Hierarchy of integrable geodesic flows

Publicacions Ma em`a iques, Vol. 44 (2000), 257–276
HIERARCHY OF INTEGRABLE GEODESIC FLOWS
Pe e Topalo
Abs ac
A amily o in eg able geodesic flows is ob ained. Any such a
amily co esponds o a pai o geodesically equi alen me ics.
1. In oduc ion
In pape s [1], [2], [3] a simple app oach was sugges ed o ob aining
fi s in eg als o Hamil onian sys ems i a ajec o ial diffeomo phism is
gi en. This app oach is closely ela ed o he ideas p esen ed in [4].
Recall b iefly he main cons uc ion ( o de ails see [1], [2]). Le
and ¯ be Hamil onian sys ems on he symplec ic mani olds (M2n,ω)
and ( ¯
M2n,¯ω) wi h Hamil onians Hand ¯
H espec i ely. Conside he
isoene gy su aces
Qde
={x∈M2n|H(x)=h},¯
Qde
={x∈¯
M2n|¯
H(x)=¯
h},
whe e hand ¯
ha e egula alues o he unc ions Hand ¯
H.
Defini ion 1. A diffeomo phism φ:Q→¯
Qis said o be ajec o ial,i
i akes he ajec o ies o he sys em o he ajec o ies o he sys em ¯ .
Le φ:Q→¯
Qbe a ajec o ial diffeomo phism. Le us deno e he
es ic ions ω|Qand ¯ω|¯
Qalso by he le e s ωand ¯ω. Then he pull-
back anish (o cou se, also L¯ ω= 0). I is ob ious ha he ke nels
o he o ms ωand φ∗¯ωcoincide wi h he linea span o he ec o .
The e o e, hese o ms induce wo non-degene a e enso fields on he
quo ien bundle TQ/  . Thus, he cha ac e is ic polynomial o he
ope a o ω−1◦φ∗¯ωis p ese ed by he flow .
In pape s [2], [3] he cons uc ion was applied o a classical example
whe e such a diffeomo phism exis s — he geodesic flows co esponding
o a pai o geodesically equi alen me ics.
Le gand ¯gbe Riemannian me ics on he mani old Mn.
258 P. Topalo
Defini ion 2. The pseudo-Riemannian me ics gand ¯ga e called geo-
desically equi alen iff hey ha e he same geodesics (conside ed as un-
pa ame e ized cu es).
Fo geodesically equi alen Riemannian me ics, a ajec o ial diffeo-
mo phism Φ is gi en by he o mula Φ : (x, ξ)→ x, ||ξ||g
||ξ||¯gξ, whe e
ξ∈T
xMn.
Deno e by Mq(g, ¯g)(1≤q≤n) he se o poin s y∈Mnsuch
ha he Riemannian me ics gand ¯gha e exac ly qdis inc eigen alues
in an open neighbo hood o he poin y. I he se Mnis e e ywhe e
dense in Mnwe say ha he Riemannian me ics gand ¯ga e s ic ly
non-p opo ional on Mn.
In he pape s [2], [3] he ollowing was p o ed.
Theo em 1. Suppose he Riemannian me ics gand ¯ga e geodesically
equi alen ; hen
a) he geodesic flow o he me ic gadmi s nin eg als (k≥0)
Ik
de
=(−1)kde (g)
de (¯g)k+2
n+1
¯g(Sk , ),(1)
whe e
Sk
de
=¯
Gk−σ1(¯
G)¯
Gk−1+···+(−1)kσk(¯
G)(2)
and ¯
Gde
=(gij ¯gjk),σi(¯
G)a e he elemen a y symme ic polynomi-
als o deg ee i;
b) he in eg als Ik(k≥0) a e in in olu ion1;
c) i Mq=∅, hen he ank o he diffe en ials dI0,... ,dI
n−1is equal
o qalmos e e ywhe e in TMq. Mo eo e , on TMqwe ha e
k(dI0,... ,dI
n−1) = k(dI0,... ,dI
q−1)≤q.(3)
This heo em is closely ela ed o some esul s p o ed by U. Dini,
P. Painle ´e, T. Le i-Ci i a and R. Liou ille (see [5], [6]).
In he p esen pape we assign o any pai o geodesically equi alen
me ics a hie a chy o in eg able geodesic flows (Theo em 5). The pape
is o ganized as ollows.
In Sec ion 2 we p esen some impo an ac s om he heo y o geo-
desic mappings needed o he sequel.
In Sec ion 3 we assign o any pai o s ic ly non-p opo ional geo-
desically equi alen me ics gand ¯ga amily o comple ely in eg able
1I.e., making he Legend e ans o ma ion we ob ain n unc ions commu ing wi h
espec o he canonical symplec ic s uc u e on T∗M.
Hie a chy o in eg able geodesic lows 259
Riemannian me ics S(g, ¯g) (see Theo em 5). Some simple p ope ies o
hese amilies a e ob ained.
In Sec ion 4 he esul s ob ained in Sec ion 3 a e applied o he me ics
dg2de
=
n

