Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Integrable hamiltonian systems and spectral theory
.pdf
§1. Integrable Hamiltonian Systems 71
Another example is a system near a stationary point, say x = y =0where
dH(0) = 0. Assume that the Taylor expansion of H begins with
n
H =
1
2
k=1
αk(x
2
2
+ y
)+···
k
k
with real numbers α
, α2, ..., αn. By a theorem of G. D. Birkhoff one can
1
approximate this system locally by an integrable one (see [3], [4]): Given any
large integer N, assume that j
integers implies that j1= j2= ... = jn=0; then there exist canonical
j
k
variables x
∗
, ..., x
1
∗
n
, y
∗
, ..., y
1
H = ϕ(F
where R
∗j2
= x
vanishes with derivatives of order N at the origin and Fj=
N+1
∗j2
+ y
. Thus if we drop the term R
be precise, one has to excise the hyperplanes x
+ ···+jnαn=0, |j1|+ |j2|+ ···+|jn| N ,
1α1
∗
such that
n
, ..., Fn)+R
1
N+1
the system is integrable. (To
N+1
∗
∗
= y
k
=0where the dFjare
k
linearly dependent.)
On the other hand one can show that in general, even if the α
, α2, ..., α
1
are rationally independent the system is not integrable in any neighborhood of
the origin.
c) The structure of the integrable system is particularly simple. Given the
integrals one considers the manifolds N
defined by F1= c1, ..., Fn= c
n
with appropriate constants c1, ..., cn. These manifolds are invariant not only
under X
Thus X
fields commute each component of N
(because of (i) of definition 1) but also under X
H
, X
F
, ..., X
F
1
2
span the tangent space of Nn. Since these vector
F
n
is topologically a cylinder — and in case
n
(because of (ii)).
F
j
it is compact, a torus. Thus in the latter case D is foliated by n-dimensional tori.
By a theorem due to Arnold [1], [2] and Jost one can near a compact
component of N
H = H (y
1
(x, y) with the integer (x
The y
, xkare called the «action-angle» variables, respectively. In other words,
k
introduce canonical coordinates, called x, y again, such that
n
, ..., yn),thaty =0corresponds to Nnand that points (x, y),
− xj)(2π)−1correspond to the same points in D.
j
the example 2 is typical.
2
2
+ y
In Example 1 these tori are given by x
The flows generated by the commuting X
x
, ykplane.
k
k
, ..., X
F
1
= ckif the ckare positive.
k
are the n rotations in the
F
n
n
n
72 Various Aspects of Integrable Hamiltonian Systems
In action-angle variables the differential equations become
= H
˙x
k
(y), ˙yk=0.
y
k
Thus the differential equations are linear on N
H
, ..., H
y
1
If one solution on N
H
/jk= ρ is independent of k with some integers j1, j2, ..., jn. Thus for an
y
k
are rationally independent then the orbits are dense on Nn.
y
n
is periodic then all are. This occurs if and only if
n
. If the frequencies
n
integrable system the periodic solutions form (n −1) dimensional families. The
proof that Hamiltonian systems generally are not integrable is based on the fact
that generically the periodic solutions on a fixed energy surface are isolated.
References
[1] V. I. Arnold and A. Avez, Probl`emes Ergodiquies de la M´ecanique classique,
Gauthier-Villars, Paris, 1967.
[2] V. I. Arnold, Mathematical Methods in Classical Mechanics, Moscow, 1974
(Russian). English translation to appear.
[3] C.L.Siegeland J.Moser, Lectures on Celestial Mechanics, Springer, 1971.
[4] J. Moser, Stable and random motions in dynamical systems, Ann. Math.
Studies, 77, 1973.
§ 2. Examples of Integrable Systems, Isospectral Deformations
In spite of their exceptional character a number of integrable Hamiltonian
systems have been discovered recently for which the underlying symmetries are
highly unexpected. The most interesting ones are given by partial differential
equations and hence of infinite degrees of freedom. For lack of space they will
not be discussed here, but some of the following systems can be viewed as
discretized versions of those of infinite degrees of freedom.
a) Toda lattice. Consider n mass points on the line with coordinates x
x
, ..., xnand satisfying the differential equations
2
∂U
= −
¨x
j
∂x
j
(1)
,
1
§2. Examples of Integrable Systems, Isospectral Deformations 73
where
This system possesses n integrals in involution F
in y
=˙xkand exp(xk− x
k
k+1
by Flaschka. The solutions of this system can be expressed in rational form: e
are rational functions of e
λ1t
, ..., e
n
xk−x
k+1
U =
e
k=1
. (2)
, ..., Fnwhich are rational
1
). This fact was discovered by Henon and then
λnt
with some distinct constants λk.
x
The original Toda lattice refers to infinitely many particles, but we will
restrict ourselves to n<∞ (see [1]–[4]).
