As written in ([ma:equad:Arnold83]) it is very powerfull not to
solve differential equations but to tranform them into a simpler
differential equation. ([ma:equad:Arnold83],[ma:equad:Guckenheimer83],[ma:equad:Berry78])
Let the system:
![{\displaystyle {\dot {x}}=F(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c830d06050725255b7686c3daafb606b2f4b6d5)
and
a fixed point of the system:
. Without lack of
generality, we can assume
.
Assume that application
can be develloped around
:
![{\displaystyle F(x)=Ax+\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1739c86131a310c8836d9b085da88d371c2ead56)
where the dots represent polynomial terms in
of degree
.
There exists the following lema:
![{\displaystyle {\dot {x}}=(I+\partial _{y}h)A(x-h(x))+\dots =Ax+[{\frac {\partial }{\partial x}}Ax-Ah(x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4972a75f91a3ad912a5d61021673d74b4ee0b44d)
Note that
is the Poisson
crochet between
and
.
We note
and we call the
following equation:
![{\displaystyle L_{A}h=v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4558824c69a621c5d575f11e7840b1c46071dea8)
the homological equation associated to the linear operator
.
We are now interested in the reverse step of theorem lemplo: We
have a nonlinear system and want to find a change of variable that
transforms it into a linear system.
For this we need to solve the homological equation, {\it i.e.} to
express
as a function of
associated to the dynamics.
Let us call
,
the basis of eigenvectors of
,
the associated
eigenvalues, and
the coordinates of the system is this basis.
Let us write
where
contains the monoms of degree
, that is the terms
,
being
a set of positive integers
such that
.
It can be easily checked (see [ma:equad:Arnold83]) that the monoms
are eigenvectors of
with eigenvalue
where
:
![{\displaystyle L_{A}x^{m}e_{s}=[(m,\lambda )-\lambda _{s}]x^{m}e_{s}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a73a310e5f961d7192c982989fa12244b38fa6)
One can thus invert the homological equation to get a change of
variable
that eliminate the nonom considered.
Note however, that one needs
to invert
previous equation.
If there exists a
with
and
such that
, then the set of eigenvalues
is
called resonnant.
If the set of eigenvalues is resonant, since there exist such
,
then monoms
can not be eliminated by a change of variable.
This leads to the normal form theory ([ma:equad:Arnold83]).
An hamiltonian system is called integrable if there exist
coordinates
such that the Hamiltonian
doesn't depend on the
.
![{\displaystyle {\frac {dI_{i}}{dt}}=-{\frac {\partial H}{\partial \phi _{i}}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed0f9fbcc755150adfdb9ed521c09f4c31ba2fa3)
![{\displaystyle {\frac {d\phi _{i}}{dt}}=-{\frac {\partial H}{\partial I_{i}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c77ac259b7e4c8d0d721ae72488638615e5dc89)
Variables
are called action and variables
are called
angles.
Integration of equation eqbasimom is thus immediate and leads to:
![{\displaystyle I=I_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9afbec02cd7b902046185ffe331abdd1b8b54cc0)
and
where
and
are the initial conditions.
Let an integrable system described by an Hamiltonian
in the
space phase of the action-angle variables
.
Let us perturb this system with a perturbation
.
![{\displaystyle H(I,\phi )=H_{0}(I)+\epsilon H_{1}(I,\phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4564d682ce801fc776e4b38fec79672676e16ed6)
where
is periodic in
.
If tori exist in this new system, there must exist new action-angle
variable
such that:
![{\displaystyle H(I,\phi )=H^{\prime }(I^{\prime })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92009a96294272d5933d600f766d6c63d5913f63)
Change of variables in Hamiltonian system can be
characterized ([ph:mecac:Goldstein80]) by a
function
called generating function that
satisfies:
![{\displaystyle I={\frac {\partial S}{\partial \phi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20dc4d9f581df5353e5525b2b33e7a69df61f263)
![{\displaystyle \phi ^{\prime }={\frac {\partial S}{\partial I^{\prime }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e14cc2a92fa0b8d39c7e84c2f758b955e3646e87)
If
admits an expension in powers of
it must be:
![{\displaystyle S=\phi I^{\prime }+\epsilon S_{1}(\phi ,I^{\prime })+\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/83cd80ae0b0869dbd7853ee608f0d20dbdb3b047)
Equation eqdefHip thus becomes:
![{\displaystyle H_{0}(I^{\prime })+\epsilon \partial _{I_{i}^{\prime }}H(I^{\prime })\partial _{\phi _{i}}+H_{1}(I^{\prime },\phi )=H^{\prime }(I^{\prime })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e312eb413af770bdf3b6981449e35d32bb7db93)
Calling
the frequencies of the unperturbed Hamiltionan
:
![{\displaystyle \omega _{0,i}(I^{\prime })=\partial _{i}H_{0}(I^{\prime })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a08215e2aa0f6c523ea4700a5e1c30168d9c18f)
Because
and
are periodic in
, they can be decomposed
in Fourier:
![{\displaystyle H_{1}(I,\phi )=\sum _{m}H_{1,m}(I)e^{im\phi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7feb6c6095134254c102e17c1bdc480e41a8b76)
![{\displaystyle S_{1}(I,\phi )=\sum _{m\neq 0}S_{1,m}(I)e^{im\phi }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f893d3b2621e253b835c846606db5402f8ef7a8a)
Projecting on the Fourier basis equation equatfondKAM one gets
the expression of the new Hamiltonian:
![{\displaystyle H^{\prime }(I^{\prime })=H_{0}^{\prime }(I^{\prime })+\epsilon H^{\prime }(I^{\prime })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8513e73fdba65248657019197d5ce136497416)
and the relations:
![{\displaystyle i.m.\omega _{0}(I^{\prime })S_{1,m}(I^{\prime })=H_{1,m}(I^{\prime })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dba731d289b00f34294af07e41d56bc18ef7b4e7)
Inverting formally previous equation leads to the generating function:
![{\displaystyle S(\phi ,I^{\prime })=\phi I^{\prime }+\epsilon i\sum _{m\neq 0}{\frac {H_{1,m}(I^{\prime })}{m.\omega _{0}(I^{\prime })}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9790ba0b7319a880365f9a0b3aa92a7e17abdef)
The problem of the convergence of the sum and the expansion in
has been solved by KAM.
Clearly, if the
are resonnant (or commensurable), the serie
diverges and the torus is destroyed.
However for non resonant frequencies, the denominator term can be very
large and the expansion in
may diverge.
This is the {\bf small denominator
problem}.
In fact, the KAM theorem states that tori with ``sufficiently
incommensurable frequencies[1]
are not destroyed: The series converges[2].
- ↑
In the case two dimensional
case the KAM
theorem proves that the tori that are not destroyed are those with two
frequencies
and
whose ratio
is sufficiently irrational for the following relation to hold:
![{\displaystyle \left|{\frac {\omega _{0,1}}{\omega _{0,2}}}-{\frac {r}{s}}\right|>{\frac {K(\epsilon )}{s^{2.5}}}{\mbox{ , for all integers }}r{\mbox{ and }}s,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d69bcd0420d637f08c3231547c16a364d7e6e2)
where
is a number that tends to zero with the
.
- ↑
To prove the convergence, KAM use an accelerated convergence method
that, to calculate the torus at
order
uses the torus calculated at order
instead of the
torus at order zero like an classical Taylor expansion. See ([ma:equad:Berry78])
for a good analogy with the relative speed of the Taylor
expansion and the Newton's method to calculate zeros of functions.