i=1
dxi2
(4)
and
d¯g2de
=1
n
i=1 xi
ai2
n

i=1
dxi2
ai
(5)
which a e geodesically equi alen on he s anda d ellipsoid Eh
de
=
n
i=1
xi2
ai=h,h>0. We gi e explici o mulae o he amily
S(dg2|Eh,d¯g2|Eh) in e ms o some me ics on he whole Rn(Theo-
em 6). I is in e es ing ha his amily con ains he me ic on he
Poisson sphe e, well-known in classical mechanics. I immedia ely pe -
mi s us o p o e ha he geodesic flow on he s anda d ellipsoid and
he geodesic flow on he Poisson sphe e ha e he same Liou ille olia ion
(Co olla y 3). This heo em is a mul idimensional gene aliza ion o he
well-known one in he wo-dimensional case. In addi ion, we gi e an
explici o mula o a me ic geodesically equi alen o he me ic on he
Poisson sphe e (Co olla y 2).
In Sec ion 5 we ob ain a amily o comple ely in eg able Hamil onians
wi h espec o he s anda d Lie-Poisson b acke on e(3)∗( he dual space
o he Lie algeb a e(3)). In pa icula , we ob ain an algeb aic Hamil-
onian H(1) such ha he co esponding Hamil onian sys em sg ad H(1)
is o bi ally equi alen o he Eule case o he ee mo ion o he igid
body (see Co olla y 6).
Th oughou he pape he s anda d ag eemen holds, i.e. Riemannian
me ic means a posi i e-defini e symme ic o m and pseudo-Riemannian
me ic means a non-degene a e symme ic o m. We conside mainly
Riemannian me ics; ne e heless many esul s hold in he pseudo-
Riemannian case.
The au ho is g a e ul o A. V. Bolsino , A. T. Fomenko, V. V. Kozlo ,
V. S. Ma ee , S. Tabachniko and I. A. Taimano o use ul discussions.
This pape was w i en du ing my s ay a he Max-Planck-Ins i u ¨u
Ma hema ik in Bonn. I is pleasu e o hank he Ins i u o i s hospi al-
i y and financial suppo . The au ho is pa ially suppo ed by MESC
g an MM-810/98.
260 P. Topalo
2. Geodesically equi alen me ics and he
co esponding 1-pa ame e amily
The ollowing lemma is needed o he sequel.
Lemma 1 (see [8]).
1) Suppose he pseudo-Riemannian me ics gand ¯ga e geodesically
equi alen ; hen he enso s aij and λi
aij
de
=Aα
igαj,(6)
λi
de
=−Aα
iψα,(7)
whe e (n+1)ψi=1
2∂iln ¯g
g,g= de (gij), and he ope a o Ais
gi en by o mula
Ai
j(g, ¯g)de
=
¯g
g
1
n+1
¯giαgαj,(8)
sa is y he equa ion
aij,k =λigjk +λjgik.(9)
He e aij,k deno es he co a ian de i a i e ∇kaij, whe e ∇is he
Le i-Ci i a connec ion co esponding o he me ic g.
2) Con e sely, i a non-degene a e symme ic enso field aij and an
1- o m λisa is y equa ion (9), hen he me ic
¯gij
de
=ˆg
gˆgij,(10)
whe e ˆgij
de
=giαaαβgβj, is geodesically equi alen o g.
I is ob ious ha Ais sel -adjoin wi h espec he bo h me ics.
Now, using Lemma 1 we can p o e
P oposi ion 1. Suppose he pseudo-Riemannian me ics gand ¯ga e
geodesically equi alen ; hen i o some pa ame e s αand β he ope a-
o (αA+β)is in e ible on Mn, hen he pseudo-Riemannian me ics g
and
¯gα,β(X,Y )de
=1
de (αA +β)g((αA +β)−1X,Y ),(11)
a e geodesically equi alen .
Rema k 1.Rema k he e ha ¯g0,1=gand ¯g1,0=¯g.
Hie a chy o in eg able geodesic lows 261
Rema k 2.I he mani old Mnis compac , hen o mula (11) gi es a
1-pa ame e amily o geodesically equi alen me ics. Mo eo e , in a
small neighbo hood o each poin o Mnwe also ob ain 1-pa ame e
amily.
Rema k 3.Le us ake wo ep esen a i es ¯gα,β and ¯gλ,µ. Now, we a e
able o apply P oposi ion 1 again. The co esponding amily o geodes-
ically equi alen me ics is
¯ga,b =1
de ((aλ +bα)A+(aµ +bβ))g((aλ +bα)A+(aµ +bβ))−1.(12)
The e o e, i (α:β)=(λ:µ), hen we ob ain he same amily. Rema k
he e also ha
A(¯gα,β,¯gλ,µ)= λA +µ
αA +β.(13)
P oo o P oposi ion 1: Le gand ¯gbe geodesically equi alen me ics.
Using (6), we ob ain ha he symme ic o m a=gA sa isfies equa-
ion (9) o some λ. Hence, he o m ac
de
=gA +cg (c= cons ) also
sa isfied his equa ion. We use he e ha ∇is he Le i-Ci i a connec ion
co esponding o he me ic g. Assume ha acis non-degene a e on
Mn. We ge ˆgde
=g(gA +cg)−1g=g(A+c)−1and using he con e se
o mula (10) ob ain ha he pseudo-Riemannian me ic
¯gc=1
de (A+c)g(A+c)−1
(14)
is geodesically equi alen o g.
Now, using P oposi ion 1 we a e able o p esen ano he ‘non-sym-
plec ic’ p oo o he fi s i em o Theo em 1.
Co olla y 1. I he Riemannian me ics gand ¯ga e geodesically equi -
alen , hen he geodesic flow o he me ic gadmi s a 1-pa ame e amily
o fi s in eg als
Iα,β(g, ¯g)de
= de (αA +β)g(αA +β)−1.(15)
P oo o Co olla y 1: The co olla y easily ollows om he nex heo em
(see [6]).