b) The inverse square potential of Calogero is also given by (1) but (2) is
replaced by
U =
(xk− xl)−2.
k<l
This system possesses n rational integrals in involution [5] (see Section 4). The
solutions x
(t) are algebraic functions of t, in fact they are the eigenvalues of a
k
matrix depending linearly on t.
c) The oldest example of this type is the geodesic flow on an n dimensional
ellipsoid in R
n+1
with different axis. This fact was discovered by Jacobi (see
Section 6). He showed, moreover, that a particle moving under the influence of
the radial force ax on the ellipsoid is integrable. For a =0one gets the geodesic
flow as a special case. The solutions are expressible in terms of hyperelliptic
functions.
d) A related problem is that of the motion of a mass point on the n
dimensional sphere |x| =1, x ∈ R
n+1
under the influence of a quadratic
potential V (x) with distinct eigenvalues. The solutions are also expressible in
terms of hyperelliptic functions (Section 7).
How does one recognize the integrable character of these systems and the
underlying hidden symmetries? There is no systematic approach. We will find
different reasons underlying the existence of the integrals. In case (a) and (b) the
integrals are found as eigenvalues of some classes of matrix, and the differential
equations correspond to deformations of these matrices leaving the spectrum
fixed (isospectral deformations).
In case (b) we will see that coadjoint representation of the unitary group is
the ultimate reason for the symmetries (see also [10]).
In cases (c) and (d) we will use a geometrical fact about confocal quadrics
to find rational integrals in involution. The geometrical fact is the following: If a
k
74 Various Aspects of Integrable Hamiltonian Systems
line touches n confocal quadrics in R
of the quadrics at these points P
n+1
at points P1, ..., Pnthen the normals
are mutually perpendicular.
j
It is likely that a common reason underlies all these phenomena. One
argument in this direction is that all these examples are related to the Korteweg –
de Vries equation. For (a) and (b) this is well known [6–9] but we will show
such a connection of (d) with Hill’s equation (which is closely related to the
Korteweg –de Vries equation) in Section 8. This observation seems new.
We conclude this section with a description of the integrals for problem (a),
following Flaschka. He constructed the Jacobi matrices
L =
⎛
b
1a1
⎜
a
1b2
⎜
⎜
⎜
⎝
0 a
·
·· ·
··a
n−1bn
0
n−1
⎞
⎟
⎟
⎟
; B =
⎟
⎠
⎛
0 a
⎜
−a
0 ·
1
⎜
⎜
⎜
⎝
0 −a
1
·· ·
· 0 a
n−1
0
n−1
0
⎞
⎟
⎟
⎟
⎟
⎠
.
He showed that the differential equations (1), (2) are equivalent to the system
d
L =[B, L]
dt
if we set
2a
(xk−x
= e
k
)/2
k+1
, 2bk= −yk= −˙xk.
From this one deduces that the eigenvalues of L are integrals — which turn out
to be in involution. Thus any functions of these, e. g. a symmetric function or
the traces
p
,p=1, 2, ..., n
tr L
are also integrals in involution.
A similar construction succeeds in (b) but we will describe a different
approach in Section 4.
References
[1] M. Toda, Wave propagation in anharmonic lattices, Jour. Phys. Soc. Japan
23 (1967) 501–506.
[2] M. Henon, Phys. Rev. B9, 1974, 1921–1923.
[3] H. Flaschka, The Toda lattice,I,Phys.Rev.B9, (1974) 1924–1925.
§3. Reduction of a Hamiltonian System with Symmetries 75
[4] J. Moser, Finitely many mass points on the line under the influence of an
exponential potential — An integrable system, Lecture Notes in Physics 38,
Springer, 1975, 467–497.
[5] J. Moser, Three integrable Hamiltonian systems connected with isospectral
deformation,Adv.Math.16 (1975) 197–220.
[6] H. Airault, H.P. McKean and J. Moser, Rational and elliptic solutions of the
Korteweg – de Vries equation and a related many body problem, Comm.
Pure Appl. Math. 30 (1977) 95–148.
[7] M. Adler and J. Moser, On a class of polynomials connected with the Ko-
rteweg – de Vries equation, Comm. Math. Phys. (1978) 1–30.
[8] F. Calogero, Motion of poles and zeroes of special solutions of nonlinear and
linear differential equations and related «solvable» many-body problems,
preprint, Univ. di Roma, 1977.
[9] D. V. Choodnovsky, and G. V. Chodnovsky, Pole expansions of nonlinear
partial differential equations, II Nuovo Cimento, 40B, 2, 1977.
[10] M. A. Olshanetsky and A. M. Perelomov, Completely integrable Hamilto-
nian systems connected with semisimple Lie algebras, Inv. Math. 37 (1976)
93–109.
