16
2 Fundamental Concepts
and
J i = J i −
∞
n 1 =−∞
∞
n 2 =−∞
n i V n 1 ,n 2 cos(n 1 1 + n 2 2 )
(n 1 ω 1 + n 2 ω 2 )
+ O((
2 ).
(2.11)
To lowest order in , this is the solution to the problem. New actions, J i , have
been obtained that contain corrections due to the perturbation. If, for example, =
0.01, then by retaining only first-order corrections, we neglect terms of order 2 =
0.0001. To first order in , J i is a constant and i varies linearly in time. At least,
this is the hope. However, there is a catch! For the expansion in Eq. (2.11) to have
meaning, we must have
|n 1 ω 1 + n 2 ω 2 | | V n 1 ,n 2 .
(2.12)
However, the condition in Eq. (2.12) breaks down when internal nonlinear resonances occur and causes the perturbation expansion to diverge. Poincaré showed
that perturbation expansions of this type can generally be expected to diverge and
therefore, cannot be used for long-time predictions.
2.3 Integrable Systems
Integrable systems form an important reference point when discussing the behavior
of dynamical systems. We define an integrable system as follows. Consider a
dynamical system with N degrees of freedom. Its phase space has 2N dimensions.
The system is integrable if there exist N independent isolating integrals of motion,
I i , such that
I i (p 1 , . . . , p N , q 1 , . . . , q N ) = C i ,
(2.13)
for i = 1, . . . , N, where C i is a constant and p i and q i are the canonical
momentum and position associated with the ith degree of freedom. The functions I i
are independent if their differentials, dI i , are linearly independent.
It is important to distinguish between isolating and non-isolating integrals
(Wintner 1947). Non-isolating integrals (an example is the initial coordinates of
a trajectory) generally vary from trajectory to trajectory and usually do not provide
useful information about a system. On the other hand, isolating integrals of motion,
by Noether’s theorem, are due to symmetries (some “hidden”) of the dynamical
system and define surfaces in phase space.
The condition for integrability may be put in another form. A classical system
with N degrees of freedom is integrable if there exist N independent globally
defined functions, I i (p 1 , . . . , p N , q i , . . . , q N ), for i = 1, . . . , N , whose mutual
Poisson brackets vanish,
2 Fundamental Concepts
and
J i = J i −
∞
n 1 =−∞
∞
n 2 =−∞
n i V n 1 ,n 2 cos(n 1 1 + n 2 2 )
(n 1 ω 1 + n 2 ω 2 )
+ O((
2 ).
(2.11)
To lowest order in , this is the solution to the problem. New actions, J i , have
been obtained that contain corrections due to the perturbation. If, for example, =
0.01, then by retaining only first-order corrections, we neglect terms of order 2 =
0.0001. To first order in , J i is a constant and i varies linearly in time. At least,
this is the hope. However, there is a catch! For the expansion in Eq. (2.11) to have
meaning, we must have
|n 1 ω 1 + n 2 ω 2 | | V n 1 ,n 2 .
(2.12)
However, the condition in Eq. (2.12) breaks down when internal nonlinear resonances occur and causes the perturbation expansion to diverge. Poincaré showed
that perturbation expansions of this type can generally be expected to diverge and
therefore, cannot be used for long-time predictions.
2.3 Integrable Systems
Integrable systems form an important reference point when discussing the behavior
of dynamical systems. We define an integrable system as follows. Consider a
dynamical system with N degrees of freedom. Its phase space has 2N dimensions.
The system is integrable if there exist N independent isolating integrals of motion,
I i , such that
I i (p 1 , . . . , p N , q 1 , . . . , q N ) = C i ,
(2.13)
for i = 1, . . . , N, where C i is a constant and p i and q i are the canonical
momentum and position associated with the ith degree of freedom. The functions I i
are independent if their differentials, dI i , are linearly independent.
It is important to distinguish between isolating and non-isolating integrals
(Wintner 1947). Non-isolating integrals (an example is the initial coordinates of
a trajectory) generally vary from trajectory to trajectory and usually do not provide
useful information about a system. On the other hand, isolating integrals of motion,
by Noether’s theorem, are due to symmetries (some “hidden”) of the dynamical
system and define surfaces in phase space.
The condition for integrability may be put in another form. A classical system
with N degrees of freedom is integrable if there exist N independent globally
defined functions, I i (p 1 , . . . , p N , q i , . . . , q N ), for i = 1, . . . , N , whose mutual
Poisson brackets vanish,