262 P. Topalo
Theo em 2 (Painle ´e).I he me ics gand ¯ga e geodesically equi a-
len , hen he unc ion
I0(g)de
=de (g)
de (¯g)2
n+1
¯g(16)
is an in eg al o he he geodesic flow o he me ic g.
In a small neighbo hood o each poin o Mnwe can find a 1-pa ame e
amily ¯g1,β o geodesically equi alen Riemannian me ics. The e o e,
locally he Riemannian me ics gand ¯g1,β a e geodesically equi alen ,
whe e β∈(a, b), a<b. I pe mi s us o apply Theo em 2. We ha e
I1,β(g)de
=de (g)
de (¯g1,β)2
n+1
¯g1,β
(17)
= de (A+β)g(A+β)−1
(18)
=I0+I1β+···+In−1βn−1,(19)
whe e he unc ions Ika e he same as in Theo em 1. The las equali y in
his chain can easily be p o ed. The e o e, he unc ions Ika e in eg als
in a small neighbo hood o each poin o Mn. Bu hey a e globally
defined on Mn. Thus, hey a e in eg als.
Rema k 4.I we apply Co olla y 1 o he geodesically equi alen me -
ics gand ¯gα,β ((α:β)= (0 : 1)), hen we’ll ob ain he same amily o
in eg als.
Rema k 5.Le us conside he map Φ : ξ→ ||ξ||g
||ξ||¯gξ2,
Φ:(TM)0→(TM)0.(20)
I is ob ious ha Φ maps each geodesic ajec o y o he me ic gin o a
geodesic ajec o y o he me ic ¯g. The e o e, he pull-back Φ∗
(Iα,β(¯g,g))
gi es a amily o in eg als o he geodesic flow o he me ic g. I can
easily be checked ha
Φ∗(Iα,β(¯g,g)) = g(ξ,ξ)
I1,0(g, ¯g)Iβ,α(g, ¯g).(21)
Thus, we won’ be able o ob ain new amily o in eg als.
2Some imes we will deno e his mapping by Φ(g, ¯g).
Hie a chy o in eg able geodesic lows 263
3. Hie a chy o in eg able flows
As we ha e seen i he Riemannian me ics gand ¯ga e geodesically
equi alen , hen hey a e con ained in a amily o geodesically equi alen
me ics
¯gα,β =1
de (αA +β)g(αA +β)−1,(22)
whe e αand βa e pa ame e s.
Rema k 6.In he ollowing sec ions we conside only Riemannian me -
ics al hough he mos o he cons uc ions pass in pseudo-Riemannian
case.
I is in e es ing ha using geodesically equi alen Riemannian me -
ics gand ¯gwe can p oduce ano he amilies o geodesically equi alen
me ics. We need
Theo em 3 (see [8]).I he Riemannian me ics gand ¯ga e geode-
sically equi alen , hen o each in ege k he Riemannian me ics g(k)de
=
gAkand ¯g(k)de
=¯gAka e also geodesically equi alen .
This heo em may be p o ed by di ec calcula ions. Fo de ails see
pape [8]. We will sligh ly gene alize his esul la e using some o he
a gumen s. A fi s we need some no a ions.
Le Bbe a sel -adjoin ope a o on he connec ed Riemannian mani-
old (Mn,g). By defini ion, pu
(B)de
= in
x∈Mmin{spec B(x)}(23)
and
R(B)de
= sup
x∈M
max{spec B(x)}.(24)
Deno e by I he se ob ained by adding o he in e al ( , R) i s endpoin s
iff hey a e achie ed o some x∈Mn. Le us conside he se o all
eal Lau en se ies La(x)=kck(x−a)kwhich a e con e gen on
some open neighbo hood o I.Deno e by ω+(B) he cone o all fini e
linea combina ions o such se ies which gi e posi i e unc ions on I.O
cou se, we a e able o conside mo e gene al se o unc ions bu i will
only complica e ou cons uc ion.
264 P. Topalo
Theo em 4. Suppose he Riemannian me ics gand ¯ga e geodesically
equi alen and F(x)∈ω+(A); hen he me ics gFde
=gF(A)and ¯gFde
=
¯gF(A)a e also geodesically equi alen . These me ics a e con ained in
a amily o geodesically equi alen Riemannian me ics
¯gF
α,β
de
=1
de (αA +β)gF(A)(αA +β)−1,(25)
whe e αand βa e pa ame e s such ha he ope a o (αA +β)is non-
degene a e.
P oo o Theo em 4: Le gand ¯gbe geodesically equi alen Riemannian
me ics on a mani old Mn. Deno e by ρ1,... ,ρ
m(1 ≤m≤n) he com-
mon eigen alues o he me ics gand ¯g. Suppose he unc ions ρ1,... ,ρ
m
a e diffe en a e e y poin o an open domain D⊂Mn. In he pape [6],
T. Le i-Ci i a p o ed ha o e e y poin P∈D he e is an open neigh-
bo hood U(P)⊂Dand a coo dina e sys em ¯x=(¯x1,... ,¯xm) (in U(P)),
whe e ¯xi=(x1
i,... ,x
ki
i), (1 ≤i≤m), such ha he quad a ic o ms o
he me ics gand ¯gha e he ollowing o m:
g(˙
¯x, ˙
¯x)=
m