[11] F. Calogero, Exactly solvable one-dimensional many body problems, Lettres
al Nuovo Cimento, 13, №11 (1975) 411–416.
§ 3. Reduction of a Hamiltonian System with Symmetries
a) Integrals of a system are in close relation to invariance groups of a
system, as is well known. For integrable systems the relevant group action is
Abelian — but in the following we will study more generally arbitrary — also
noncommutative group actions. A good example is the rotation group SO(3)
and a Hamiltonian system in R
reduced to a system in the radial variable only, i.e. to a system of one degree of
freedom.
How this reduction can be effected is the topic of this section. There are
excellent presentations of this reduction, see [1]–[4]. Therefore we describe the
approach informally without proof and stress simple examples illustrating the
approach.
3×R3
invariant under the rotation group can be
76 Various Aspects of Integrable Hamiltonian Systems
EXAMPLE 1. Let H(x, y) be a Hamiltonian in R2ninvariant under the
translation x
→ xk+ s, yk→ yk, i. e. satisfying
k
n
∂
H =0.
x
k
k=1
Then, clearly
F =
n
k=1
y
k
is an integral generating the above group action and
{F, H} =0.
This system can be reduced to a system of n − 1 degrees of freedom, e. g., by
introducing «relative coordinates»,
ξ
= xk− xn,ηk= ykfor k =1, ..., n− 1;
k
ξ
= xn,η
n
This is a canonical transformation and the integral becomes η
= y1+ ···+ yn.
n
, so that the new
n
Hamiltonian
Γ(ξ, η)=H(x, y)
is independent of ξ
can be solved for η
. Thus if we fix ηn= c the system
n
˙
ξ
=Γ
k
= c. Afterwards one can solve the equation for
n
, ˙ηk= −Γ
η
k
˙
ξ
n=Γη
(k n − 1)
ξ
k
.
n
Thus one integral allows us to reduce the phase-space by 2 dimensions: One
dimension is lost by fixing the integral F = c and the second by ignoring the
variable ξ
= xnalong the orbit of the group action.
n
This example is rather typical for integrals in involution: If a system admits
r integrals in involution, one can reduce the system by 2r dimensions to a system
of n − r degrees of freedom. However, if the integrals are not in involution the
reduction is more complicated.
§3. Reduction of a Hamiltonian System with Symmetries 77
E
XAMPLE 2. Let H(x, y)=
1
|y|2+ V (|x|) be a Hamiltonian in R6invari-
2
ant under the orthogonal group SO(3):
x → Rx, y → Ry where R ∈ SO(3).
This group is three dimensional and generated by the vector fields
˙x = Ax, ˙y = Ay
where
A =
⎛
0 −a
⎝
a
−a2a
3
3a2
0 −a
1
⎞
⎠
= a
1
+ a2I2+ a3I3.
1I1
0
The vector fields are Hamiltonian with the Hamiltonian
H = Ax, y = a, x ∧ y
where a =(a
1,a2,a3
) and x ∧ y is the vector product of x and y. Thus the
components
F
= x2y3− x3y2,F2= x3y1− x1y3,F3= x1y2− x2y
1
1
are integrals of the motion, defining the angular momentum vector.
Since the number of integrals is 3, one may expect that one can reduce the
phase space by 6 dimensions but, in fact, in this case the integrals are not in
involution and the reduced phase space is of 2 dimensions. One proceeds as
follows: We fix the angular momentum vector to, say,
x ∧ y = μ
where μ =0. We may assume that μ = λe
where e3=(0, 0, 1) and λ>0.
3
Then it follows from
− x3y2=0
x
2y3
x
3
1
1
3
that x
the quadric x
= y3=0and the problem is reduced to one in R2×R2, where we have
3
1y2−x2y1
= λ. This problem is still rotation invariant under SO(2)
and we may use polar coordinates r, ϕ and conjugate variables p
be defined by the canonical transformation
= r cosϕ = W
x
1
x2= r sinϕ = W
y
1
y
2
, pϕ. They can
r
78 Various Aspects of Integrable Hamiltonian Systems
with W = r(y1cos ϕ + y2sin ϕ).Thenpr= Wr, pϕ= Wϕgives
p
ϕ
= prcos ϕ −
y
1
= prsin ϕ +
y
2
sin ϕ
r
p
ϕ
cos ϕ
r
and x
− x2y1= λ. Thus pϕis the integral and H independent of ϕ.The
1y2
reduced Hamiltonian is
H =
1
p
2
2
λ
2
+
r
r
2
+ V (r)
and the ϕ-dependence is obtained separately from
∂H
˙ϕ =
∂p
This type of reduction was known to Jacobi who «reduced» the 3 body
problem in R
3
by using the invariance of this system under the Galilei group
λ
=
.
