}).$

The two terms in the middle are zero since $L$ is Lagrangian, whereas by $(i)$
the first and last terms are $\geq 0$ and vanish only if $x_{1}$ and $x_{2}$
are both zero. ∎

######  Example 1.9.

Take $E=\mathbb{R}^{2}$ with the standard symplectic structure

$\omega:((x,y),(x^{\prime},y^{\prime}))\mapsto xy^{\prime}-x^{\prime}y.$

A matrix defines an almost complex structure if its eigenvalues are $\pm i$.
Such a matrix is of the form

$J=\begin{pmatrix}a&b\\\ c&-a\end{pmatrix}$

with $a^{2}+bc=-1$. Therefore the set of complex structures is a two-sheeted
hyperboloid. One sheet corresponds to the complex structure for which
$\omega(-,J-)$ is positive definite and the other one for which it is negative
definite.

######  Theorem 1.10.

Let $(E,\omega)$ be a linear symplectic space. The subset
${\mathcal{J}}_{\omega}$ of $GL(E)$ consisting of complex structures adapted
to $\omega$ is a non-empty contractible manifold.

###### Proof.

We fix two transverse linear Lagrangian subspace $L_{1},L_{2}\subset E$. As
$J$ is adapted to $\omega$, the quadratic form

$q(\eta):=\omega(\eta,J\eta),\;\;\;\eta\in L_{1}$

is positive definite and $L_{1}$ and $JL_{1}$ provide a direct sum
decomposition of $E$:

$E=L_{1}\oplus JL_{1}.$

If conversely $L^{\prime}$ is any linear Lagrangian subspace transverse to
$L_{1}$ and

$q:L_{1}\longrightarrow\mathbb{R}$

is a positive definite quadratic form, we can recover an adapted complex
structure $J$ on $E$ as follows: the $q$-orthogonal space to $x\in L_{1}$ is a
hyperplane $H\subset L_{1}$. Therefore there is a unique $y\in L^{\prime}$
which vanishes on $H$ such that

$\omega(x,y)=q(x)$

Now define a complex structure $J$ by mapping $x$ to $y$ and $y$ to $-x$.

As the Lagrangian subspace $L_{2}$ is transverse to $L_{1}$, all other
transverse Lagrangian subspaces $L^{\prime}$ are obtained as graphs of the
derivative of a quadratic function

$S:L_{2}\longrightarrow\mathbb{R}.$

Thus the space of all adapted complex structures on our symplectic vector
space is in one-to-one correspondence with the contractible set of pairs of
quadratic forms $(S,q)$ such that $q$ is positive definite. ∎

Given a manifold $M$, we have a principal bundle

$GL(TM)\longrightarrow M,$

whose fibre above $x\in M$ is the linear group $GL(T_{x}M)$. A section of this
bundle

$J:M\longrightarrow GL(TM)$

is called an almost complex structure if for any $x\in M$, we have

$J(x)^{2}=-{\rm Id\,}.$

If now $M$ is equipped with a symplectic form $\omega$, the almost complex
structure $J$ is called adapted (or also tamed) if $\omega(-,J-)$ is a
Riemannian metric. From the above lemma one deduces the existence of adapted
almost complex structures on symplectic manifolds. Indeed we have a bundle

${\mathcal{J}}_{\omega}(M)\longrightarrow M$

whose fibre above $x\in M$ is the set of $\omega_{x}$-adapted structures on
$T_{x}M$. This space is contractible and therefore the bundle admits sections.

######  Lemma 1.11.

Let $\omega_{1}$ and $\omega_{2}$ be two symplectic forms on a manifold $M$.
Assume that there exists a complex structure $J$ adapted both to $\omega_{1}$
and $\omega_{2}$. Then all two-forms lying on the positive cone

$\alpha_{1}\omega_{1}+\alpha_{2}\omega_{2},\ \alpha_{i}\geq 0$

are symplectic.

###### Proof.

Put

$\omega=\alpha_{1}\omega_{1}+\alpha_{2}\omega_{2},\ \alpha_{i}\geq 0.$

Assume that $\omega(\xi,-)$ vanishes then

$\omega(\xi,J\xi)=\alpha_{1}\omega_{1}(\xi,J\xi)+\alpha_{2}\omega_{2}(\xi,J\xi)=0$

But the $\omega_{i}(-,J-)$’s are euclidean scalar product thus

$\omega_{1}(\xi,J\xi)\geq 0,\ \omega_{2}(\xi,J\xi)\geq 0$

therefore the sum cannot be zero unless both terms vanish. Consequently
$\xi=0$. This proves the lemma. ∎

We can now prove the Darboux-Weinstein theorem.

###### Proof of theorem.

The abstract normal bundle $NL$ to $L$ is the quotient of the restriction of
the tangent bundle of $M$ to $L$, $TM_{|L}$, by the tangent bundle $TL$ to
$L$. By the tubular neighbourhood theorem, there is a diffeomorphism $\phi$
from a neighbourhood of $L$ in $M$ to a neighbourhood of $L$ in the normal
bundle $NL$. The interior product of vectors $v$ based at a point $p$ of $L$
with the symplectic form induces an isomorphism of $NL$ with the cotangent
bundle $T^{*}L$ of $L$:

$\nu:N_{p}L\longrightarrow T^{*}_{p}L,\;\;v\mapsto i_{v}\omega.$

Therefore we reduced the theorem to the case $(M,\omega)=(T^{*}L,\omega)$
where $\omega$ is a priori not the standard symplectic form $\omega_{std}$ on
$T^{*}L$.

We now choose almost complex structures $J$ and $J_{std}$ on $M=T^{*}L$
adapted to the symplectic forms $\omega$ and $\omega_{std}$.

First observe that the tangent space $T_{q,0}M$ is a direct sum

$T_{q,0}M=T_{q}L\oplus JT_{q}L.$

The inner product with the symplectic form gives an identification of
$JT_{q}L$ with $T_{q}^{*}L$.

Using these data, we will now define a map

$f:M\longrightarrow M$

in the following way. The inclusion of bundles over $L$

$i:T^{*}L\longrightarrow TM_{|L},$

is given pointwise by the inclusion:

$T^{*}_{q}L\hookrightarrow T_{q,0}M.$

Its composition with the linear bundle map

$JJ_{std}^{-1}:TM_{|L}\longrightarrow TM_{|L}$

defines a map

$JJ_{std}^{-1}\circ i:M\longrightarrow TM_{|L}.$

The map $f$ is obtained by composing this map with the exponential map

$exp:TM_{|L}\longrightarrow M,$

defined pointwise by the exponential maps

$exp_{q}:T_{(q,0)}M\longrightarrow M$

for the Riemannian metric $\omega(-,J_{std}-)$. So

$f:=exp\circ JJ_{std}^{-1}\circ i:M=T^{*}L\stackrel{{\scriptstyle
i}}{{\hookrightarrow}}TM_{|L}\stackrel{{\scriptstyle
JJ_{std}^{-1}}}{{\longrightarrow}}TM_{|L}\stackrel{{\scriptstyle
exp}}{{\longrightarrow}}M.$

Clearly $f$ restricted to $L$ is the identity.

The derivative of this map is easily computed at points of $L$:

$\displaystyle df_{\mid TL}$ $\displaystyle={\rm Id\,}$ $\displaystyle
df_{\mid J_{std}TL}$ $\displaystyle=JJ_{std}^{-1}$

We assert that the form $f^{*}\omega$ is $J_{std}$-adapted in any sufficiently
small neighbourhood of $L$. Indeed, for any point $q\in L$ and any $\xi\in
T_{q}L$ , we have

$\displaystyle f^{*}\omega_{(q,0)}(\xi,J_{std}\xi)$
$\displaystyle=\omega_{(q,0)}(df(x)\xi,df(x)J_{std}\xi)$
$\displaystyle=\omega_{(q,0)}(\xi,JJ_{std}^{-1}J_{std}\xi)$
$\displaystyle=\omega_{(q,0)}(\xi,J\xi)>0$

As the condition of being adapted is open, this proves the assertion. Lemma
1.11 shows that the path

$\omega_{t}:=t\omega_{std}+(1-t)f^{*}\omega$

remains inside the space of symplectic forms. Therefore, using Theorem 1.6
with $M=T^{*}L$, we get that $(T^{*}L,\omega_{std})$ and
$(T^{*}L,f^{*}\omega)$ are symplectomorphic, in a sufficiently small
neighbourhood of $L$. This concludes the proof of the theorem. ∎

###  1.5. The relative Darboux-Weinstein theorem

As is customary in algebraic geometry, it is useful to look for a relative
version of the Darboux-Weinstein theorem, where one considers families of
manifolds over a base $S$ and which specialise to it in the case that $S$
reduces to a point.

A fibration of $C^{\infty}$-manifolds

$\pi:M\longrightarrow S$

is called symplectic, if it carries a two-form which restricts to a symplectic
form on all the fibres $M_{s}=\pi^{-1}(s)$. In the spirit of Grothendieck, $M$
is viewed as a symplectic manifold over the base $S$. Similarly, a Lagrangian
manifold $L$ over $S$ is a fibration $\rho:L\hookrightarrow S$, sitting in a
diagram

$\textstyle{L\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{S}$

Thus the fibre $L_{s}$ at $s$ is a Lagrangian submanifold of $M_{s}$. Recall
that for such a relative manifold $\rho:L\longrightarrow S$ there is a natural
surjective map of vector bundles on $L$:

$TL\longrightarrow\rho^{*}(TS),\;\;v\mapsto d\rho(v).$

The kernel of this map is $T_{S}L$, called the relative tangent space. It
consists of those tangent vectors of $L$ that are tangent to the fibres of
$\rho$. In particular, if $\rho$ is a trivial fibration with fibre $L_{0}$
then $T_{S}L$ is simply $TL_{0}\times S$.

The dual of the relative tangent bundle we denote by $T^{*}_{S}L$: it sits in
an exact sequence

$0\longrightarrow\rho^{*}(T^{*}S)\longrightarrow T^{*}L\longrightarrow
T^{*}_{S}L\longrightarrow 0$

of vector bundles on $L$. The restriction of $T^{*}_{S}L$ to the fibre $L_{s}$
is just the cotangent bundle of $L_{s}$:

$T^{*}_{S}L_{|L_{s}}=T^{*}L_{s}$

The following parametric version of the Darboux-Weinstein theorem expresses
the fact that there are no local invariants for families of symplectic
manifolds.

######  Theorem 1.12.

Let $L\longrightarrow S$ be a proper Lagrangian submanifold of a symplectic
manifold $M\longrightarrow S$ over some base $S$. The germ of
$M\longrightarrow S$ along $L\longrightarrow S$ is, locally on the base $S$,
symplectomorphic to a the germ of $T_{S}^{*}L$ along the zero section.

###### Proof.

To adapt the proof the Darboux-Weinstein theorem to this parametric case, we
use the relative variant of differential forms. The relative de Rham complex
is defined by

$\Omega^{\bullet}_{\pi}:=\Omega^{\bullet}_{M}/(\pi^{*}\Omega^{1}_{S}\wedge\Omega^{\bullet-1}_{M}),$

where $\Omega_{N}^{p}$ denotes the space of $C^{\infty}$ $p$-forms on a
manifold $N$. If we take local coordinates $s_{1},\dots,s_{n}$ which
trivialise the fibration, two differential forms on $M$ are equal as elements
of $\Omega^{\bullet}_{\pi}$ if they are equal modulo a form of the type

$\sum\alpha_{i}\wedge ds_{i},\ \alpha_{i}\in\Omega^{\bullet}_{M}.$

The cohomology of this relative de Rham complex is, locally on the base, the
cohomology of the fibre tensored with the $C^{\infty}$-functions on the base.
In particular, the cohomology class of a symplectic form vanishes in a
neighbourhood of a Lagrangian fibre. The deformation argument remains the same
with the only difference that we now consider the forms $\omega_{t}$ entering
the equation:

$i_{X_{t}}\omega_{t}=-\dot{\alpha}_{t}$

as relative differential forms with vanishing de Rham class. ∎

###  1.6. Liouville integrability

The classical language of Poisson brackets is a very useful way to express the
main features of symplectic geometry.

For two smooth functions

$f,g:M\longrightarrow\mathbb{R},$

the Poisson-bracket of $f$ and $g$ is defined by

$\\{f,g\\}=\omega(X_{f},X_{g})=-\\{g,f\\}.$

It is readily seen that

$\\{f,g\\}={\mathcal{L}}_{X_{f}}(g),\;\;\;[X_{f},X_{g}]=X_{\\{f,g\\}}.$

The Poisson-bracket is a bi-derivation

$\\{f,gh\\}=\\{f,g\\}h+\\{f,h\\}g,\;\;\\{fg,h\\}=f\\{g,h\\}+g\\{f,h\\}$

which satisfies the Jacobi identity:

$\\{f,\\{g,h\\}\\}+\\{g,\\{h,f\\}\\}+\\{h,\\{f,g\\}\\}=0$

The Darboux coordinates provide a local model for the Poisson bracket:

$\\{f,g\\}=\sum_{i=1}^{n}\partial_{p_{i}}f\partial_{q_{i}}g-\partial_{q_{i}}f\partial_{p_{i}}g.$

More generally, a Poisson structure on a manifold is such an anti-symmetric
bi-derivation satisfying the Jacobi identity. The dynamics of a Hamiltonian
$H:M\longrightarrow\mathbb{R}$ can be expressed concisely by the statement

$\dot{G}=\\{H,G\\},$

which indeed reduce to Hamilton’s equations of motion for Darboux coordinates.
From this we see that a quantity

$G:M\longrightarrow\mathbb{R}$

is preserved by the flow of $H$ if and only if it Poisson-commutes with $H$:

$\\{H,G\\}=0.$

Such a quantity is called a first integral. Any function $\phi$ of $H$ is a
first integral:

${\mathcal{L}}_{X_{H}}\phi(H)=\\{H,\phi(H)\\}=0$

In some cases the converse is true as well. For instance, endow
$M=\mathbb{R}^{2}$ with its canonical symplectic structure $\omega=dq\wedge
dp$ and take $H=p$. Then any first integral is a function $G$ with the
property that

$\\{H,G\\}=0\iff\partial_{q}G=0.$

so depends only on the variable $p$. A Hamiltonian $H$ is called Liouville
integrable or simply integrable if there exists Poisson commuting functions
$H=f_{1},\dots,f_{n}$ with Hamiltonian vector fields that are linearly
independent

$df_{1}\wedge df_{2}\wedge\dots\wedge df_{n}\neq 0$

on an open dense subset of $M$. The map $f=(f_{1},f_{2},\dots,f_{n})$ is
called a moment mapping. The smooth fibres of this map are automatically
Lagrangian. Indeed, the Hamiltonian vector fields $X_{i}$ of the $f_{i}$’s
generate the tangent bundle and

$\omega(X_{i},X_{j})=\\{f_{i},f_{j}\\}=0.$

In particular if $f$ is smooth and proper, then it defines what is called a
Lagrangian fibration. The Hamiltonian flows then induce a cocompact
$\mathbb{R}^{n}$-action, and therefore, by general properties of
$\mathbb{R}^{n}$-actions, the moment mapping is a fibration whose fibres are
tori.

A moment mapping defined over some $n$-dimensional base

$f=(f_{1},\dots,f_{n}):X\longrightarrow S$

gives rise to a Lagrangian subspace in a symplectic manifold in the following
way: the projection on the second factor gives a symplectic manifold

$M:=X\times S\longrightarrow S$

over the base $S$. We denote the graph of $f$ by

$L_{f}\subset M=X\times S.$

It is a Lagrangian submanifold of $M$ over $S$. By the relative Darboux-
Weinstein theorem, locally on $S$, a neighbourhood of $L_{f}$ in $M$ is
symplectomorphic to a neighbourhood of the zero section its cotangent bundle.

###  1.7. Action-angle variables

If a moment mapping $f$ is proper, then its fibres have a co-compact
$\mathbb{R}^{n}$-action coming from the commuting flows of
$f_{1},\dots,f_{n}$. The Lagrangian manifold $L_{f}$ is therefore a fibration
by tori over its base $S$. Action-angle variables provide Darboux coordinates
adapted to this situation. The action variables $I_{1},I_{2},\ldots,I_{n}$ are
functions of the $f_{i}$, so they are constant on the tori of the fibration
and have the further property that the integral curves of the corresponding
Hamiltonian fields $X_{i}$ all have a fixed period. The angle variables
$\theta_{1},\theta_{2},\ldots,\theta_{n}$ are the corresponding ’time’
variables, so have the property that

$\omega=d\theta_{1}\wedge dI_{1}+d\theta_{2}\wedge
dI_{2}+\ldots+d\theta_{n}\wedge dI_{n}.$

Before we explain this in detail, let us start with some examples.