i=1
Πi(¯x)Ai(¯xi,˙
¯xi),(26)
¯g(˙
¯x, ˙
¯x)=
m

i=1
ρiΠi(¯x)Ai(¯xi,˙
¯xi),(27)
whe e Ai(¯xi,˙
¯xi) a e posi i e-defini e quad a ic o ms in he eloci ies ˙
¯xi
wi h coefficien s depending on ¯xi,
Πi
de
=(φi−φ1)...(φi−φi−1)(φi+1 −φi)...(φm−φi),(28)
ρi=1
φ1...φ
m
1
φi
(29)
and φ1,φ
2,... ,φ
m(0 <φ
1<φ
2<···<φ
m) a e smoo h unc ions such
ha
φi=φi(¯xi),i ki=1
cons an ,else.
Hie a chy o in eg able geodesic lows 265
Defini ion 3. Le me ics gand ¯gbe gi en by o mulae (26) and (27)
in a coo dina e cha U. Then we say ha he me ics gand ¯gha e Le i-
Ci i a local o m o ( ype m), and he coo dina e cha Uis Le i-Ci i a
coo dina e cha (wi h espec o he me ics).
In he pape [6], Le i-Ci i a p o ed ha he me ics gand ¯ggi en
by o mulae (26) and (27) a e geodesically equi alen .
Deno e by M he se o all poin in Mnwhich a e con ained in some
Le i-Ci i a cha , i.e. x∈Miff he e is a Le i-Ci i a cha Uwhich
con ains x. By defini ion, Mis an open subse o Mn.In[2], [3]was
p o ed ha Mis e e ywhe e dense in Mn.
In e e y Le i-Ci i a coo dina e cha we ha e
¯g
g1
n+1
=1
φk1−1
1...φ
km−1
m1
φ1...φ
m
= cons .1
φ1...φ
m
,(30)
(¯gikgkj) = diag 1
ρ1,... , 1
ρ1
  