2
r
ϕ
which contains the rotation group SO(3) as a subgroup. Eliminating the integrals
of the center of mass and the angular momentum one can reduce this system of 9
degrees of freedom to one of 4 degrees of freedom, i. e. reduce the phase-space
from 18 dimensions to 8 dimensions. Using, in addition, the conservation of
energy one has a vector field on a seven dimensional manifold.
b) The moment map. We describe the generalization of this reduction in
abstract form. We consider a manifold M with a one-form θ for which ω = dθ is
nondegenerate, so that (M, ω) is a symplectic manifold. One sometimes refers
to (M, θ) as an «exact symplectic» manifold. An example is the cotangent
∗
bundle M = T
If G is a Lie group we speak of a symplectic group action (ϕ
g ∈ G there exists a diffeomorphism ϕ
g, h ∈ G, ϕ
N of a manifold N with the natural 1-form.
: M → M such that ϕg◦ϕh= ϕghfor
g
=Id,andϕgis symplectic, i. e. ϕ
e
∗
ω = ω. If, moreover, ϕ
g
) if for every
g
∗
θ = θ
g
we speak of an exact symplectic group action.
For G = R this concept is the same as that of symplectic flow. Every such
flow is generated by a Hamiltonian vector field, say X for which
ω is closed.
X
We will assume that this one form is even exact and set
ω = dF,
X
§3. Reduction of a Hamiltonian System with Symmetries 79
F being a Hamiltonian for this vector field. For example, in the exact symplectic
case we can define F by
θ = F.
X
If A is the Lie algebra of G,anda ∈A, exp(ta)=g(t) ∈ G then for any
function on M — the relation
f(ϕ
)
= Xf
g
t=0
d
dt
defines a symplectic vector field on M. This vector field depends linearly on A.
If ϕ
, hence X are exact symplectic then we can define the corresponding
g
Hamiltonian F = F(p, a) by
F = X
θ.
This Hamiltonian depends linearly on a ∈Aand therefore defines an ele-
∗
ment ψ ∈A
in the dual of the Lie algebra via
F (p, a)=ψ(p),a.
The mapping ψ: M →A
These concepts are easily illustrated and motivated by Example 2, where
G = SO(3), A = R
symplectic space (M, θ)=R
and the corresponding vector fields for R = e
∗
so defined is called the moment map (of Souriau).
3
with [a, b]=a ∧ b. The group action on the exact
3
6
,
i=1
(x, y)=(Rx, Ry),R∈ SO(3)
ϕ
R
yidx
is given by
i
tA
, Ax = a ∧ x is obtained by
differentiation as
X = a
1(x2∂x
− x3∂
3
)+ cyclic permutation.
x
2
This vector field is Hamiltonian with Hamiltonian
3
F =
The moment map ψ: R
vector (F
1,F2,F3
ajFj,F1= x2y3− x3y2(and cyclic perm).
j=1
6
→ R3takes p =(x, y) into the angular moment
).
80 Various Aspects of Integrable Hamiltonian Systems
If H0is invariant under the group action ϕg,i.e. H0◦ ϕg= H0then the
corresponding flow ϕ
t
generated by H0leaves ψ invariant, i. e.
0
t
ψ ◦ ϕ
= ψ.
0
This generalizes the fact that the angular momentum vector is an integral. The
proof of the above statement is immediate (see [3]).
d) The coadjoint representation of a group. An important group action,
which is independent of symplectic structure, is the adjoint representation of a
Lie group G which is given by
−1
x.
g
Here L
ϕ
: x → g ×g−1= LgR
g
, Rgdenote left and right multiplication. This mapping takes the unit
g
element ∈ G into itself and the linearized map at = e is defined as
−1
Ad (g)=dL
It maps A→Aand satisfied Ad (g
dR
g
g
)=Ad(g1)Ad (g2). It is the adjoint
1g2
representation of G. The induced mapping Ad
.
x=e
∗
(g−1) on A∗, the dual of A,is
called the coadjoint representation.
∗
For a fixed μ ∈A
tation as
one defines the orbits O(μ) of the coadjoint represen-
O(μ)={(Ad
∗
g)μ | g ∈ G}.
It is a basic result due to Kirillov and Kostant that these orbits O(μ) carry a
symplectic structure. We illustrate this fact with one example, referring for the
full discussion to [5].
Let G = GL(n, R), then one sees at once that
−1
where T ∈ G; A ∈A.
We can identify A
Ad (T ): A → TAT
∗
with A by representing any linear functional on Aby tr(AB)
with B ∈A. Thus we can identify the adjoint and coadjoint representation.
In this case the orbits O(μ) are the matrices similar to μ.Ifμ has distinct
eigenvalues the orbit consists precisely of the isospectral matrices. The symplectic structure is defined by giving the values of the two-form on the tangent
space of
−1
(A)={TAT−1,T∈ G}.
ψ
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]