######  Example 1.13.

The restriction of the map

$\mathbb{R}^{2n}\longrightarrow\mathbb{R}^{n},\
(q,p)\mapsto(\frac{1}{2}(q_{1}^{2}+p_{1}^{2}),\dots,\frac{1}{2}(q_{n}^{2}+p_{n}^{2}))$

to the preimage $X$ of any open subset $S\subset\mathbb{R}_{>0}^{n}$ produces
a Lagrangian fibration

$f:X\longrightarrow S.$

The Hamiltonian vector field of $f_{i}=\frac{1}{2}(q_{i}^{2}+p_{i}^{2})$ is

$X_{i}=p_{i}\partial_{q_{i}}-q_{i}\partial_{p_{i}},\ i=1,\dots,n.$

These vector fields $X_{i}$ span the tangent spaces to the fibres. We get
action-angle coordinates by putting

$I_{i}=f_{i},\;\;\;\;\theta_{i}=\arctan\frac{p_{i}}{q_{i}}$

or equivalently

$p_{i}:=\sqrt{I_{i}}\cos\theta_{i},\ q_{i}:=\sqrt{I_{i}}\sin\theta_{i}.$

Note that the function $\theta_{i}$ is multi-valued, but its differential
$d\theta_{i}$ is equal to the closed one-form

$\alpha_{i}:=\frac{-q_{i}}{p_{i}^{2}+q_{i}^{2}}dp_{i}+\frac{p_{i}}{p_{i}^{2}+q_{i}^{2}}dq_{i}$

These closed one-forms $\alpha_{i}$ form a dual basis to commuting vector
fields $X_{i}$:

$\alpha_{i}(X_{j})=\delta_{ij}$

######  Example 1.14.

The above example is archetypical for any integrable system, but untypical in
the sense that the action variables were very simple. In general the
determination of the action-angle variables lead to integrals defining
transcendental functions. Consider for example the case $n=1$ and:

$f(q,p)=p^{2}+\frac{1}{2}q^{2}-\frac{1}{3}q^{3}.$

Let $S$ be a pointed disc of radius $r<1/6$ and $X$ be the preimage of $S$
under the polynomial map $f$. This defines a locally trivial
$S^{1}$-fibration, that we denote in the same way:

$f:X\longrightarrow S.$

In that case, the action is the function

$I=\int_{\gamma_{\varepsilon}}pdq$

where $\gamma_{\varepsilon}$ is the small loop inside the level set

$\\{(q,p)\in X:|\;f(q,p)=\varepsilon\\}\approx S^{1}.$

By Stokes’ formula, this is just the area of the disc bounded by
$\gamma_{\varepsilon}$ and is thus given as an elliptic integral of the second
kind. If we regard $I$ as a function of $q,p$, then the angle $\theta$ is the
time of this vector field. It is given by the indefinite integral

$\theta=\int_{*}^{(q,p)}\frac{dp\wedge dq}{dI}$

and is defined only modulo a period. This is a generic feature: computing
action-angle coordinates involves abelian integrals and theta functions.

Let us now consider the general case. We consider a proper moment map

$f=(f_{1},\dots,f_{n}):X\longrightarrow S,$

pick a reference point $0\in S$ and let $L_{0}:=f^{-1}(0)$ be the
corresponding reference torus. By shrinking $S$, we may assume that $S$ is
contractible and the fibration is trivial. The total space $X$ retracts to the
torus $L_{0}$ and because the symplectic form $\omega$ vanishes on $L_{0}$,
the form $\omega$ is exact on $X$, so we can find an action form $\alpha$ for
$\omega$:

$\omega=d\alpha.$

For each cycle $\gamma(0)\in H_{1}(L_{0},\mathbb{Z})$, we obtain, by parallel
transport to neighbouring tori, a family of cycles $\gamma(s)\in
H_{1}(L_{s},\mathbb{Z})$. This defines an action integral:

$I_{\gamma}:S\longrightarrow\mathbb{R},\;\;s\mapsto
I_{\gamma}(s):=\int_{\gamma(s)}\alpha.$

If we pick a basis $\gamma_{1}(s),\gamma_{2}(s),\ldots,\gamma_{n}(s)$ for
$H_{1}(L_{s},\mathbb{Z})$, we obtain the action functions, or action
variables, $I_{j}$ by composing the action integrals

$\int_{\gamma_{j}(s)}\alpha.$

with the map $f$:

$I_{j}(q,p)=I_{\gamma_{j}}(f(p,q)).$

Note that a change of the homology basis induces an $GL(n,\mathbb{Z})$-action
on the possible choices for the action variables.

The classical Arnold-Liouville-Mineur theorem can be stated as follows:

######  Theorem 1.15.

Let $f:X\longrightarrow S$ be a proper integrable system on a symplectic
manifold $(M,\omega)$. There exists functions

$\theta_{j}:X\longrightarrow S^{1},$

defined locally near a fibre, called the angles, such that the action-angle
functions induce a local symplectomorphism

$X\longrightarrow(S^{1})^{n}\times\mathbb{R}^{n},$

where $(S^{1})^{n}\times\mathbb{R}^{n}=T^{*}(S^{1})^{n}$ is equipped with the
standard symplectic form.

###### Proof.

The Darboux-Weinstein theorem implies that we may assume that
$M=(S^{1})^{n}\times\mathbb{R}^{n}=\\{(q,p)\\}$ equipped with its standard
symplectic structure and that the Lagrangian fibre of $f$ over $s_{0}\in S$ is
the zero section. Up to a linear change of coordinates we may also assume that

$f_{i}(q,p)=p_{i}+\dots$

where the dots stand for higher order terms in the $p_{i}$’s depending on the
$q$-variables.

By the implicit function theorem, the fibres of $f$ are the graphs of maps

$g_{s}:L\longrightarrow\mathbb{R}^{n}$

with

$f(q,p)=0\iff p=g_{s}(q).$

Comparing the Taylor series of $f$ and $g_{s}$, we see that

$g_{s}=s+o(\|s\|).$

We choose the cycle $\gamma_{j}(s)$ which projects to the $j$-th coordinate
circle in the torus. Then the actions are defined by

$\displaystyle I_{j}(q,p)$
$\displaystyle=\int_{\gamma_{j}(s)}\sum_{i=1}^{n}p_{i}dq_{i}$
$\displaystyle=\int_{\gamma_{j}(s)}\sum_{i=1}^{n}(g_{i})_{s}(q)dq_{i},$
$\displaystyle=(g_{j})_{s}(q=0)$ $\displaystyle=s_{j}+o(\|s\|)$
$\displaystyle=p_{j}+o(\|p\|)$

The associated angles $\theta_{j}$ are of the form

$\theta_{j}=q_{j}+o(\|q\|)$

Thus the map

$(q,p)\mapsto(\theta,I)$

is a diffeomorphism and therefore a symplectomorphism. ∎

###  1.8. Integrable systems and the Gauss-Manin connection

The above proof for the existence of action-angle variables relies on blind
computations. One of the beautiful aspects of classical integrable systems is
that the construction of action-angle coordinates can also be explained in
terms symplectic geometry of the action integrals $I_{\gamma}$ that we defined
above.

We consider a torus bundle

$f:X\longrightarrow S$

defined by a proper smooth moment mapping over a contractible open set
$S\subset\mathbb{R}^{n}$. We denote by $X_{i}$ the Hamiltonian fields of its
components and choose a Lagrangian section of the torus bundle

$\sigma:S\longrightarrow X,\ \sigma^{*}\omega=0.$

The Hamiltonian flows of the $X_{i}$’s induce an $\mathbb{R}^{n}$-action on
the fibres of $f$, which is, as any such $\mathbb{R}^{n}$-action, transitive
on the fibres. Thus translating the section $\sigma$ by the Hamiltonian vector
fields define sections starting at arbitrary points on a fibre. Thus for each
point $x\in X$, we get a vector space $H_{x}\subset T_{x}X$ complementary to
the tangent space of the fibre, that is, we get an Ehresmann connection on
$X$.

In particular, each vector field $v$ on the base $S$ is lifted in a unique way
to a vector field $\tilde{v}$ on $X$.

######  Lemma 1.16.

One has

$\omega=\sum_{i=1}^{n}\alpha_{i}\wedge df_{i}$

where $\alpha_{i}:=\omega(\tilde{v}_{i},-)$ and

$\tilde{v}_{i}:=\tilde{\partial}_{s_{i}}$

###### Proof.

The vector fields $\tilde{v}_{1},\dots,\tilde{v}_{n}$ lifting
$\partial_{s_{1}},\dots,\partial_{s_{n}}$ satisfy

$\omega(X_{i},\tilde{v}_{j})=df_{i}(\tilde{v}_{j})=ds_{i}(\partial_{s_{j}})=\delta_{ij}.$

Moreover as the section $\sigma$ is Lagrangian, we have:

$\omega(\tilde{v}_{i},\tilde{v}_{j})=0.$

Since the $X_{i}$ pairwise commute, we also have

$\omega(X_{i},X_{j})=0.$

This means exactly that the symplectic form is
$\sum_{i=1}^{n}df_{i}\wedge\alpha_{i}$. ∎

######  Corollary 1.17.

The lifted vector field $\tilde{v}$ is symplectic:

${\mathcal{L}}_{\tilde{v}}\omega=0.$

###### Proof.

Any lifted vector field is a linear combination of the vector fields
$\tilde{v}_{i}$ with coefficients that are pulled back from $S$. ∎

By taking the interior product with the symplectic form $\omega$, a vector
field $v$ is lifted to a one-form $\alpha_{v}$. As symplectic vector fields
correspond to closed one-forms, we obtain a map

$\Theta(S)\longrightarrow\Omega^{1}(X)_{closed},\;\;\;v\mapsto\alpha_{v}$

For each $s\in S$, this induces a map $T_{s}S\longrightarrow
H^{1}(L_{s},\mathbb{R}),v\mapsto[\alpha_{v}]$.

######  Lemma 1.18.

For each $s\in S$, the map

$T_{s}S\longrightarrow H^{1}(L_{s},\mathbb{R}),v\mapsto[\alpha_{v}].$

is an isomorphism.

###### Proof.

The forms $\alpha_{1},\dots,\alpha_{n}$ symplectically associated to the
vector fields $\tilde{v_{1}},\tilde{v_{2}},\ldots,\tilde{v_{n}}$ lifting
$\partial_{s_{1}},\dots,\partial_{s_{n}}$ form at each point the dual basis to
hamiltonian vector fields $X_{1},\dots,X_{n}$ and are therefore linearly
independent. ∎

######  Lemma 1.19.

The one-form $\alpha_{v}$ lifting a vector field $v\in\Theta_{S}$ satisfies

${\mathcal{L}}_{\tilde{v}}\alpha=\alpha_{v}+d(i_{\tilde{v}}\alpha).$

###### Proof.

By Cartan’s formula

$\displaystyle{\mathcal{L}}_{\tilde{v}}\alpha$
$\displaystyle=i_{\tilde{v}}d\alpha+d(i_{\tilde{v}}\alpha)$
$\displaystyle=i_{\tilde{v}}\omega+d(i_{\tilde{v}}\alpha)$
$\displaystyle=\alpha_{v}+d(i_{\tilde{v}}\alpha)$

∎

The derivative of an action-integral $I_{\gamma}$ with respect to the variable
$s_{i}=f_{i}$ is given by

${\mathcal{L}}_{v}I_{\gamma(s)}=\int_{\gamma(s)}{\mathcal{L}}_{\tilde{v}}\alpha=\int_{\gamma(s)}\alpha_{v}$

which means that the Ehresmann connection induces the usual Gauss-Manin
connection on relative differential forms.

As a corollary to our discussion, we get the

######  Theorem 1.20.

If $\gamma_{1}(s),\gamma_{2}(s),\ldots,\gamma_{n}(s)$ form a basis for the
lattice $H_{1}(L_{s},\mathbb{Z})$, then the associated map

$\Psi:S\longrightarrow
T\subset\mathbb{R}^{n},s\mapsto(I_{\gamma_{1}}(s),I_{\gamma_{2}}(s),\ldots,I_{\gamma_{n}}(s))$

is a local diffeomorphism.

###### Proof.

By De Rham duality, if the $\gamma_{i}(s)$’s form a basis then integration
over the $\gamma_{i}(s)$’s are linearly independent linear forms. ∎

###  1.9. Affine structures and integrable systems

Let us keep the notation of the previous section.

The base space of a proper smooth moment map carries an affine structure, that
we shall now define. Integration over a cycle $\gamma(s)\in
H_{1}(L_{s},\mathbb{Z})$ defines a linear function

$\int_{\gamma(s)}:H^{1}(L_{s},\mathbb{R})\longrightarrow\mathbb{R}.$

Identifying $H^{1}(L_{s},\mathbb{R})$ with $T_{s}S$, its kernel defines a
hyperplane $\gamma^{\perp}(s)$ in $T_{s}S$. In this way, we obtain a
distribution $\gamma^{\perp}$ associated to a cycle and the level sets of
$I_{\gamma}$ are tangent to this distribution.

If $\gamma_{1}(s),\gamma_{2}(s),\ldots,\gamma_{n}(s)$ form a basis for the
lattice $H_{1}(L_{s},\mathbb{Z})$ then the level-sets of the functions
$I_{\gamma_{i}}$ are tangent to the distribution and define an affine
structure in the base of the moment map.

If we now consider these integrals of functions depending on the variables
$(q,p)$:

$I_{j}(q,p)=I_{\gamma_{j}}(f(q,p)).$

We get a new integrable system $I=(I_{1},\dots,I_{n})$ for which the affine
structure is now linear:

$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{I}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{T}$

meaning that all action integrals $I_{\gamma}$ become linear functions in the
standard coordinates $t_{1},t_{2},\ldots,t_{n}$ of $\mathbb{R}^{n}\supset T$.

Via that map $\Psi$, the distribution $\gamma^{\perp}$ becomes the parallel
distribution of linear hyperplanes.

Our original Lagrangian section of $X\longrightarrow S$ induces a section of
$X\longrightarrow T$ by composition with the inverse of $\Psi$. In this way,
we lift the vector fields $\partial_{t_{i}}$ to closed one-forms
$\beta_{1},\beta_{2},\ldots,\beta_{n}$, using the new Ehresmann connection.

As $I_{\gamma}(t)$ is now linear in $t$, the periods of the $\beta_{i}$’s are
all constant. Thus we define:

$\lambda_{i}:=\int_{\gamma(t)}\beta_{i}=\partial_{t_{i}}I_{\gamma}(t)\in\mathbb{R}.$

The indefinite integral gives a well-defined map modulo the periods:

$\int_{\sigma(t)}^{*}:L_{t}\longrightarrow\oplus_{i=1}^{n}\mathbb{R}/\lambda_{i}\mathbb{Z},\
x\longrightarrow(\int_{\sigma(t)}^{x}\beta_{1},\int_{\sigma(t)}^{x}\beta_{2},\ldots,\int_{\sigma(t)}^{x}\beta_{n})$

As the $\beta_{i}$’s are linearly independent this map is a local
diffeomorphism and by definition of the periods $\lambda_{i}$, it is also one-
to-one. Hence the induced torus fibration $X\longrightarrow T$ is trivialised.

Such a trivialisation is, in general, not possible if the base $S$ of the
torus fibration is not contractible. Such examples occurs when we remove
singular fibres from a larger family

$\overline{X}\longrightarrow\overline{S}.$

As a rule, the base $S$ of the fibration is then not contractible, and
interesting global issues arise.

The cycles $\gamma(s)$ may undergo monodromy under parallel transport along a
loop in the base $S$, leading to a non-trivial representation of the
fundamental group:

$\rho:\pi_{1}(S,0)\longrightarrow Aut(H_{1}(L_{0},\mathbb{Z})).$

The lattices $\Lambda_{s}=H_{1}(L_{s},\mathbb{Z})$ then form a non-trivial
lattice bundle $\Lambda$ over $S$, commonly referred to as a local system.
Although this phenomenon obstructs the existence of action-angle variables,
the above constructions can be done locally on $S$. This leads to a version of
action angle coordinates using a certain tautological symplectic torus bundle
$T^{*}S/\Lambda$ associated to a Lagrangian section of $f:X\longrightarrow S$
that we will describe now.