k1
;... ;1
ρm,... , 1
ρm
  
km
.(31)
The e o e,
A(g, ¯g) = cons .diag(φ1,... ,φ
1
  
k1
;... ;φm,... ,φ
m
  
km
).(32)
Hence,
g(F(A)˙
¯x, ˙
¯x)=
m

i=1
Πi(¯x)A
i(¯xi,˙
¯xi),(33)
and
¯g(F(A)˙
¯x, ˙
¯x)=
m

i=1
ρiΠi(¯x)A
i(¯xi,˙
¯xi),(34)
whe e A
i= i(¯xi)Aiand i(¯xi) is a smoo h unc ion depending o he
a iable ¯xi. The e o e, in any Le i-Ci i a cha he me ics gF(A) and
¯gF(A) a e geodesically equi alen . Le us conside he map Φ : ξ→
||ξ||g
||ξ||¯gξ,
Φ:(TM)0→(TM)0.(35)
272 P. Topalo
This cons uc ion gene alize he well-known me ic on he Poisson sphe e
conside ed in he classical mechanics (see [10]).
Co olla y 2. The es ic ion o he me ic dg2
(1) o he ellipsoid Ehhas
he same geodesic lines as he me ic on he Poisson sphe e.
Using P oposi ion 2, we ob ain
Co olla y 3. The geodesic flows on he ellipsoid and he Poisson sphe e
ha e iden ical Liou ille olia ions.
I n= 2 he las esul is well-known. Mo eo e , in [11] A. T. Fomenko
and A. V. Bolsino using he heo y o o bi al equi alence o he in e-
g able Hamil onian sys ems p o ed ha o e e y 2-dimensional ellipsoid
we can ake a sui able Poisson sphe e such ha he co esponding geo-
desic flows a e con inuously o bi ally equi alen . I is in e es ing o sol e
he same p oblem in mul idimensional case.
The s anda d sphe e and he Poisson sphe e. In he case k=−1
we ha e
dg2
(−1) =A(x)dx, dx
(55)
=A−1dx, dx−2A−1x, dxA−2x, dx
A−1x, A−1x.(56)
The las e m on he igh side anish on TEh. The e o e, we can hink
ha dg2
(−1) =A−1dx, dx. Changing he a iables x=√Ay we see
ha g(−1) is he s anda d me ic on he sphe e. Fu he ,
d¯g2
(−1) =!A−1dx, A−1dx−A−1x,A−1dx2
A−1x,A−1x"
A−1x, A−1x.(57)
Co olla y 4. The es ic ion o he me ic
d¯g2
(−1) =A−1dy, dy−A−1y,dy2
A−1y,y
A−1y,y
(58)
o he sphe e Sn={y,y=h>0}has he same geodesic lines as he
s anda d me ic on he sphe e.