As the fibre $L_{s}$ is a torus with transitive $\mathbb{R}^{n}$-action, the
choice of a point $x\in L_{s}$ determines a map

$\mathbb{R}^{n}\longrightarrow L_{s},\;\;0\mapsto x$

and as this map is the universal covering map, the kernel can be identified
with $H_{1}(L,\mathbb{Z})$. In this way $L_{s}$ is, after choosing a point as
origin, identified with $H_{1}(L_{s},\mathbb{R})/H_{1}(L_{s},\mathbb{Z})$.

When we dualise the isomorphism

$T_{s}S\longrightarrow H^{1}(L_{s},\mathbb{R}),\ v\mapsto[\alpha_{v}]$

we get an isomorphism

$T_{s}S\longrightarrow H_{1}(L_{s},\mathbb{R})\longrightarrow T^{*}_{s}S$

Thus we get a natural lattice

$\Lambda_{s}\subset T_{s}^{*}S$

in the cotangent space at $s$, isomorphic to first homology group
$H_{1}(L_{s},\mathbb{Z})$ of the fibre. When we let $s$ run in $S$, we get a
natural symplectic torus bundle attached to the situation:

$T^{*}S/\Lambda\longrightarrow S.$

isomorphic to

$X\longrightarrow S.$

This dicussion can be summarised as follows

######  Theorem 1.21.

Let $f:X\longrightarrow S$ be a proper smooth integrable system with a
Lagrangian section

$\sigma:S\longrightarrow X.$

The identification of the fibre of $f$ at $s$ with
$H_{1}(L_{s},\mathbb{R})/H_{1}(L_{s},\mathbb{Z})$ induces a symplectomorphism
of torus bundles over $S$

$\textstyle{T^{*}S/\Lambda\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$

where $\Lambda_{s}$ is identified with $H_{1}(L_{s},\mathbb{Z})$ via the
isomorphism $T^{*}_{s}S\longrightarrow H_{1}(L_{s},\mathbb{R})$ induced by the
section $\sigma$.

###  1.10. Bibliographical notes

The technique used to prove the Darboux theorem is called Moser’s path
homotopy method. It was introduced in:

J. Moser, On the volume elements on a manifold, Trans. of the Am. Math.
Soc.120(2), 286-294, (1965).

The Darboux-Weinstein theorem is proved in:

A. Weinstein, Lagrangian submanifolds and Hamiltonian systems, Annals of Math.
98, 377-410, (1973).

Almost complex structures were introduced in symplectic geometry by Gromov:

Gromov, M. Pseudo holomorphic curves in symplectic manifolds, Inventiones
Mathematicae, 82(2), 307-347, (1985).

The existence of action-angle coordinates for smooth proper integrable systems
was first proved in:

H. Mineur, Sur les systèmes mécaniques admettant n intégrales premieres
uniformes et l’extension à ces systèmes de la méthode de quantification de
Sommerfeld, C. R. Acad. Sci., Paris 200, 1571 - 1573, (1935).

For a long time, the work of Mineur was forgotten and the result was
rediscovered by Arnold in:

V.I. Arnold, A theorem of Liouville concerning integrable problems of
dynamics, Sibirsk. Math . Zh. 4 , 471 - 474, (1963).

Theorem 1.21 is due to Duistermaat. In his paper, he also discusses monodromy
obstructions to the global existence of action-angle variables:

H. Duistermaat, On global action-angle variables, Communications on Pure and
Applies Mathematics, 33(6), 687-706, (1980).

The beautiful effect of monodromy on the semi-classical spectrum for the
spherical pendulum was described in:

R. Cushman, H. Duistermaat,The quantum mechanical spherical pendulum, Bull.
Amer. Math. Soc. (N.S.), Volume 19, Number 2, 475-479, (1988)

and for more general integrable systems in:

S. Vu Ngoc, Quantum monodromy in integrable systems, Communications in
mathematical physics, 203(2), 465-479, (1999).

The proof of the existence of action-angle variables and its generalisation
using a relative Darboux-Weinstein theorem over a base was introduced to prove
the absence of obstruction to quantisation, see:

M. Garay and D. van Straten Classical and quantum integrability. Mosc. Math.
Journal 10, 519-545, (2010).
There are some misprints in the paper which we (hopefully!) corrected here.

Classical sources on classical mechanics in the language of global analysis
are:

V.I. Arnold, Mathematical methods of classical mechanics, Graduate texts in
Mathematics (Vol. 60). Springer Verlag.

R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin/Cummings
Publishing Company, (1978).

For more detail on symplectic geometry, we also refer to:

D. McDuff, D. Salamon Introduction to Symplectic Topology, Oxford mathematical
monographs, Clarendon Press, 1998.

##  Chapter 2 The KAM problem

We now come to a central theme of classical KAM-theory, namely that of the
persistence of quasi-periodic motions after perturbation of an integrable
Hamiltonian system. It is convenient to work with an algebraic model and
formulate the problem in terms of certain Poisson-algebras associated to the
algebraic torus, like the algebra of analytic Fourier series. We discuss the
problem first on the level of formal power series and then discuss the special
features that arise at the analytic level and identify the first order
analytic obstruction.

###  2.1. Quasi-periodic motions

Let us now look at the motion for an integrable Hamiltonian $H=f(I)$ in action
angle coordinates:

$(\theta_{1}\dots,\theta_{n},I_{1},\dots,I_{n}),\
\\{I_{j},\theta_{k}\\}=\delta_{jk}.$

Hamiltons equations of motion are

$\left\\{\begin{matrix}\dot{I}_{j}&=&0\\\
\dot{\theta}_{j}&=&\partial_{I_{j}}f(I).\end{matrix}\right.$

These equations with initial condition

$I(0)=c,\;\;\;\theta(0)=\beta$

can easily be integrated:

$\left\\{\begin{matrix}I_{j}&=&c_{j}\\\
\theta_{j}&=&\omega_{j}t+\beta_{j}\end{matrix}\right.$

where

$\omega_{i}:=\partial_{I_{j}}f(c)\;.$

The trajectory lies on the torus $I=c$ and such a motion is called quasi-
periodic. A quasi-periodic motion is dense on its torus, if the the frequency
vector of the motion

$\omega:=(\omega_{1},\omega_{2},\ldots,\omega_{n})$

has $\mathbb{Z}$-independent coordinates. Of course, the vector $\omega$ will
depend on $c$.

The question from which KAM-theory originates is the following:

Given an integrable system in action-angle coordinates, do such quasi-periodic
motions persist after turning on a perturbation of the original Hamiltonian?

In order to formulate this problem more precisely, it is convenient to
introduce first an algebra-geometric model of the above situation. Then we
will consider the formal and the analytic versions.

Rather then working with the angular variables $\theta_{j}$ it is convenient
to use variables

$q_{j}:=e^{i\theta_{j}}.$

For real values of $\theta_{j}$, the variable $q_{j}$ runs over the unit
circle in the complex plane. Adding an imaginary part to $\theta_{j}$ changes
the radius of the circle traced by $q_{j}$. So we are led to consider the
algebraic torus

$X:=(\mathbb{C}^{*})^{n}=\\{(q_{1},\dots,q_{n})\in\mathbb{C}^{n}:q_{i}\neq
0,i=1,2,\ldots,n\\}.$

Note that $X$ has a anti-holomorphic involution

$q_{i}\mapsto\frac{1}{\bar{q}_{i}}$

and the usual torus $(S^{1})^{n}$ traced by all the angular variables
$\theta_{j}$ is the part of the algebraic torus $X$ that is fixed by this
involution, so it may be considered as the ’real part’ of $X$ for the real
structure defined by this anti-homomorphic involution.

Note that

$\\{I_{j},q_{k}\\}=\\{I_{j},e^{i\theta_{k}}\\}=ie^{i\theta_{k}}\delta_{jk}.$

We also will write $p_{j}$ for the conjugate action variables $-iI_{j}$, and
note that then

$\\{p_{j},q_{k}\\}=-i\\{I_{j},e^{i\theta_{k}}\\}=e^{i\theta_{k}}\delta_{jk}=q_{k}\delta_{jk}.$

###  2.2. Algebraic model

We started our discussion of symplectic geometry in the context of
differential geometry, but now we wish to take a more algebraic viewpoint. In
differential geometry, the basic notion is that of a manifold, but in algebra
one considers rings of functions on the manifold under consideration. Usually
there are several natural choices for the types of functions one might want to
consider: $C^{\infty}$, analytic, polynomial, convergent or formal power
series.

The Poisson bracket of functions on a symplectic manifold gives the rings of
functions the additional structure of a Poisson algebra. We recall that a
Poisson algebra over a (commutative) ring $R$ is an $R$-algebra with an
additional anti-symmetric operation $\\{-,-\\}$ which satisfy the Jacobi
identity

$\\{\\{f,g\\},h\\}+\\{\\{g,h\\},f\\}+\\{\\{h,f\\},g\\}=0$

and which is a bi-derivation, that is

$g\mapsto\\{f,g\\}$

is a derivation for any fixed $f$:

$\\{f,gh\\}=\\{f,g\\}h+g\\{f,h\\}.$

As the bracket is anti-symmetric, it is also a derivation on the first
variable. For a given real symplectic manifold $M$, the $\mathbb{R}$-algebra
$C^{\infty}(M)$ is naturally a Poisson $\mathbb{R}$-algebra.

The algebraic model of the family of tori is the Laurent polynomial ring

$A:=\mathbb{C}[q,q^{-1},p]:=\mathbb{C}[q_{1},q_{2},\ldots,q_{1}^{-1},q_{2}^{-1},\ldots,q_{n}^{-1},p_{1},p_{2},\ldots,p_{n}]$

with the Poisson-bracket

$\\{p_{j},q_{k}\\}=q_{k}\delta_{jk}.$

The Poisson-algebra $A$ can be seen as the affine coordinate ring of the
cotangent bundle $T^{*}X$ of our torus $X$. The canonical symplectic form on
that space is

$\omega:=\sum_{j=1}^{n}\frac{dq_{j}}{q_{j}}\wedge dp_{j}$

and leads precisely to the above Poisson-bracket. The fibres of the map

$T^{*}X\longrightarrow\mathbb{C}^{n},(q,p)\mapsto
p=(p_{1},p_{2},\ldots,p_{n})$

are copies of $X$ and the flow of any Hamiltonian
$H=f(p)\in\mathbb{C}[p_{1},p_{2},\ldots,p_{n}]$ preserves these fibres.

We also note that in the sense of algebraic geometry, quasi-periodic motion
can be defined “over any field $K$” by replacing the coefficients $\mathbb{C}$
by $K$, and then considering the Poisson-algebra

$A=K[q,q^{-1},p]$

with the Poisson-bracket as defined before.

###  2.3. Perturbation theory

To describe a perturbation of a Hamiltonian $H\in A$ we “add a parameter $t$
to the ring $A$” and consider the ring of Laurent polynomials

$B:=K[q,q^{-1},p,t].$

Here $t$ is a parameter, which is supposed to be a central element, i.e. an
element that Poisson-commutes with all elements $f\in B$:

$\\{t,f\\}=0.$

By a deformation or perturbation of $H$ we mean any element $H^{\prime}\in B$
which reduces to $H$ when we put $t=0$, which means that one can write it in
the form

$H^{\prime}=H+tQ,\;\;Q\in B.$

Note that we can consider $A$ as a subring of $B$, consisting of functions
that do not depend on $t$, but also as factor ring of $B$: $A=B/tB$.

One of the main ideas of perturbation theory is that by applying
systematically appropriate automorphisms of the algebra $B$, one may try to
bring the perturbed Hamiltonian $H+tQ$ to a simpler form or reduce it to a
specific normal form.

If we consider two $C^{\infty}$ symplectic manifolds $(M,\omega)$ and
$(N,\omega^{\prime})$, then any symplectic map $\varphi:M\longrightarrow N$
produces a map of Poisson-algebras

$\phi:=\varphi^{*}:C^{\infty}(N)\longrightarrow C^{\infty}(M),f\mapsto
f\circ\varphi,$

which is a Poisson morphism. In general a Poisson morphism is defined as a map
between two Poisson $R$-algebras $A$ and $B$

$\phi:A\longrightarrow B$

such that

$\begin{array}[]{c}\phi(\lambda
f)=\lambda\phi(f),\;\;\;\phi(f+g)=\phi(f)+\phi(g),\\\
\phi(fg)=\phi(f)\phi(g),\;\;\phi(\\{f,g\\})=\\{\phi(f),\phi(g)\\}\end{array}$

for all $\lambda\in R,\;f,g\in A$.

A Poisson automorphism is an invertible Poisson morphism

$\phi:A\longrightarrow A,$

and can be seen as a generalisation of the notion of symplectomorphism.

A Casimir element is an element $c$ that Poisson-commutes with all elements of
$A$:

$\\{c,a\\}=0\;\;\;\textup{for all}\;\;\;a\in A.$

We call a Poisson automorphism $\phi$ central, if $\phi(c)=c$ for all Casimir
elements.

The rings $A$ and $B$ introduced above however is too small for many of the
constructions we wish to perform. We will encounter several Poisson algebras
similar to $A$ and $B$, that consist of certain power series rather than
polynomials. These power series can be formal or convergent in some of the
variables. We will introduce them in the sequel, but will usually call them
$A$ and $B$.

######  Definition 2.1.

If $A$ is a Poisson algebra over a field of characteristic $0$ and $S\in
B:=A[[t]]$, then the series

$\varphi:=e^{t\\{-,S\\}}={\rm
Id\,}+t\\{-,S\\}+\frac{t^{2}}{2!}\\{\\{-,S\\},S\\}+\dots$

is called the formal flow of $S$.

######  Proposition 2.2.

The formal flow

$\varphi:B\longrightarrow B$

is a central Poisson-automorphism.

###### Proof.

As the terms in the series of $\varphi$ consist of Poisson-brackets, it is
clear that $\varphi$ is central. One has

$\begin{array}[]{rcl}\varphi(f)\cdot\varphi(g)&=&(f+t\\{f,S\\}+\ldots)(g+t\\{g,S\\}+\ldots)\\\
&=&f\cdot g+t(\\{\\{f,S\\}g\\}+f\\{g,S\\})+\ldots\\\
&=&\\{f,g\\}+t\\{fg,S\\}+\ldots\\\ &=&\varphi(f\cdot g),\\\ \end{array}$

where the derivation property of the Poisson-bracket is used. Furthermore

$\begin{array}[]{rcl}\\{\varphi(f),\varphi(g)\\}&=&\\{f+t\\{f,S\\}+\ldots,g+t\\{g,S\\}+\ldots\\}\\\
&=&\\{f,g\\}+t(\\{\\{f,S\\},g\\}+\\{f,\\{g,S\\}\\})+\ldots\\\
&=&\\{f,g\\}+t\\{\\{f,g\\},S\\}+\ldots\\\ &=&\varphi(\\{f,g\\}),\\\
\end{array}$

where one uses the Jacobi-identity.

∎

###  2.4. The formal model

In this section and in the next one, we denote by $K$ any field of
characteristic zero and consider the rings

$A:=K[q,q^{-1}][[p]],\;\;\;B:=A[[t]]=K[q,q^{-1}][[p,t]]$

of formal power series in $p=(p_{1},\dots,p_{n})$ and a central element $t$
whose coefficients are Laurent polynomials in $q=(q_{1},\dots,q_{n})$ with the
Poisson bracket

$\\{p_{j},q_{k}\\}=q_{k}\delta_{jk}.$

Note that $K[[p]]\subset A$ and thus we can and will consider $A$ as a
$K[[p]]$-algebra. There is also a natural averaging map

$av:A\longrightarrow K[[p]]$

of ’taking the average over the torus’. It is the $K[[p]]$-linear map that
maps each monomial $q^{I}$, $I\neq 0$ to $0$, whereas $av$ is the identity on
$K[[p]]\subset A$.

The powers of the maximal ideal of $K[[t]]$ filter the ring $B$. We write

$f=g+(t)^{n}$

if $f-g$ is of order $n$, i.e., it is a power series with monomials of degree
not smaller than $n$ in the $t$ variable. More generally, given an ideal we
write $f=g+I$ if $f-g$ belongs to some ideal $I$. We also write $(p)$ for the
ideal $(p_{1},p_{2},\ldots,p_{n})$ generated by the $p_{i}$’s.

We will consider a special class of Hamiltonians. We have seen that a
Hamiltonian written in action-angle variables $(q,p)$ depends only on the
action variables $p$. Therefore we start with a Hamiltonian $H\in K[[p]]$
without constant term:

$H=\sum_{I\in\mathbb{N}^{n}}\omega_{I}p^{I}=\sum_{i=1}^{n}\omega_{i}p_{i}+(p)^{2}.$

We will call such a Hamiltonian integrable, as clearly

$\\{H,p_{i}\\}=0,\;\;i=1,2,\ldots,n.$

The vector $\omega:=(\omega_{1},\omega_{2},\ldots,\omega_{n})$ is called the
frequency vector of $H$ (at $0$):

$\omega=(\frac{\partial H}{\partial p_{1}}(0),\frac{\partial H}{\partial
p_{2}}(0),\ldots,\frac{\partial H}{\partial p_{n}}(0)).$

If $I\in\mathbb{Z}^{n}$ is an integral vector, we write

$(\omega,I):=\sum_{i=1}^{n}\omega_{i}I_{i}$

for the euclidean scalar product of $\omega$ and $I$.

The Hamiltonian $H$ is called non-resonant, if the
$\omega_{1},\omega_{2},\ldots,\omega_{n}$ are $\mathbb{Z}$-independent:

$(\omega,I)\neq 0\;\;\;\textup{for
all}\;\;\;I\in\mathbb{Z}^{n}\setminus\\{0\\}$

Otherwise it is called resonant and in that case a non-zero vector
$I\in\mathbb{Z}^{n}$ such that

$(\omega,I)=0$

is called a resonance of $H$.

For instance,

$H=p_{1}+ap_{2}$

is resonant if and only if $a\in\mathbb{Q}$.

######  Proposition 2.3.

For a non-resonant $H\in K[[p]]$ the map

$\\{H,-\\}:A\longrightarrow A$

has $K[[p]]$ as kernel and the $K[[p]]$-module generated by the monomials
$q^{I}$, $I\neq 0$ as image.

###### Proof.

We can consider $A=K[q,q^{-1}][[p]]$ as a $K[[p]]$-algebra with the monomials
$q^{I}$, $I\in\mathbb{Z}^{n}$ as $K$-basis. As $H\in K[[p]]$, the map
$\\{H,-\\}:A\longrightarrow A$ is $K[[p]]$-linear. Furthermore, due to the
Poisson-commutation rule

$\\{p_{k},q_{j}\\}=q_{j}\delta_{kj},$

the map $\\{H,-\\}$ is diagonal in the monomial basis and we can write

$\\{H,q^{I}\\}=((\omega,I)+(p))q^{I},$

where $(-,-)$ denotes the Euclidean scalar product. As by assumption

$(\omega,I)\neq 0$

and all elements of the form

$c+(p),\ c\neq 0,\ c\in K$

are invertible in the local ring $K[[p]]$, it follows that the eigenvalue of
$\\{H,-\\}$ associated to $q^{I}$ is invertible for $I\neq 0$. As a
consequence, the kernel of $\\{H,-\\}$ is $K[[p]]$ and furthermore the image
contains all monomials $q^{I}$, $I\neq 0$. ∎

So in the non-resonant case, the image of the map $\\{H,-\\}$ consists exactly
of the series with average equal to zero. One can express the above
proposition by saying that for a non-resonant $H\in K[[p]]$ we have an exact
sequence of the form

$0\longrightarrow K[[p]]\longrightarrow
A\stackrel{{\scriptstyle\\{H,-\\}}}{{\longrightarrow}}A\stackrel{{\scriptstyle
av}}{{\longrightarrow}}K[[p]]\longrightarrow 0,$

which says that kernel and cokernel of $\\{H,-\\}$ are both isomorphic to
$K[[p]]$.

###  2.5. Formal stability

The following proposition states that a formal deformation of an integrable
Hamiltonian remains integrable in a formal neighborhood of a non-resonant
torus.

######  Proposition 2.4.

Let $H+tQ\in B$ be a deformation of an integrable Hamiltonian

$H=\sum_{I\in\mathbb{N}^{n}}\omega_{I}p^{I}=\sum_{i=1}^{n}\omega_{i}p_{i}+(p)^{2}.$

If $H$ is non-resonant, then there exists a central Poisson automorphism

$\varphi:B\longrightarrow B$

such that

$\varphi(H+tQ)\in K[[p,t]].$

###### Proof.

Let us write $(t^{n})\subset B$ for the terms that are divisible by $t^{n}$,
so we can write

$H+tQ=H(p)+(t)$

We will construct inductively a sequence of central Poisson automorphisms
$\varphi_{0},\varphi_{1},\varphi_{2},\ldots$ of $B$ such that

$\varphi_{n}(H+tQ)=P_{n}+(t^{n+1})$

and where

$P_{n}\in K[[p,t]],\;P_{n}=P_{n-1}+(t^{n})$

We take $\varphi_{0}=Id$, $P_{0}=H$. Assume we have constructed the sequence
up to $\varphi_{n}$. Then we look to the next order in $t$ and write:

$\varphi_{n}(H)=P_{n}(p,t)+t^{n+1}Q_{n+1}(q,q^{-1},p)+(t^{n+2}).$

By transferring the part of $Q_{n+1}$ that does not contain $q$ to the $P_{n}$
part (“averaging”), we arrive at the form

$\varphi_{n}(H+tQ)=\widetilde{P}_{n}(p,t)+t^{n+1}\widetilde{Q}_{n+1}(q,q^{-1},p)+(t^{n+2}),$

where we can assume that $\widetilde{Q}_{n+1}$ does not contain pure
$p$-monomials. As $H$ is assumed to be non-resonant, we can find $S_{n+1}\in
A=K[q,q^{-1}][[p]]$ such that

$\\{H,S_{n+1}\\}+\widetilde{Q}_{n+1}(p,q,q^{-1})\in K[[p]].$

As $K$ is assumed to be of characteristic zero, the map

$\psi_{n+1}=e^{t^{n+1}\\{-,S_{n+1}\\}}$

is a well-defined Poisson automorphism of $B$. As

$\psi_{n+1}(f)=f+t^{n+1}\\{f,S_{n+1}\\}+\frac{t^{2n+2}}{2!}\\{\\{f,S_{n+1}\\},S_{n+1}\\}+\dots.$

the automorphism

$\varphi_{n+1}:=\psi_{n+1}\varphi_{n}$

has the property that

$\varphi_{n+1}(H+tQ)=P_{n+1}+(t^{n+2}).$

This proves the statement by induction on $n$. ∎

Given a mechanical system defined by an Hamiltonian of the form described by
the proposition, the change of variables constructed above reduces the motion
to a quasi-periodic one. In fact the proposition gives the classical method
for performing computations in celestial mechanics. However, one should notice
that this reduction to the normal form is done by means of formal power series
and convergence issues are not considered. In practice, one performs this
reduction only up to a certain order, and the solution one obtains by
truncation of higher order terms can only be expected to be asymptotic to the
real solution.

######  Example 2.5.

Consider the Hamiltonian function $H=p\in\mathbb{C}[q,q^{-1},p]$ and its
deformation

$H^{\prime}=p+tp^{2}+tpq+tpq^{-1}.$

Keeping in mind that $\\{p,q\\}=q,\\{f(p),q\\}=f^{\prime}(p)q$ the equations
of motions are:

$\displaystyle\dot{q}$ $\displaystyle=\\{H^{\prime},q\\}=q+t(q^{2}+2pq+1),$
$\displaystyle\dot{p}$ $\displaystyle=\\{H^{\prime},p\\}=t(pq^{-1}-pq)$

We put

$S=pq-pq^{-1}$

so that

$p^{2}+\\{S,H\\}=p^{2}+pq+pq^{-1}$

is the perturbation. The symplectomorphism $\varphi=e^{t\\{-,S\\}}$ eliminates
the first order term in $H^{\prime}$, but creates terms of higher order in
$t$:

$\varphi(H^{\prime})=p+tp^{2}-2t^{2}(p^{2}q^{-1}+p^{2}q+p)+\ldots$

Hence, neglecting the terms of order $2$ in $t$, we get the quasi periodic
motion

$\displaystyle\dot{q}$ $\displaystyle=(1+2tp)q,$ $\displaystyle\dot{p}$
$\displaystyle=0$

which can easily be integrated

$\displaystyle q(\tau)$ $\displaystyle=ae^{\tau+2b\tau t},$ $\displaystyle
p(\tau)$ $\displaystyle=b$

where $\tau$ denotes the time.

Another difficulty which appears in celestial mechanics is the occurrence of
resonances. In the resonant case, there exist resonance vectors $I\neq 0$ with
$(\omega,I)=0$ and in that case the corresponding resonant monomial $q^{I}$ is
not in the image of the map

$\\{H,-\\}:A\longrightarrow A.$

The above procedure then still gives a sequence of Poisson automorphisms, but
it is stopped when we arrive at the first resonant monomial: if $H$ is of the
form

$H=P_{n}(p)+t^{n+1}Q_{n+1}(p,q,q^{-1})+(t^{n+2})$

and $Q_{n+1}(p,q,q^{-1})$ contains a resonant monomial, then the equation

$\\{H,S_{n+1}\\}+Q_{n+1}(p,q,q^{-1})\in K[[p]].$

cannot be solved, hence we see:

######  Proposition 2.6.

Let

$H=\sum_{I\in\mathbb{N}^{n}}\omega_{I}p^{I}=\sum_{i=1}^{n}\omega_{i}p_{i}+(t^{2})$

be such that the frequencies $\omega_{1},\dots,\omega_{n}$ are
$\mathbb{Z}$-dependent. There exists perturbations $H+tQ$ of $H$ which cannot
be reduced to a function of the $p_{i}$’s (and $t$) by a Poisson automorphism.

The proposition is of course a very weak statement, but nevertheless, the
above simple fact already shows a dichotomy between the different tori of an
integrable system. For instance the Hamiltonian

$H=p_{1}-2p_{2}$

is resonant, $I=(2,1)$ is a resonance vector and we have the resonant monomial
$q_{1}^{2}q_{2}$. With the perturbation $p_{1}-2p_{2}+tq_{1}^{2}q_{2}$ we are
in the situation of Proposition 2.6: the resonant monomial $q_{1}^{2}q_{2}$
cannot be suppressed by a formal symplectomorphism. But if we make a scaling

$(p_{1},p_{2})\mapsto((1+\lambda)p_{1},p_{2})$

with $\lambda$ irrational, we are in the situation of Proposition 2.4, and the
monomial can be transformed away. So already at the formal level, the
situation is subtle. This is one of the manifestations of formal KAM theory
that we will explore more in detail in a later section.

The beautiful non-integrability theorem of Poincaré goes far beyond this
elementary remark. It states that over $\mathbb{R}$ and $\mathbb{C}$, the only
first integrals are, in general, the functions of the Hamiltonian itself.

This non-integrability theorem can be understood in terms of global analysis
using transversality arguments: we fix a time $T$ and consider the space
$\Omega(M)$ of $T$-periodic loops in $M$. These $T$-periodic orbits are
exactly the critical points of the action functional

$\Omega(M)\longrightarrow\mathbb{R},\;\;\;\gamma\mapsto\int_{\gamma}pdq.$

Using Smale’s transversality theorem, one can show that for a general $H$, the
critical points of this function are isolated, hence the periodic orbits with
period $T$ are isolated. If there were an independent first integral $F$, the
flow of $F$ would transform each $T$-periodic orbit into a one-parameter
family of $T$-periodic orbits, contradiction the fact that they are isolated.
Therefore the flow of $F$ should be stationary at all periodic orbits and
Poincaré’s theorem reduces to showing that $F$ is then a function $H$.

###  2.6. Analytic model and holomorphic Fourier series

Our analysis of the formal case reveals the inherent algebraic structures
involved. But the problem we really want to deal with, is the analytic and not
the formal case. We will describe a particular Poisson-algebra

$\mathbb{C}\\{q,q^{-1},p\\}$

that is of importance for the Kolmogorov invariant torus theorem. We start
with some preparations.

Consider a $2\pi$-periodic function $f$ that is holomorphic on a neighbourhood
of the real axis. It has an expansion into a (convergent) Fourier series

$f(\theta)=\sum_{n\in\mathbb{Z}}a_{n}e^{in\theta}.$

Due to the compactness of the interval $[0,2\pi]$, the function $f$ is in fact
holomorphic on a strip

$B_{t}:=\\{\theta=x+iy\in\mathbb{C}:x\in\mathbb{R},\ |y|<t\\}.$

There exist a well-known relation between the rate of vanishing of the
coefficients $a_{n}$ and the width $t$ of the strip.

######  Proposition 2.7.

The function $f$ extends holomorphically to $B_{t}$ if and only if

$|a_{n}|=O(e^{-|n|s})$

for any $s<t$.

###### Proof.

Let $s<t$ and pick any $s^{\prime}\in]s,t[$. Consider the Hilbert space
$L^{2}([0,2\pi]\times[-s^{\prime},s^{\prime}],\mathbb{C})$ with Hermitian
scalar product

$(f,g):=\frac{1}{2\pi}\int_{[0,2\pi]\times[-s^{\prime},s^{\prime}]}f\cdot\overline{g}\
dxdy\;.$

The functions $e_{n}:=e^{in\theta}=e^{inx-ny}$ form an orthogonal set with

$(e_{n},e_{n})=\frac{1}{2\pi}\int_{-s^{\prime}}^{s^{\prime}}\int_{0}^{2\pi}e^{inx-
ny}e^{-inx-ny}dxdy=\frac{e^{2ns^{\prime}}-e^{-2ns^{\prime}}}{2n}\;.$

From the expansion of $f$ as Fourier-series

$f=\sum_{n\in\mathbb{Z}}a_{n}e^{in\theta}$

we get

$(f,e_{n})=a_{n}(e_{n},e_{n}).$

Hence we obtain

$|a_{n}|(e_{n},e_{n})=|(f,e_{n})|\leq\|f\|\|e_{n}\|$

or

$|a_{n}|\leq\frac{\|f\|}{\|e_{n}\|}\leq Ce^{-|n|s}$

for an appropriate choice of $C$. Conversely, if the coefficients decrease
exponentially, then the Fourier series is convergent inside some compact
subset of the strip and thus defines a holomorphic function. ∎

Now if the Fourier series $f=\sum_{n\in\mathbb{Z}}a_{n}e^{in\theta}$ is
analytic in a strip $B_{t}$, the associated series

$\sum_{n\in\mathbb{Z}}a_{n}q^{n}$

is analytic in the annulus

$\\{q\in\mathbb{C}\;\;|\;\;e^{-t}<|q|<e^{t}\\}$

These annuli form a fundamental system of neighbourhoods of the circle

$S^{1}:=\\{q\in\mathbb{C}\;\;|\;\;|q|=1\\}.$

######  Definition 2.8.

We put

$\mathbb{C}\\{q,q^{-1}\\}:=\\{\sum_{n=-\infty}^{\infty}a_{n}q^{n}\;\;|\;\exists
t>0\;\;\forall n:|a_{n}|\leq O(e^{-|n|t})\;\\}$

and call it the ring of holomorphic Fourier series.

The above discussion shows that $\mathbb{C}\\{q,q^{-1}\\}$ can be identified
with algebra of germs of holomorphic functions along the unit circle.

This notion extends to $n$ variables $q_{1},\dots,q_{n}$ in an obvious way. We
will be concerned with the Poisson algebras

$A:=\mathbb{C}\\{q,q^{-1},p\\},\;\;\;B:=\mathbb{C}\\{q,q^{-1},p,t\\}$

with Poisson-bracket

$\\{p_{k},q_{j}\\}=q_{j}\delta_{kj}$

and $t$ a central element. In terms of the cotangent space $T^{*}X$ to the
algebraic torus $X:=(\mathbb{C}^{*})^{n}$, we can say that the algebra $A$
consists of the germs of holomorphic functions on $T^{*}X$ along the central
$n$-torus

$(S^{1})^{n}=\\{(q_{1},q_{2},\ldots,q_{n})\in(\mathbb{C}^{*})^{n}\;\;|\;\;|q_{j}|=1\\}\subset
X.$

###  2.7. First order analytic obstruction

In a previous section we saw that for a non-resonant $H\in K[[p]]$ the
operator

$L=\\{H,-\\}:K[q,q^{-1}][[p]]\longrightarrow K[q,q^{-1}][[p]]$

is diagonal in the monomial basis with kernel and cokernel isomorphic to
$K[[p]]$ and series with vanishing average

$\\{\sum_{I\neq 0,J}a_{I}q^{I}p^{J}\\}$

as image. We showed that this fact implies that any perturbation of a non-
resonant integrable Hamiltonian can formally be transformed to the integrable
normal form.

As Poincaré already observed, an analogous statement does not hold at an
analytic level. In fact, if we consider again an analytic integrable
Hamiltonian $H\in\mathbb{C}\\{p\\}$, then the analogous operator

$L^{an}:=\\{H,-\\}:\mathbb{C}\\{q,q^{-1},p\\}\longrightarrow\mathbb{C}\\{q,q^{-1},p\\}$

is of great complexity. Let us give an example and consider the Hamiltonian

$H=p_{1}+\sqrt{2}p_{2}+\frac{1}{2}p_{2}^{2}$

The frequency vector

$\omega:=(\frac{\partial H}{\partial p_{1}},\frac{\partial H}{\partial
p_{2}})=(1,\sqrt{2}+p_{2})$

depends linearly on $p_{2}$ and we find:

$L^{an}(q_{1}^{m}q_{2}^{-n})=(m-n(\sqrt{2}+p_{2}))q_{1}^{m}q_{2}^{-n}$

and so the preimage under $L^{an}$

$\frac{1}{m-n(\sqrt{2}+p_{2})}q_{1}^{m}q_{2}^{-n}$

of the monomial $q_{1}^{m}q_{2}^{-n}$ has a pole at

$p_{2}^{0}:=\frac{m}{n}-\sqrt{2}$

which is near the origin if $m/n$ is close to $\sqrt{2}$. By taking any (non-
polynomial) convergent power series with such monomials:

$\sum_{m,n\geq 0}\alpha_{nm}q_{1}^{m}q_{2}^{-n}$

we get an analytic series which is not in the image of $L^{an}$. So contrary
to what happens in the formal case, the image of $L^{an}$ misses many non-
trivial analytic functions and not only the the series with non-zero average,
that is, series depending only on the $p$ variables. The image is therefore
much smaller and it is non-trivial to see if a series belongs to it.

Let us return to the general case and consider a Hamiltonian

$H=\sum_{i=1}^{n}\omega_{i}p_{i}+(p)^{2}\in\mathbb{C}\\{p\\}.$

and a given perturbation

$H+tQ,\;\;\;S\in B$

We try to find, as in the formal case, a Poisson-automorphism of the form

$\varphi:B\longrightarrow
B,\;\;\;\varphi(f)=e^{t\\{f,S\\}}=f+t\\{f,S\\}+(t^{2})$

that transforms $H+tQ$ to an element of $B$ that is independent of the
$q_{i}$. So we find

$\varphi(H+tQ)=H+t\\{H,S\\}+tQ+(t^{2}),$

that is, we need to find $S\in B$ such that

$\\{H,S\\}=-Q.$

As we indicated above, this is very delicate condition.

However, this equation reduces drastically by restricting both sides to $p=0$:
we get a new equation

$\\{H,S\\}_{\mid p=0}=-Q(q,p=0)$

where $Q(q,p=0)\in\mathbb{C}\\{q,q^{-1}\\}$. This equation reduces to

$L_{0}(S)=\sum_{i=1}^{n}\omega_{i}q_{i}\partial_{q_{i}}S=-Q(q,p=0)=:g$

where

$L_{0}=\\{\sum_{i=1}^{n}\omega_{i}p_{i},-\\}$

is the linear part of the Hamiltonian derivation.

We will consider the ring of holomorphic Fourier series as sitting inside the
space of all formal Fourier series.

$\mathbb{C}\\{q,q^{-1}\\}\subset\mathbb{C}[[q,q^{-1}]]:=\\{\sum_{I\in\mathbb{Z}^{n}}a_{I}q^{I}\;\;|\;\;\;a_{I}\in\mathbb{C}\\}\,.$

In general, two series in $\mathbb{C}[[q,q^{-1}]]$ can not be multiplied in
the usual way, so it is not a ring, but only a vector space. But it will still
be useful to to equip $\mathbb{C}[[q,q^{-1}]]$ with the coefficient-wise
product $\star$:

$\sum_{I}a_{I}q^{I}\star\sum_{I}b_{I}q^{I}=\sum_{I}a_{I}b_{I}q^{I}$

called the Hadamard or convolution product.

We see that the operator $L_{0}$ is equal to taking the Hadamard product with
the function

$l=\sum_{I\in\mathbb{Z}^{n}\setminus\\{0\\}}(\omega,I)q^{I}.$ $L_{0}(S)=l\star
S=g$

This equation is solved for $S$ as

$S=h\star g$

with

$h:=\sum_{I\in\mathbb{Z}^{n}\setminus\\{0\\}}(\omega,I)^{-1}q^{I},$

the Hadamard-inverse of $l$.

There is an obvious restriction: $(\omega,I)$ should not be equal to zero for
any $I\neq 0$, otherwise $h$ is not defined. This is exactly the non-resonance
condition. But even if this condition is satisfied, it might happen $h\star g$
is not an element of the ring $\mathbb{C}\\{q,q^{-1}\\}$ of holomorphic
Fourier series.

We thus see that for homomorphic Fourier series there is an obstruction:
$h\star f$ has to be holomorphic for any $f\in\mathbb{C}\\{q,q^{-1}\\}$.

This first order obstruction of analytic nature can be easily identified:

######  Proposition 2.9.

The Hadamard product

$h\star:\mathbb{C}[[q,q^{-1}]]\longrightarrow\mathbb{C}[[q,q^{-1}]],\ f\mapsto
h\star f$

with the series

$h=\sum_{I\in\mathbb{Z}^{n}}a_{I}q^{I}\in\mathbb{C}[[q,q^{-1}]]$

maps the sub-algebra $\mathbb{C}\\{q,q^{-1}\\}$ to itself if and only for any
$s>0$ there exists a constant $C=C(s)$ such that the Fourier coefficients of
$h$ satisfy the estimate:

$|a_{I}|\leq Ce^{|I|s}.$

###### Proof.

Assume that the coefficient $a_{I}$ of $h$ satisfy the above estimate and take
any element

$f(q)=\sum_{I}b_{I}q^{I}\in\mathbb{C}\\{q,q^{-1}\\}.$

By definition of $\mathbb{C}\\{q,q^{-1}\\}$, there exists $t$ such that

$|b_{I}|=O(e^{-|I|t}).$

Take $s<t$, we get that

$|a_{I}b_{I}|=O(e^{-|I|(t-s)}).$

Thus $h\star f\in\mathbb{C}\\{q,q^{-1}\\}$. Conversely if $h\star$ preserves
the sub-algebra $\mathbb{C}\\{q,q^{-1}\\}$, then consider for any $s>0$ the
holomorphic series

$f(q)=\sum_{I}e^{-|I|s}q^{I}\in\mathbb{C}\\{q,q^{-1}\\}.$

We have $h\star f\in\mathbb{C}\\{q,q^{-1}\\}$, thus

$|a_{I}e^{-|I|s}|=O(1).$

Hence the coefficients of $h$ satisfy an estimate $|a_{I}|\leq Ce^{|I|s}$. ∎

###  2.8. Bibliographical notes

It is difficult to trace the birth of perturbation theory, but it is clear
that the book of Poincaré has been an essential step:

Poincaré, H., Les Méthodes Nouvelles de la Mécanique Céleste I, Gauthier
Villars. Paris (1892).

Poincaré discovered the non-integrability of the three body problem in:

H. Poincaré, Sur le problème des trois corps et les équations de la dynamique,
Acta mathematica 13(1), 3-270, (1890).

Fermi gave a generalisation of the Poincaré divergence theorem in :

E. Fermi, Dimostrazione che in generale un sistema meccanico è quasi ergodico,
Il Nuovo Cimento 25, 267 -269, (1923).

See also :

C.L. Siegel, On the integrals of canonical systems, Annals of Mathematics 42
(3), 806-822, (1941).

G. Benettin, G. Ferrari, L. Galgani L et A. Giorgilli, An extension of the
Poincaré-Fermi theorem on the nonexistence of invariant manifolds in nearly
integrable Hamiltonian systems. Il Nuovo Cimento B, 72(2), 137 - 148, (1982).

##  Chapter 3 The Kolmogorov invariant torus theorem

In this chapter we take a closer look at the the non-resonance condition and
introduce the so called Diophantine condition, and show that the set of
Diophantine frequency vectors has full measure. Then we formulate Kolmogorov’s
theorem on the existence of invariant tori and reinterpret the theorem in
terms of infinite dimensional group actions.

###  3.1. Kolmogorov’s Diophantine condition

We have seen that in the context of perturbation theory we are given a
frequency vector

$\omega=(\omega_{1},\omega_{2},\ldots,\omega_{n})$

and we have to consider the Euclidean scalar product

$(\omega,I)$

with lattice vectors $I\in\mathbb{Z}^{n}$. In the non-resonant case, the
hyperplane $\omega^{\perp}$ orthogonal to $\omega$ intersects the lattice
$\mathbb{Z}^{n}$ only at the origin. But of course, there will always be be
lattice points that are very close to the hyperplane. As a result, the series

$h=\sum_{I}(\omega,I)^{-1}q^{I}$

may have coefficients that grow very fast if $|I|$ becomes big: the problem of
small denominators. We saw that the Hadamard product with a formal Fourier
series

$h=\sum a_{I}q^{I}$

will preserve the space of analytic Fourier series if the coefficients $a_{I}$
grow not faster than exponentially with $|I|$.

Rather than studying types of sub-exponential growth, we consider the simpler
case of polynomial growth:

######  Definition 3.1.

A vector $\omega=(\omega_{1},\dots,\omega_{n})$ satisfies Kolmogorov’s
Diophantine condition $K(C,\nu)$ if for all $0\neq I\in\mathbb{Z}^{n}$

$|(\omega,I)|\geq\frac{C}{\|I\|^{n-1+\nu}}.$

Here $C>0$ and $\nu\in\mathbb{R}$. (The shift by $n-1$ is conventional). We
say that $\omega$ satisfies Kolmogorov’s condition if it satisfies $K(C,\nu)$
for some $C$ and $\nu$.

The condition $K(C,\nu)$ means that the growth is bounded by a polynomial
function:

$|(\omega,I)|^{-1}\leq\frac{\|I\|^{n-1+\nu}}{C}.$

In particular, Proposition 2.9 implies that if $\omega$ satisfies Kolmogorov’s
Diophantine condition, then the endomorphism

$h\star:\mathbb{C}[[q,q^{-1}]]\longrightarrow\mathbb{C}[[q,q^{-1}]],\ f\mapsto
h\star f,\ h:=\sum_{I\neq 0}(\omega,I)^{-1}q^{I}$

preserves the sub-space $\mathbb{C}\\{q,q^{-1}\\}$ of holomorphic Fourier
series: the first analytic obstruction vanishes.

###  3.2. Diophantine approximation

The question how small $|(\omega,I)|$ can be belongs to the field of
Diophantine approximation . Let us first look at the case $n=2$ and
$\omega=(1,\alpha)$, $I=(p,-q)$. Then we have

$|(\omega,I)|=|q\alpha-p|,$

so this becomes small if the rational number $\frac{p}{q}$ is a good
approximation to $\alpha$. If we subdivide the unit interval in $N+1$ sub-
intervals of length $1/(N+1)$, then the fractional part of the $N$ numbers

$\alpha,2\alpha,3\alpha,\ldots,N\alpha$

all fall in different intervals if $1/(N+1)<\alpha$. As a consequence, at
least one falls in the first or last interval, which means that one of these
numbers differs by less than $\leq 1/(N+1)$ from an integer.111This is
Dirichlet’s pigeon hole principle. So for all $\alpha$ there exist infinitely
many integers $p,q$ so that

$|q\alpha-p|<\frac{1}{q},\;\;|\alpha-\frac{p}{q}|<\frac{1}{q^{2}}.$

In fact, such good rational approximations can be obtained from the continued
fraction expansion of $\alpha$. In higher dimension, one can show similarly
that for any $\omega\in\mathbb{R}^{n}$, there are always approximations such

$\forall C>0,\exists I\in\mathbb{Z}^{n},\
|(\omega,I)|\leq\frac{C}{\|I\|^{n-1}}.$

In the theory of Diophantine approximation, one usually says that the vector
admits very good rational approximations if one can find an exponent bigger
than $n-1$. Kolmogorov’s Diophantine condition goes in the opposite direction.

It is easy to construct vectors which do not satisfy Kolmogorov’s Diophantine
condition. Take:

$\alpha=\sum_{j\geq 0}10^{-j!}.$

The rational numbers:

$r_{k}=\sum_{j=0}^{k}10^{-j!}$

satisfy:

$|\alpha-r_{k}|\leq 2\cdot 10^{-(k+1)!}.$

The vector

$\omega=(1,\alpha)$

admits very good rational approximations. To see it put:

$\beta_{k}=(\sum_{j=0}^{k}10^{k!-j!},10^{k!})\in\mathbb{Z}^{2}.$

Then

$(\omega,\beta_{k})=10^{k!}\sum_{j\geq k+1}10^{-j!}=O(10^{k!-(k+1)!})$

and

$\|\beta_{k}\|=O(10^{k!})$

therefore, for any $\nu>0$, the sequence

$(\omega,\beta_{k})\|\beta_{k}\|^{1+\nu}=O(10^{(\nu-k)k!})$

converges to zero.

Thus the vector $\omega$ does not satisfy Kolmogorov’s Diophantine condition.
On the other hand, a similar computation shows that for any
$I\in\mathbb{Z}^{n}$ with

$10^{k!}\leq|I|<10^{(k+1)!}$

we have

$\left|(\omega,I)\right|\geq 10^{-kk!}.$

Therefore

$\left|(\omega,I)^{-1}\right|\leq 10^{-kk!}\leq e^{10^{k!}}\leq e^{|I|}.$

In particular, Proposition 2.9 implies that although $\omega$ does not
satisfies Kolmogorov’s Diophantine condition, the endomorphism

$h\star:\mathbb{C}[[q,q^{-1}]]\longrightarrow\mathbb{C}[[q,q^{-1}]],\ f\mapsto
h\star f,\ h:=\sum_{I\neq 0}(\omega,I)^{-1}q^{I}$

still preserves the space of holomorphic Fourier series.

###  3.3. Vectors satisfying the Kolmogorov Diophantine condition

Kolmogorov’s Diophantine condition is not a necessary condition for having an
invariant torus theorem, but is sufficiently weak to allow the vectors
satisfying it to form a set of positive measure. The condition also has also
an abstract meaning in terms of operators as we shall see later. For the
moment, we continue to investigate how many vectors satisfy the condition.

According to the well-known theorem of Liouville, for any algebraic non-
rational number

$\alpha\in\overline{\mathbb{Q}}\setminus\mathbb{Q}$

of degree $d$ then for some $C>0$ one has

$|\alpha-\frac{p}{q}|\geq\frac{C}{q^{d}}\,,$

so the vector $(1,\alpha)$ satisfies Kolmogorov’s Diophantine condition.
Therefore the set of vectors which satisfy the condition obviously form a
dense subset. But in fact one can prove that this set even has full measure.
Let us denote by

$\Omega(C,\nu):=\\{\omega\in\mathbb{R}^{n}:\forall 0\neq I\in\mathbb{Z}^{n},\
|(\omega,I)|\geq\frac{C}{\|I\|^{n-1+\nu}}\\}$

the set of vectors that satisfy Kolmogorov’s condition $K(C,\nu)$.

######  Proposition 3.2.

Assume that $\nu>0$. Then for all $R,C>0$ there is a constant $k:=k(\nu,R)$
such that

$Vol(B_{R}\setminus\Omega(C,\nu))\leq k\cdot C$

Here $Vol$ is the Lebesgue measure and $B_{R}$ the ball of radius $R$.

###### Proof.

We consider a slightly more general situation.
Let

$f:\mathbb{Z}^{n}\longrightarrow\mathbb{R}$

be a function that only depends on the Euclidean length $\|I\|$ of
$I\in\mathbb{Z}^{n}$. Consider the set

$D(C):=\\{\omega\in B_{R}\;|\;\;\;\forall 0\neq
I\in\mathbb{Z}^{n}:\;\;\;|(\omega,I)|\geq Cf(I)\\}.$

The complement of this set in $B_{R}$ can be written as

$D(C)^{c}=\bigcup_{0\neq I\in\mathbb{Z}^{n}}B(C,I)$

where $B(C,I)$ is the set of $\omega\in B_{R}$ that are ’bad’ for $C$ and $I$:

$B(C,I):=\\{\omega\in B_{R}\;|\;|(\omega,I)|<Cf(I)\\}$

These $\omega$’s lie in a band of width

$2Cf(I)/\|I\|$

around the intersection of hyperplane $I^{\perp}$ with the ball $B_{R}$. So
the volume of $B(C,I)$ can be bounded by

$Vol(B(C,I))\leq 2V_{n-1}(R).C.f(I)/\|I\|,$

where $V_{n-1}(R)$ is the volume of the $n-1$-dimensional radius $R$-ball. Now
if the lattice sum

$\sum_{0\neq I\in\mathbb{Z}^{n}}f(I)/\|I\|$

exists and is finite, then the volume of $\cup_{I\in\mathbb{Z}^{n}}B(C,I)$ is
bounded by

$Vol\left(\bigcup_{0\neq I\in\mathbb{Z}^{n}}B(C,I)\right)\leq
k.C,\;\;\;k=2V_{n-1}(R)\sum_{0\neq I\in\mathbb{Z}^{n}}f(I)/\|I\|.$

Now taking the intersection over $C>0$ gives

$Vol(D^{c})\leq\lim_{C\longrightarrow 0}KC=0.$

Applying this to

$f(I):=1/\|I\|^{n-1+\nu}$

gives the result, as for any $\nu>0$ one has

$\sum_{I}1/\|I\|^{n+\nu}<\infty.$

∎

######  Corollary 3.3.

For $\nu>0$ the complement of the set

$\Omega(\nu)=\bigcup_{C>0}\Omega(C,\nu)$

has measure zero.

###### Proof.

That complement $\Omega(\nu)$ in the ball $B_{R}$ of radius $R$ is the set
$D(R,\nu):=\bigcap_{C>0}B_{R}\setminus\Omega(C,\nu)$. According to the
proposition, the volume of $B_{R}\setminus\Omega(C,\nu)$ goes to zero linearly
with $C$, hence the measure of $D(R,\nu)$ is zero. By choosing a sequence of
radii going to infinity, we see that the complement of $\Omega(\nu)$ is the
union of countably many sets of measure zero, hence has itself measure zero. ∎

Similar considerations hold, of course, over the field of complex numbers.
Note that for $\nu\leq 0$, the set $\Omega({\nu})$ is empty.

###  3.4. Kolmogorov’s invariant torus theorem: Statement

We now come to the formulation of the first important result in KAM theory,
namely the theorem of Kolmogorov concerning the stability of invariant tori
under certain conditions. The proof of the theorem will be given in chapter
11, after we have all technical machinery in place.

######  Theorem 3.4.

Consider the Poisson algebras $A=\mathbb{C}\\{q,q^{-1},p\\}$ and
$B=\mathbb{C}\\{q,q^{-1},p,t\\}$. Let $I=(p_{1},p_{2},\ldots,p_{n})\subset A$
the ideal generated by the $p_{i}$’s. Consider an element $H\in A$ of the form

$H=\sum_{i=1}^{n}\omega_{i}p_{i}+\sum_{i=1}^{n}\alpha_{ij}p_{i}p_{j}\
+(p)^{3}$

If

(D) the vector $(\omega_{i})$ satisfies the Kolmogorov Diophantine condition.

(N) the matrix $(\alpha_{ij})$ is invertible.

then the pair $(H,I)$ is homotopically stable.

Here by homotopic stability we mean the follwing: for any deformation

$H+tQ\in\mathbb{C}\\{q,q^{-1},p,t\\}$

of $H$, there exists a central Poisson automorphism $\varphi$ of
$\mathbb{C}\\{q,q^{-1},p,t\\}$, series $c(t)\in t\mathbb{C}\\{t\\}$ and
$a_{ij}\in tB$ such that:

$\varphi(H+tQ)=H+c(t)+\sum a_{ij}p_{i}p_{j}.$

Condition (N) is called Kolmogorov’s non-degeneracy condition.

As each hamiltonian of the form at the right hand side has
$p_{1}=p_{2}=\ldots=p_{n}=0$ as invariant torus, we see that under the
assumptions of the theorem, any hamiltonian $H+tQ$ admits a family of
invariant Lagrangian manifolds parametrised by $t$, for $t$ small enough. As
we shall see, the theorem also holds in the real analytic context, because all
our constructions commute with the conjugation $q_{i}\mapsto
1/\overline{q_{i}}$, the real part of the invariant manifolds are tori.

###  3.5. The one dimensional case

Although the case $n=1$ of the Kolmogorov theorem is rather trivial, it
already shows the origin of the non-degeneracy condition and its relation to
symplectic but non-hamiltonian vector fields.

Let us take a closer look at the Hamiltonian

$H(q,p)=\omega p+\alpha p^{2}.$

The conditions of the theorem become:

(D) $\omega\neq 0$,

(N) $\alpha\neq 0$.

The theorem produces for any perturbation $H+tQ$ a Poisson morphism $\varphi$
and an element $c(t)\in\mathbb{C}\\{t\\}$ such that

$\varphi(H+tQ)=c(t)+\omega p+(p)^{2}.$

As the Hamiltonian flow preserves the level sets of $H$, we know that, in a
neighbourhood of the zero section, the motion will take place along the curves

$H=\text{constant}.$

These curves are diffeomorphic to circles, that is one-dimensional tori.
Kolmogorov’s theorem tells us that for each $t$, we may select a circle on
which the period of the motion is precisely $\omega$, as is the case for
$H_{0}$. If we omit the non-degeneracy condition, then this is obviously
wrong: take for instance

$H=(\omega+t)p.$

For fixed value of $t$, the motion along the circles $p=\text{constant}$ has
frequency $\omega+t$, so it cannot be equal to $\omega$ unless $t=0$.

Let us consider the specific deformation

$H^{\prime}=\omega p+\alpha p^{2}+tp$

of $H$ and let us compute the function $c$ and the Poisson automorphism
$\varphi$ in this case.

First if we consider a Poisson automorphism induced by a Hamiltonian vector
field

$\varphi=e^{t\\{-,S\\}}$

then the effect is

$\varphi(H^{\prime})=\omega p+\alpha p^{2}+tp+t\\{\omega p+\alpha
p^{2},S\\}+(t^{2}).$

As $p$ is not in the image of

$S\mapsto\\{\omega p+\alpha p^{2},S\\}$

we must use a non-hamiltonian vector field to get rid of the deformation:

$v=d(t)\partial_{p}.$

We have

$e^{tv}H^{\prime}=\omega p+\alpha p^{2}+tp+d(t)(\omega+2\alpha p)+(t^{2}).$

Therefore we take

$d(t)=-\frac{t}{2\alpha}.$

The associated Poisson automorphism $\varphi$ maps $H$ to

$\varphi(H^{\prime})=\omega p+\alpha
p^{2}-\frac{\omega}{\alpha}t+\frac{1}{4\alpha}t^{2}.$

In general, we will have an infinite series, but here the process stops at the
first step.

The circle

${\mathcal{C}}_{t}:=\\{p+\frac{1}{2\alpha}t=0\\}$

is mapped to

${\mathcal{C}}_{0}:=\\{p=0\\}$

The frequency of $H^{\prime}$ along ${\mathcal{C}}_{t}$ and $H$ along
${\mathcal{C}}_{0}$ are equal:

$\partial_{p}H^{\prime}_{|{\mathcal{C}}_{t}}=\omega+p+\frac{1}{2\alpha}t=\omega=\partial_{p}(H)_{|{\mathcal{C}}_{0}}.$

This trivial example shows how the non-degeneracy condition works: it enables
us to choose among the invariant circles the one for which the frequency is
precisely $\omega$. If $\alpha$ is equal to $0$, such a choice is no longer
possible.

###  3.6. Kolmogorov’s theorem and group actions

We will now give a slight reformulation Kolmogorov’s invariant torus theorem
as a statement about a certain group action in the infinite dimensional vector
space

$E:=\mathbb{C}\\{q,q^{-1},p,t\\}$

of analytic Fourier series depending on a parameter $t$. Clearly, the group of
central Poisson automorphisms acts on $E$, but we we will consider the
subgroup $G$ of elements $\varphi$ which are tangent to the identity, i.e.,
whose restriction to $t=0$ is the identity on $\mathbb{C}\\{q,q^{-1},p\\}$.
When we fix an element $H\in E$, the group $G$ acts on the affine space

$H+M\subset E,$

of perturbations of $H$, where $M:=tE\subset E$ consist of all the elements
$tQ$ we can add to $H$.

We will take $H$ to be of the special form

$H=\sum_{i=1}^{n}\omega_{i}p_{i}+\sum_{i=1}^{n}\alpha_{ij}p_{i}p_{j},$

where the frequency $\omega=(\omega_{i})$ and $(\alpha_{ij})$ are fixed. If we
add to $H$ a special perturbation from the linear space

$F=t(\mathbb{C}\\{t\\}\oplus I^{2})\subset M,$

we obtain a Hamiltonian of the form

$H+tc(t)+\sum_{ij}a_{ij}p_{i}p_{j}$

The elements of this affine space $H+F\subset H+M$ have clearly the special
property that their flow preserves the subset $p_{1}=p_{2}=\ldots=p_{n}=0$
that defines the torus.

Kolmogorov’s theorem can be formulated as follows:

######  Theorem 3.5.

If $(\omega_{i})$ is Diophantine and $(\alpha_{ij})$ is invertible, then any
element of $H+M$ lies in the $G$-orbit of an element in $F$. In other words,
the map

$G\times F\longrightarrow H+M,\ (\varphi,\alpha)\mapsto\varphi(H+\alpha)$

is surjective.

It means that any perturbation of $H$ can be transformed into a normal form
belonging to $F$. As all elements of the normal form posses an invariant
torus, any perturbation of $H$ admits an invariant torus, obtained from the
torus $p_{1}=p_{2}=\ldots=0$ in the normal form via the automorphism
$\varphi\in G$.

The Kolmogorov theorem gives an answer to a very general question in a
particular case: given a group action on a space $M$, how can we ensure that a
subset $F\subset M$ is a local transversal to the action?

When such a property holds we say that the corresponding element in $F$ is a
normal form and $H+F$ is called a transversal to the action. Such a choice for
$F$ is usually not unique and correspondingy there are different normal forms.
The approporiate choice of $F$ is determined by utilitarian considerations.

In the next chapters of this book we will develop this point of view and
define a general iteration scheme to bring elements to normal form.
Furthermore, we will formulate a general statement of convergence that implies
the above theorem as a special case.

###  3.7. Bibliographical notes

Kolmogorov’s original paper on the invariant torus theorem:

A.N. Kolmogorov, On the conservation of quasi-periodic motions for a small
perturbation of the Hamiltonian function, Dokl. Akad. Nauk SSSR 98, 527 - 530,
(1954).

Note that the paper was not even translated in the english version of the
journal. Arnold gave a first proof in:

V.I. Arnold, Proof of a theorem of A. N. Kolmogorov on the preservation of
conditionally periodic motions under a small perturbation of the Hamiltonian,
Uspehi Mat. Nauk 18(5), 13 - 40, (1963).

Arnold studied the stability of the three body problem in:

V.I. Arnold,  Small denominators and problems of stability of motion in
classical and celestial mechanics, Uspehi Mat. Nauk 18(6), 91-192, (1963).

Moser proposed the general scheme of group actions in:

J. Moser, A rapidly convergent iteration method and non-linear partial
differential equations II, Ann. Scuola Norm Sup. Pisa - Classe di Scienze Sér.
3,20(3), 499-535, (1966).

For details on classical KAM theory, we refer to the following papers and
references therein:

V.I. Arnold, V.V. Kozlov, A.I. Neishtadt Mathematical aspects of classical and
celestial mechanics, Encyclopedia of Mathematical Sciences 3, 1-291, Springer
Verlag, (1985)

J.-B. Bost, Tores invariants des systèmes dynamiques hamiltoniens, Séminaire
Bourbaki 27 Exposé No. 639, (1984-1985).

J. Féjoz, Introduction to KAM theory, 19 pp. Lectures given at the Jornadas on
interactions between dynamical systems and partial differential equations,
Barcelona (2013) and the Ciclo di Lezioni at the Université di Bicocca, Milano
(2011).

M. Sevryuk, The classical KAM theory at the dawn of the twenty-first century.,
Moscow Mathematical Journal, 3(3), 1113-1144, (2003)

Arnold pointed out that the set of vectors which satisfy Kolmogorov’s
Diophantine condition and which lie on a submanifold might have zero measure.
There are many results which prevent such pathologies, see for instance:

A.S. Pyartli, Diophantine approximations on submanifolds of Euclidean space,
Functional Analysis and Its Applications 3(4), 303-306, (1969).

D.Y. Kleinbock et G.A. Margulis, Flows on homogeneous spaces and Diophantine
approximation on manifolds, Annals of Mathematics 148 339 - 360, (1998).

See also:
M. Garay, Arithmetic density, Proceedings of the Edinburgh Mathematical
Society, Vol. 59, 691-700, 2016.

##  Chapter 4 The generalised KAM problem

The original theorem of Kolmogorov pertains to the very special problem of
perturbations of quasi-period motion on tori in a Hamiltonian system given in
action-angle variables. It is of considerable interest to look for a more
intrinsic and coordinate independent understanding of the theorem. This is
indeed possible and a first step consists of replacing tori to more general
Lagrangian subvarieties or coisotropic subvarieties that are invariant under
the flow of a Hamiltonian $H$. This can be done in a general context of
Poisson-algebras.

###  4.1. Invariant ideals

In this section we start with the standard symplectic space $M=K^{2n}$, with
canonical coordinates $q,p$ and we assume that the field $K$ is algebraically
closed of characteristic $0$, e.g. $K=\mathbb{C}$. The ring of polynomial
functions $A:=K[q,p]$ is a Poisson-algebra with the standard Poisson-bracket

$\\{F,G\\}=\sum_{i=1}^{n}\partial_{p_{i}}F\partial_{q_{i}}G-\partial_{q_{i}}F\partial_{p_{i}}G.$

To a collection of polynomials

$f_{1},\dots,f_{k}\in K[q,p]$

one associates the affine variety:

$V:=\\{(p,q)\in M:f_{1}(q,p)=\dots=f_{k}(q,p)=0\\}\subset M.$

This variety in fact only depends on the ideal $I=(f_{1},\dots,f_{k})\subset
A$ generated by the polynomials $f_{i}$, hence the notation $V=V(I)$. More
generally, given a ring $A$ and an ideal $I$ there is an associated subscheme
$V(I)\subset{\rm Spec\,}(A)$.

Recall that the radical $\sqrt{I}$ of an ideal $I$ consists of those elements
of the ring for which a power belongs to the ideal $I$. Hilbert’s
Nullstellensatz is the statement that $\sqrt{I}$ is equal to the ideal of all
polynomials that vanish on $V(I)$. A radical ideal is an ideal such that
$\sqrt{I}=I$. So radical ideals $I$ can alternatively be characterised by the
the property that any polynomial which vanishes on $V(I)$ belongs to $I$.

If $G$ is a first integral of $H$, then the level sets of $G$ are preserved by
the Hamiltonian flow of $H$. One idea of Kolmogorov was to search for lower
dimension invariant manifolds.

######  Proposition 4.1.

Let $I\subset A$ be a radical ideal. The following assertions are equivalent :

  1. i)

$\\{H,I\\}\subset I$

  2. ii)

$V(I)$ is invariant under the formal flow of $H$.

If this applies, we say that $I$ is $H$-invariant.

###### Proof.

We denote by $\varphi$ the formal flow of the Hamiltonian $H$, which was
defined as the automorphism of $B:=A[[t]]$ defined by the series
$e^{t\\{-,H\\}}$:

$\varphi(f)=f+t\\{f,H\\}+\frac{t^{2}}{2!}\\{\\{f,H\\},H\\}+\ldots$

The ideal $I\subset A$ generates an ideal $IB\subset B$ consisting of series

$h_{0}+h_{1}t+h_{2}t^{2}+\ldots$

where the $h_{i}\in I$. The invariance of $V(I)$ under $\varphi$ means, by
definition:

$\varphi(\sqrt{IB})\subset\sqrt{IB}$

But if $I$ is radical, $IB$ is radical too, $\sqrt{IB}=IB$. To show this, it
is sufficient to show that if $f^{2}\in IB$ then in fact $f\in IB$. When we
write $f$ as a series

$f=f_{0}+f_{1}t+f_{2}t^{2}+\ldots,\;\;f_{i}\in A$

then $f^{2}\in IB$ leads to equations

$f_{0}^{2}\in I,\;\;2f_{0}f_{1}\in I,\;\;2f_{0}f_{2}+f_{1}^{2}\in I,\ldots$

If $I$ is radical, the first equation gives $f_{0}\in I$, the second gives no
information on $f_{1}$, but from the third we deduce $f_{1}^{2}\in I$, so
$f_{1}\in I$. Continuing this way, we get $f_{i}\in I$ for all
$i=0,1,2,\ldots$, that is $f\in IB$, hence $IB$ is radical.

It follows that $V(I)$ is $\varphi$-invariant if and only if

$f\in I\Longrightarrow\varphi(f)\in IB$

By looking at the coefficient of $t$ at the right hand side we see that
$\varphi$-invariance of $V(I)$ implies that $\\{I,H\\}\subset I$. On the other
hand, if $\\{I,H\\}\subset I$, then for $f\in I$, all terms

$f,\;\;\\{f,H\\},\;\;\\{\\{f,H\\},H\\},\;\;\ldots$

of the series for $\varphi(f)$ belong to $I$, so $\varphi(f)\in IB$. Hence the
conditions $i)$ and $ii)$ are equivalent. ∎

The $K$-algebra

$K[q,p]/I$

can be interpreted as the ring of polynomial functions on the variety $V(I)$
defined by the ideal $I$. It is usually called the affine coordinate ring of
$V(I)$.

The invariance condition $\\{H,I\\}\subset I$ implies that the map

$K[q,p]/I\longrightarrow K[q,p]/I,\ f\mapsto\\{H,f\\}$

is well-defined. It is a derivation of the coordinate ring $K[q,p]/I$ and
represents the restriction of the Hamiltonian field $X_{H}$ to $V(I)$. We use
the same name for this induced map and consider $\\{H,-\\}\in Der(K[q,p]/I).$

Recall that an ideal $I\subset K[q,p]$ is called involutive (or co-isotropic)
if

$\\{I,I\\}\subset I.$

If a radical ideal $I$ is involutive and $V(I)$ is non-empty, then $\dim
V(I)\geq n$, and if $V(I)$ is of pure dimension $n$, then $V(I)$ is a
Lagrangian subvariety, possibly with singularities.

######  Proposition 4.2.

For any involutive ideal $I\subset K[q,p]$ there is a well-defined map

$I/I^{2}\longrightarrow{\rm Der\,}(K[q,p]/I),\ H\mapsto\\{H,-\\}.$

###### Proof.

As $\\{I,I\\}\subset I$, the ideal $I$ is $H$-invariant for any $H\in I$, so
for each $H\in I$ we have a well-defined derivation

$\\{H,-\\}:K[q,p]/I\longrightarrow K[q,p]/I$

We obtain a map

$I\longrightarrow Der(K[q,p]/I),\;H\mapsto\\{H,-\\}$

Let $(f_{1},\dots,f_{n})$ be generators of $I$ and $H\in I$. Put

$H^{\prime}=H+\sum_{ij}a_{ij}f_{i}f_{j}$

We have

$\begin{array}[]{rcl}\\{H+\sum_{ij}a_{ij}f_{i}f_{j},g\\}&=&\\{H,g\\}+\sum_{ij}a_{ij}f_{i}\\{f_{j},g\\}+\sum_{ij}a_{ij}f_{j}\\{f_{i},g\\}\\\
&=&\\{H,g\\}\;\;{\rm mod\,}I\end{array}$

So the polynomials $H$ and $H^{\prime}$ define the same derivation of the
coordinate ring $K[q,p]/I$. This shows that the above map factors over
$I/I^{2}$ and we get an induced map

$I/I^{2}\longrightarrow{\rm Der\,}(K[q,p]/I),\ H\mapsto\\{H,-\\}.$

∎

In algebraic geometry, the module $I/I^{2}$ is called the conormal module
because when $V(I)$ is smooth, it is dual to the space of sections of the
normal bundle. Geometrically, the above proposition states that the conormal
bundle to an involutive manifold maps naturally to its tangent space. As we
shall see, this elementary algebraic statement turns out to be quite
fundamental in KAM theory, as it shows that for the study of the flow of $H$
along $V(I)$ the important object is not the Hamiltonian $H$ itself, but
rather its class modulo $I^{2}$.

There is a second point: the Hamiltonian flow depends on the differential of
the Hamiltonian and not on the Hamiltonian itself. Adding a constant does not
change the dynamics, and therefore the important object is the class of the
Hamiltonian modulo the subspace $I^{2}\oplus K$.

Also, it is clear that the above algebraic set-up can be applied in a more
general context: if $I$ is an involutive ideal in an arbitrary Poisson-algebra
$A$, we obtain in precisely the same way a well defined map

$I/I^{2}\longrightarrow Der(A/I),\;\;H\mapsto\\{H,-\\}$

In the general situation the role of the constants is taken over by the
Casimir elements, i.e. the elements Poisson-commuting with everything:

$H^{0}(A):=\\{f\in A\;|\;\\{f,g\\}=0\;\;\textup{\ for all\ }g\in A\\}.$

######  Definition 4.3.

We say that two Hamiltonians $H,H^{\prime}\in A$ are $I^{2}$-equivalent, if

$H^{\prime}=H\;\;{\rm mod\,}H^{0}(A)+I^{2}$

that is, if we can write

$H^{\prime}=H+c+i$

with $c\in H^{0}(A)$ and $i\in I^{2}$.

If $H$ and $H^{\prime}$ are $I^{2}$-equivalent, then the derivations
$\\{H,-\\}$ and $\\{H^{\prime},-\\}$ of $A/I$ coincide, so $H$ and
$H^{\prime}$ define exactly the same dynamics on the variety defined by $I$.

Let us now consider some elementary examples.

######  Example 4.4.

Let $n=1$ and consider the ideal $I$ generated by $p\in K[p,q]$. The
Hamiltonians $H=p$ and $H^{\prime}=p+qp^{2}+1$ are $I^{2}$-equivalent. Both
Hamiltonian fields are equal along the line

$V(I)=\\{(q,p)\in K^{2}:p=0\\}$

although the second one is much more complicated over the whole space:

$\displaystyle X_{H}$ $\displaystyle=\partial_{q}$ $\displaystyle
X_{H^{\prime}}$ $\displaystyle=(1+2pq)\partial_{q}-p^{2}\partial_{p}$

######  Example 4.5.

Consider again the ring $K[q,q^{-1},p]$ with the Poisson structure

$\\{p_{j},q_{k}\\}=q_{k}\delta_{jk}$

The ideal $I$ generated by the $p_{i}$ is involutive and the factor ring

$K[q,q^{-1},p]/I\approx K[q,q^{-1}]$

is the coordinate ring of our algebraic torus $V(I)$. Consider a linear
Hamiltonian

$H_{0}=\sum_{i=1}^{n}\omega_{i}p_{i}.$

Any Hamiltonian $H$ of the form

$H=H_{0}\;\;{\rm mod\,}(K\oplus I^{2})$

defines the same quasi-periodic motion on the torus $V(I)$ as $H_{0}$.

Via the isomorphism

$K[q,q^{-1},p]/I\approx K[q,q^{-1}]$

the derivation

$\\{H_{0},-\\}:K[q,q^{-1},p]/I\longrightarrow K[q,q^{-1},p]/I$

is identified with the map

$\\{H_{0},-\\}:K[q,q^{-1}]\longrightarrow
K[q,q^{-1}],q^{I}\mapsto(\omega,I)q^{I}$

We see that the kernel reduces to $K$ if and only if the frequencies
$\omega_{i}$’s are $\mathbb{Z}$-independent.

From a formal viewpoint, the situation is therefore similar to that of the
dynamics along a non resonant Hamiltonian. However in our previous approach,
the qualitative behaviour of the operator varies at a resonant torus. But when
we restrict the dynamics to a specific torus we can avoid this problem.

###  4.2. Poisson cohomology and Poisson vector fields

We already encountered the space of Casimir elements:

$H^{0}(A):=\\{f\in A\;|\;\\{f,g\\}=0\;\;\textup{\ for all\ }g\in A\\}.$

As the notation already indicates, there exist also higher cohomology groups
$H^{i}(A)$ that generalise the de Rham groups of a symplectic manifold. In
particular, we will see there exists an exact sequence analogous to the exact
sequence

$0\longrightarrow{\rm Ham\,}(M)\longrightarrow{\rm Symp\,}(M)\longrightarrow
H^{1}(M)\longrightarrow 0$

of a symplectic manifold to the case of a general Poisson algebra.

Let us denote by $\Theta_{A}$ the space of derivations of $A$ and consider the
exterior algebra

${\bigwedge}^{\bullet}\Theta_{A}.$

We extend the map

$A\longrightarrow\Theta_{A},\ f\mapsto\\{-,f\\}$

to the exterior algebra so that it becomes a differential graded algebra. The
resulting map gives a complex

$C^{\bullet}(A):0\longrightarrow
A\longrightarrow\Theta_{A}\longrightarrow{\bigwedge}^{2}\Theta_{A}\longrightarrow\cdots$

called the Poisson complex. If we consider the bi-derivation $\\{-,-\\}$ as an
element $\pi\in\Lambda^{2}\Theta_{A}$, then the differential of the complex is
obtained by taking the Lie bracket with $\pi$. We denote by $H^{\bullet}(A)$
the cohomology of the Poisson complex. In the symplectic case, the Poisson
complex is isomorphic to the de Rham complex. From this definition we see that
indeed $H^{0}(A)$ is the space of Casimir elements.

A Poisson-derivation is a derivation $\theta$ that ’preserves’ the Poisson-
bracket:

$\theta\\{f,g\\}=\\{\theta(f),g\\}+\\{f,\theta(g)\\}$

and similarly, the space $H^{1}(A)$ can be interpreted as the space of such
Poisson derivations, modulo those that are Hamiltonian. It fits inside the
exact sequence:

$0\longrightarrow{\rm Ham\,}(A)\longrightarrow\textup{Poiss}(A)\longrightarrow
H^{1}(A)\longrightarrow 0$

which provides a generalisation of the de Rham type sequence for the
symplectic case.

######  Example 4.6.

Consider the two-dimensional $K$-algebra

$A:=K[q,q^{-1}][[p]]$

of formal power series in $p$ whose coefficients are Laurent polynomials in
$q$, with the Poisson bracket

$\\{p,q\\}=q.$

A direct computation shows that:

$\displaystyle H^{0}(A)=K,$ $\displaystyle H^{1}(A)=K\partial_{p}$

######  Example 4.7.

Consider the $2n$-dimensional $K$-algebra

$A:=K[q,q^{-1}][[p]]$

of formal power series in $p=(p_{1},\dots,p_{n})$ whose coefficients are
Laurent polynomials in $q=(q_{1},\dots,q_{n})$. with the Poisson bracket

$\\{p_{j},q_{j}\\}=q_{j}$

and all other variables Poisson-commuting. The Poisson complex is isomorphic
to the $n$-fold wedge product of the one considered above. According to
Künneth formula:

$H^{\bullet}(A)={\bigwedge}^{\bullet}\oplus_{i=1}^{n}K[\partial_{p_{i}}].$

So analogous to de Rham theory, we recover the cohomology of the $n$-torus.

###  4.3. Pairs and homotopic stability

We now describe a general framework in which one can formulate the problem
posed by Kolmogorov. Of course, one can not hope for very general answers, and
the answers one can give will very much depend on the particulars.
Nevertheless, we hope that this general setup provides a useful conceptual
framework.

######  Definition 4.8.

  1. A)

pair $(H,I)$ in a Poisson-algebra $A$ consists of an element $H\in A$ and an
$H$-invariant involutive ideal $I\subset A$:

$\\{H,I\\}\subset I,\;\;\;\\{I,I\\}\subset I\;.$

  2. B)

The normal space to a pair $(H,I)$, denoted $N(H,I)$, is defined by

$N(H,I)=A/\left(\\{H,A\\}+I^{2}+H^{0}(A)\right).$

Note that the subspace that is divided out is just a linear subspace and not
an ideal.

######  Example 4.9.

The ideal $I=(p_{1},p_{2},\dots,p_{n})$ is involutive in the Poisson-algebra
$A:=K[q,q^{-1}][[p]]$, which models the formal completion of the zero section
of the cotangent bundle to the algebraic torus. For any $H\in K[[p]]$ we
obtain a pair $(H,I)$ in $A$. We have seen that for a non-resonant Hamiltonian

$H=\sum_{i=1}^{n}\omega_{i}p_{i}$

the space $K[[p]]$ complements the image of the operator $\\{H,-\\}$.
Therefore the normal space to $(H,I)$ is the $n$-dimensional vector space

$N(H,I)=K[p_{1}]\oplus K[p_{2}]\oplus\dots\oplus K[p_{n}]$

where we denote by $[-]$ the class of $-$ in the factor space.

######  Example 4.10.

Consider now the case of the symplectic two-dimensional algebra $A:=K[[q,p]]$
and consider the Hamiltonian

$H=pq$

and the invariant ideal $I=(H)$. The Hamiltonian derivation

$\\{H,-\\}:A\longrightarrow A$

maps $p^{i}q^{j}$ to $(i-j)p^{i}q^{j}$ therefore $K[[pq]]$ complements the
image of the operator. Thus the normal space to $(H,I)$ is the $1$-dimensional
vector space

$N(H,I)=K[pq]$

We want to describe the behaviour of pairs under deformation. So we consider
Poisson-algebras $B$ with a non-zero divisor $t$ as central element. Hence one
has an exact sequence

$0\longrightarrow B\stackrel{{\scriptstyle
t\cdot}}{{\longrightarrow}}B\longrightarrow A\longrightarrow 0$

where $A$ is the factor ring $B/tB$. From any pair $(H^{\prime},I^{\prime})$
in $B$ one obtains a pair $(H,I)$ in $A$ by reduction modulo $t$, i.e. by
setting $t=0$. We will only consider pairs $(H^{\prime},I^{\prime})$ with the
flatness property:

$tf\in I^{\prime}\implies f\in I^{\prime}$

One then has a corresponding exact sequence

$0\longrightarrow B/I^{\prime}\stackrel{{\scriptstyle
t\cdot}}{{\longrightarrow}}B/I^{\prime}\longrightarrow A/I\longrightarrow 0$

One wants to understand properties of the map

${\rm Pairs\ in\ B}\longrightarrow{\rm Pairs\ in\
A},(H^{\prime},I^{\prime})\mapsto(H,I)$

The persistence property of invariant tori in KAM theory leads to the
following general notion.

######  Definition 4.11.

(Persistence property)

We say that a pair $(H,I)$ in $A$ is persistent, if for any deformation
$H^{\prime}$ of $H$ there exists at least one pair $(H^{\prime},I^{\prime})$
in $B$.

Of course, it is of great interest to find conditions that imply the
persistence of a given pair.

In the classical situation the parameter $t$ in $A$ provides a trivial
modification of the algebra $A$, like $B=A[[t]]$. In such cases we have the
additional property that the canonical projection $B\longrightarrow A$ has a
section $A\hookrightarrow B$ and we can consider $A$ as a subring of $B$. In
this situation one can formulate the following stronger property.

######  Definition 4.12.

(Homotopic stability property)

We say that a pair $(H,I)$ is homotopically stable, if for any deformation
$H^{\prime}$ of $H$, there exists a central Poisson morphism

$\varphi:B\longrightarrow B,\;\;\varphi(t)=t,\varphi_{|A}=Id_{A}$

such that $\varphi(H^{\prime})$ is $I^{2}$-equivalent to $H$, i.e. there exist
$c\in H^{0}(B)$, $i\in I^{2}$ such that

$\varphi(H^{\prime})=H+c+i$

######  Proposition 4.13.

If the pair $(H,I)$ is homotopically stable, then it is persistent

###### Proof.

Take the ideal $IB$. It has the same generators as $I\subset A$, but now we
look at what they generate in $B$. Then take $I^{\prime}:=\varphi^{-1}(IB)$.
Clearly $\\{I^{\prime},I^{\prime}\\}\subset I^{\prime}$ and
$\\{H^{\prime},I^{\prime}\\}\subset I^{\prime}$, so $(H^{\prime},I^{\prime})$
is a pair in $B$ that lifts $(H,I)$. ∎

###  4.4. The formal Kolmogorov theorem

The normal space $N(H,I)$ of a pair can also be understood in terms of first
order deformations, where we only look at the first order in $t$ and ignare
all higher order terms. Algebraically, this means that we work in the ring
$B=A[t]/(t^{2})$. Note that given element $Q\in A$ which maps to zero in
$N(H,I)$, then there exists $h\in A$ and $i\in I^{2}$ such that:

$\\{H,h\\}=Q+i.$

Then

$\varphi={\rm Id\,}-t\\{-,h\\}$

is an Poisson automorphism of the Poisson algebra

$B=A[t]/(t^{2})$

that maps the first order deformation

$H^{\prime}=H+tQ$

to an element $I^{2}$-equivalent to $H$.

This remark can be lifted to obtain the following generalised formal
Kolmogorov theorem.

######  Theorem 4.14.

Let $(H,I)$ be a pair in $A$ and let $B=A[[t]]$. If the map

$H^{1}(A)\longrightarrow N(H,I),\ v\mapsto[v(H)]$

is surjective, then $(H,I)$ is homotopically stable.

###### Proof.

We will construct a Poisson automorphism $\varphi$ of $B=A[[t]]$ that brings a
perturbation $H+tQ$, $Q\in B$ to the normal-form

$\varphi(H+tQ)=H+c+i$

where $c\in H^{0}(B)=H^{0}(A)[[t]]$, $i\in I^{2}B$. Surjectivity of the above
map means that all elements $Q$ of $A$ can be represented in the form

$Q=v(H)+c+i$

where $v\in Poiss(A)$, $c\in H^{0}(A)$ and $i\in I^{2}\subset A$. We work
modulo $(t^{k})$ and use induction on $k$ and construct a sequence of Poisson-
automorphisms $\varphi_{0}=Id,\varphi_{1},\varphi_{2},\ldots$ with

$\varphi_{k+1}=\varphi_{k}+(t^{k}).$

Assume that we found such a sequence of Poisson morphisms such that

$\varphi_{k}(H+tQ)=H+c_{k}+i_{k}+(t^{k}),\;\;c_{k}\in H^{0}(B),i_{k}\in I^{2}$

If we look one order in $t$ further we have

$\varphi_{k}(H+tQ)=H+c_{k}+i_{k}+a_{k}t^{k}+(t^{k+1})$

By assumption, we may find $v_{k+1}\in{{\mathcal{P}}}oiss(A)$ and 8.

$v_{k+1}(H)=$

Define

$a_{k+1}=a_{k}+t^{k}\alpha_{k}\in I^{2}+H^{0}(A)$

we get that

$e^{-t^{k}v_{k}}\varphi_{k}(H_{0})=H_{0}+a_{k+1}\ {\rm mod\,}(t^{k+1}).$

This proves the theorem. ∎

######  Example 4.15.

Let us consider the case $n=1$, $A=K[[q,p]]$ with

$H=pq$

and let $I=(H)$ be the ideal generated by $H$. The coordinate lines are
invariant under the Hamiltonian flow. We know that

$N(H,I)=K[pq]$

We have

$H^{0}(A)=K,\ H^{1}(A)=0.$

The map

$H^{1}(A)\longrightarrow N(H,I),\ v\mapsto[v(H)]$

maps a zero-dimensional space to a one-dimensional space, so it cannot be
surjective. There is indeed a non-trivial deformation of the pair $(H,I)$
namely

$H^{\prime}=(1+t)H,\ I^{\prime}=pq.$

The difference with the torus case is that here there is a single special
Lagrangian variety consisting of two lines. Along this variety there is a well
defined frequency equal to $(1+t)$.

######  Example 4.16.

We make the following variation of the previous example: we let

$A=K[[\lambda,q,p]],\ H=(1+\lambda)pq,I=pq$

where $\lambda$ is a central element and let $I=(H)$ be the ideal generated by
$H$. We have

$N(H,I)=K[[\lambda]][pq]$

but now

$H^{0}(A)=K[[\lambda]],\ H^{1}(A)=K[[\lambda]].[\partial_{\lambda}]$

The map

$H^{1}(A)\longrightarrow N(H,I),\ v\mapsto[v(H)]$

is now an isomorphism. The pair $(H,I)$ is now homotopically stable.

Like in the example, the map involved in the theorem is in general a morphism
of finite type modules over the ring of Casimir operators. Let us now work out
the torus case in details.

######  Proposition 4.17.

Let $I$ be the ideal generated by $p_{1},\dots,p_{n}$ in
$A:=K[q,q^{-1}][[p]]$. Let
$H=\sum_{i=1}^{n}\omega_{i}p_{i}+\sum_{i,j}\alpha_{ij}p_{i}p_{j}+\dots\in
K[[p]]$ be a power series in the $p$ variables such that

  1. (R)

the vector $\omega=(\omega_{1},\dots,\omega_{n})$ is non resonant,

  2. (N)

the $n\times n$ matrix $(\alpha_{ij})$ is invertible

then these conditions are respectively equivalent to:

  1. (R)

$N(H,I)$ is an $n$-dimensional vector space generated by the classes of
$p_{1},\dots,p_{n}$,

  2. (N)

the natural map $H^{1}(A)\longrightarrow N(H,I)$ is surjective.

###### Proof.

The first assertion is due to the fact that the absence of resonances implies
that $\\{H,-\\}$ is diagonal in the monomial basis and that its image consists
of functions with zero mean value. For the second assertion note that

$\displaystyle H^{0}(A)=K[1]$ $\displaystyle
H^{1}(A)=\bigoplus_{i=1}^{n}K[\partial_{p_{i}}]$

and that the map

$K^{n}\longrightarrow N(H,I),\
(c_{1},\dots,c_{n})\mapsto[\sum_{i=1}^{n}c_{i}\partial_{p_{i}}H_{0}]$

is an isomorphism if and only if the matrix $(a_{ij})$ is invertible. This
proves the theorem. ∎

In particular the pair $(H,I)$ of the proposition is homotopically stable and
we see that the formal version of Kolmogorov’s theorem is a special case of
our general theorem.

We will see that in the analytic case, that is, Kolmogorov’s invariant torus
theorem, persistence is also shown be showing homotopic stability.

###  4.5. Bibliographical notes

The Poisson complex was introduced in:
A. Lichnerowicz. Les variétés de Poisson et leurs algèbres de Lie associées.
J. Differential Geom. 12 (1977), no. 2, 253–300.

##  Chapter 5 The Lie iteration

In the previous chapter, we stated Kolmogorov’s invariant torus theorem and
its generalisations in terms of infinite dimensional group actions. In this
sense, it is a particular case of a general theorem on group actions. It is
fortunate that all basic geometrical ideas already appear in the finite
dimensional situation, namely for Lie group actions and in fact even for
linear group actions. So the natural action of a matrix Lie group $G\subset
GL(V)$ on a finite dimensional vector space $V$ is a natural starting point
for our investigation.

###  5.1. Local transversals to a group action

When a Lie group $G$ acts smoothly on a manifold $M$, we are given a map

$\sigma:G\times M\longrightarrow M,\;\;(g,x)\mapsto g\cdot x\,.$

If $a\in M$ is a point, we obtain the so-called orbit map

$\sigma_{a}:G\longrightarrow M,\;\;g\mapsto g\cdot a,$

whose image is the $G$-orbit through $a$. The derivative this map at the
identity element $e\in G$ gives a linear map

$\rho:=d\sigma_{a}:\mathfrak{g}\longrightarrow
T_{a}M,\;\;\;\xi\mapsto\xi(a):=d\sigma_{a}(\xi)$

where $\mathfrak{g}=T_{e}G$ denotes the Lie-algebra of $G$. We call this map
the infinitesimal action (at $a$). The image

$\mathfrak{g}\cdot a:=d\sigma_{a}(\mathfrak{g})$

of this map is the tangent space at $a$ to the orbit through $a$ and might be
called the $\mathfrak{g}$-orbit.

One may ask the following general question:

If $F\subset M$ is a submanifold such that $T_{a}F$ is a transversal to the
$\mathfrak{g}$-orbit of $a$, is it true that $F$ is a transversal to the
$G$-action?

In other words: does

$T_{a}F+\mathfrak{g}\cdot a=T_{a}M$

implies that the map

$G\times F\longrightarrow M,\;\;(g,b)\mapsto g\cdot b$

is locally surjective around $a$?

It follows from the implicit function theorem that the answer is positive. In
the next chapter we will describe a general iteration schema that provides a
constructive solution. KAM theory is concerned with the analogous situation in
infinite dimensions where no implicit function theorem is available, but the
same iteration scheme still can be used. For example, as we will see,
Kolmogorov’s invariant torus theorem can be proven using this iteration, if
interpreted in an appropriate sense. For that we will introduce the notion of
Kolmogorov spaces, which will be developed in the next chapter. The resulting
general theory then has many applications.

We start with some basic examples of Lie group actions.

######  Example 5.1.

Consider the action of the Lie group $G=\mathbb{R}$ on $\mathbb{R}^{2}$ by
translation along the vector $u=(1,0)$:

$\mathbb{R}\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n},\
(t,x,y)\mapsto(x+t,y).$

The orbits are horizontal lines. Any curve which does not have an horizontal
tangent at a given point is a transversal to the group action at that point.

######  Example 5.2.

Consider the action of the Lie group $G=\mathbb{R}$ acting on $\mathbb{R}^{2}$
by translation along a parabola:

$\mathbb{R}\times\mathbb{R}^{2}\longrightarrow\mathbb{R}^{2},\
(t,x,y)\mapsto(x+t,y-2xt-t^{2}).$

The orbit through $(x_{0},y_{0})$ is the parabola with equation

$y+x^{2}=y_{0}+x_{0}^{2}$

Any curve which is not tangent at a point to the corresponding parabola is a
transversal at the point.

The infinitesimal action at $a=(x_{0},y_{0})$ is the map

$\mathbb{R}\longrightarrow\mathbb{R}^{2},\ t\mapsto(1,-2x_{0})t.$

The $\mathfrak{g}$-orbit is the tangent to the parabola at $(x_{0},y_{0})$, so
any curve which is not tangent at a point to the corresponding parabola is a
transversal at the point. In particular, if $x_{0}=0$ the $\mathfrak{g}$-orbit
is an horizontal line. The situation is summarised by the first picture of
this chapter.

######  Example 5.3.

Consider the natural action of the Lie group $G=Gl(n,\mathbb{R})$ on
$\mathbb{R}^{n}$:

$GL(n,\mathbb{R})\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n},\
(A,x)\mapsto Ax.$

The action is homogeneous at any point $x\neq 0$ and the orbit is
$\mathbb{R}^{n}\setminus\\{0\\}$. Therefore there are two orbits: the origin
and its complement.

######  Example 5.4.

Consider the natural action of the Lie group $G=SO(n,\mathbb{R})$ on
$\mathbb{R}^{n}$:

$SO(n,\mathbb{R})\times\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n},\
(A,x)\mapsto Ax.$

The orbit of $a\neq 0$ is the $n-1$ dimensional sphere $S^{n-1}$ of radius
$\|a\|$ centred at the origin. Therefore the action is no longer homogeneous.
A transversal to the orbit at $a$ is given by any manifold transversal to the
sphere at $a$, for instance the straight line $\mathbb{R}a$.

The Lie algebra $\mathfrak{g}$ consists of antisymmetric matrices and its
orbit at $a$ is the tangent space to our sphere at $a$.

###  5.2. Adjoint orbits for the linear group

If $G$ is a Lie group, it acts on itself via conjugation

$G\times G\longrightarrow G,(g,h)\mapsto ghg^{-1}$

As this action preserves the unit element $e\in G$, there is an induced linear
action of $G$ on the tangent space $\mathfrak{g}=T_{e}G$

$G\times\mathfrak{g}\longrightarrow\mathfrak{g}\;.$

This is commonly called the adjoint action of a group on its Lie-algebra.

Let us take a closer look at it for the group $G=GL(n,\mathbb{R})$. Its Lie-
algebra $\mathfrak{g}$ can be identified with the space $M(n,\mathbb{R})$ of
$n\times n$ matrices and the adjoint action is given by conjugation

$GL(n,\mathbb{R})\times M(n,\mathbb{R})\longrightarrow
M(n,\mathbb{R}),(P,A)\longrightarrow PAP^{-1}$

To determine the corresponding infinitesimal action, we write down a one-
parameter family of matrices near the identity, with a given $B\in
M(n,\mathbb{R})$ as tangent vector:

$P={\rm Id\,}+tB+o(t)$

The inverse then is

$P^{-1}={\rm Id\,}-tB+o(t)$

and therefore

$PAP^{-1}=A+t[B,A]+o(t).$

Therefore the infinitesimal action at $A\in M(n,\mathbb{R})$ is the map

$M(n,\mathbb{R})\longrightarrow M(n,\mathbb{R}),B\mapsto[B,A].$

In order to describe a transversal to the $\mathfrak{g}$-orbit, it is
convenient to use the Euclidean scalar product ${\langle}-,-{\rangle}$ on
$\mathfrak{g}=M(n,\mathbb{R})$ given by the formula

${\langle}A,B{\rangle}:=Tr(AB^{T}).$

Here $-^{T}$ denotes the transposed matrix.

######  Lemma 5.5.

The commutant

$C(A):=\\{B\in M(n,\mathbb{R}):[B,A]=0\\}$

of $A$ is mapped by $B\mapsto B^{T}$ to the orthogonal space of the
$\mathfrak{g}$-orbit:

$C(A)^{T}=(\mathfrak{g}\cdot A)^{\perp}.$

###### Proof.

The $\mathfrak{g}$-orbit of $A$ consist of all elements of the form $[A,X]$,
where $X\in M(n,\mathbb{R})$. Now note the identity

$\begin{array}[]{rcl}{\langle}[A,X],B^{T}{\rangle}&=&Tr(AXB)-Tr(XAB)\\\
&=&Tr(BAX)-Tr(ABX)={\langle}[B,A],X^{T}{\rangle}\end{array}$

So we see that $B^{T}$ belongs to the orthogonal space of the
$\mathfrak{g}$-orbit of $A$ precisely when $B$ commutes with $A$. ∎

We compute a local transversal at a matrix $A$ in two extreme cases:

1) $A$ is diagonal with distinct eigenvalues
In this case, the characteristic polynomial has distinct roots. By
transversality, the matrix $A$ admits a neighbourhood in which all matrices
share the same property and are therefore all diagonalisable. In terms of
group actions: the space of diagonal matrices is a local transversal to the
adjoint action at $A$, which indeed is the transpose of the commutant. As an
example, if

$A=\begin{pmatrix}1&0\\\ 0&2\end{pmatrix}$

then the space of matrices of the form

$D_{\lambda}=\begin{pmatrix}1+\lambda_{1}&0\\\ 0&2+\lambda_{2}\end{pmatrix}$

is a local transversal. In general it means that $A$ has a neighbourhood $U$
such that for all $B\in U$ there exits
$\lambda_{1},\lambda_{2},\ldots,\lambda_{n}$ and $P\in GL(n,\mathbb{R})$ with
the property that $PBP^{-1}=D_{\lambda}$.

2) $A$ is maximally nilpotent Jordan block. So we let

$A=\begin{pmatrix}0&1&\dots&0&0\\\ 0&0&1&\dots&0\\\
\dots&\dots&\dots&\dots&\dots\\\ 0&0&\dots&0&1\\\ 0&0&\dots&0&0\\\
\end{pmatrix}$

If a matrix $B$ commutes with $A$, $[B,A]=0$, then $B$ preserves the
filtration defined by the kernels of the powers of $A$:

$\\{0\\}\subset{\rm Ker\,}A\subset{\rm Ker\,}A^{2}\subset\dots\rm
Ker\,A^{n-1}\subset{\rm Ker\,}A^{n}=\mathbb{R}^{n}.$

and thus $B$ is seen to be of the form:

$B=\begin{pmatrix}\lambda_{1}&\lambda_{2}&\lambda_{3}&\dots&\lambda_{n-1}&\lambda_{n}\\\
0&\lambda_{1}&\lambda_{2}&\dots&\lambda_{n-2}&\lambda_{n-1}\\\
\dots&\dots&\dots&\dots&\dots&\dots\\\ \dots&\dots&\dots&\dots&\dots&\dots\\\
0&0&0&\dots&\lambda_{1}&\lambda_{2}\\\
0&0&0&\dots&0&\lambda_{1}\end{pmatrix}=\sum_{i=1}^{n}\lambda_{i}A^{i-1}$

The transpose of this space is a transversal to the orbit:

$F=\left\\{\sum_{i=1}^{n}\lambda_{i}\,\left(A^{i-1}\right)^{\mathrm{T}}:\lambda_{1},\dots,\lambda_{n}\in\mathbb{C}\right\\}.$

For instance, we start with

$A=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$

The commutant consists of matrices of the form

$\begin{pmatrix}\lambda_{1}&\lambda_{2}\\\ 0&\lambda_{1}\end{pmatrix}$