Hie a chy o in eg able geodesic lows 273
Using P oposi ion 3, we ob ain
Co olla y 5. The me ic on he Poisson sphe e d¯g2
(1)|Ehand he s an-
da d me ic on he sphe e dg2
(−1)|Eha e comple ely in eg able on T∗M
such ha he co esponding in eg als a e he same.
5. A amily o in eg able Hamil onians in e(3)
∗
I is well-known ha he Eule -Poisson equa ions which desc ibe he
mo ion o he igid body can be w i en as Eule equa ions on he dual
space o he Lie algeb a e(3) (see [12], [13]). Mo e p ecisely, i he
coo dina es in e(3)∗a e deno ed by ( 1,
2,
3,s
1,s
2,s
3), hen he Lie-
Poisson b acke can be gi en by ela ions {si,s
j}=8ijksk,{ i,
j}=0,
{si,
j}=8ijk k. The e o e, i H=H( , s) is a Hamil onian unc ion,
hen he Eule equa ions ake he o m o Ki choff equa ions
˙s=s×∂H
∂s + ×∂H
∂ ,(59)
˙ = ×∂H
∂s .(60)
The Lie-Poisson b acke has wo annihila o s F1= 2
1+ 2
2+ 2
3and
F2= 1s1+ 2s2+ 3s3. The Hamil onian unc ion o he igid body is
Hde
=1
2s2
1
I1
+s2
2
I2
+s2
3
I3+mg l, ,(61)
whe e s=(s1,s
2,s
3) deno es he angula momen um o he body,
l=(l1,l
2,l
3) deno es he coo dina es o he cen e o g a i y o he
body, I= diag(I1,I
1,I
3) is he enso o ine ia, =( 1,
2,
3) — he
coo dina es o he uni e ical ec o in he space, and mg is he weigh
o he body.
The co angen bundle T∗S2supplied wi h he canonical symplec ic
s uc u e dp ∧dq is symplec omo phic o he mani old Oh
de
={F1=
h, F2=0}⊂e(3)∗(see [14]). The e o e, any in eg able Hamil onian
on T∗S2gi es an in eg able Hamil onian on Oh. Mo eo e , i he fi s
Hamil onian is polynomial in impulses, hen he second one is also poly-
nomial in momen a s=(s1,s
2,s
3) and has he same deg ee. Using he
esul s o [14] and Lemma 2 i is easy o p o e
274 P. Topalo
Lemma 3. Suppose ha he Riemannian me ic dg2=gij dxidxj,x=
(x1,x
2,x
3)is smoo h in a neighbo hood o he sphe e S2
h={x, x=h}
and he es ic ion dg2|S2
hgi es a me ic whose geodesic flow is com-
ple ely in eg able; hen he Hamil onian unc ion
Hg( , s)de
=de E
de G
1
e(ne,n
g) 
i,j
gij( )sisj,(62)
whe e de2=(dx1)2+(dx2)2+(dx3)2,G=(gij),E=(eij )and neand
nga e he uni ex e nal no mal ec o s o he sphe e S2
hcalcula ed wi h
espec o he me ics eand g espec i ely a he poin x= ∈S2
h,is
comple ely in eg able on he submani old Oh.
In Sec ion 4 we ound a amily o me ics on Rnsuch ha hei es ic-
ion on he ellipsoid Ehgi e a amily o in eg able me ics. Now, using
hese me ics and Lemma 3 we a e able o find a amily o in eg able
Hamil onians in e(3)∗.
Theo em 7. Conside he ope a o s
A( )de
=A−(√A )⊗(√A )
 , +(A−1
2 )⊗(A−1
2 )(63)
and
B(k)( )de
=√AAk( )√A.(64)
Fo any fixed in ege k he Hamil onians
H(k)de
=1
A−1 , k+1 B(k)s, s
(65)
and
¯
H(k)de
=1
A−1 , k−1B(k−1)s, s
(66)
a e Liou ille in eg able on any o bi Oh. Mo eo e , he co esponding
Hamil onian sys ems a e o bi ally equi alen .
Example. I k= 1, hen we ob ain ha he Hamil onians
H(1) =1
A−1 , 2A2s, s−A , s2
 , + , s2#
(67)
and
¯
H(1) =As, s(68)
a e o bi ally equi alen .
Hie a chy o in eg able geodesic lows 275
Co olla y 6. The Hamil onian sys em wi h Hamil onian H(1) and he
Eule case o he ee mo ion o he igid body a e o bi ally equi alen on
he o bi Oh,h>0.
Re e ences
[1] P. J.Topalo , Tenso in a ian s o na u al mechanical sys ems on
compac su aces, and he co esponding in eg als, Ma . Sb. 188(2)
(1997), 137–157.
[2] P. J. Topalo and V. S. Ma ee , Geodesic equi alence and
in eg abili y, P ep in o MPI-Bonn (1998), MPI 98-74.
[3] V. S. Ma ee and P. J. Topalo , Geodesic equi alence and Li-
ou ille in eg abili y, Regula and Chao ic Dynamics 2(1998), 29–44.
[4] S. V. Bolo in and V. V. Kozlo , Symme y fields o geodesic
flows, Russian J. Ma h. Phys. 3(3) (1995), 279–295.
[5] P. Painle ´
e, Su les in ´eg ale quad a iques des ´equa ions de la Di-
namique, C. R. Acad. Sci. Pa is S´e . I Ma h. 124 (1897), 221–224.
[6] T. Le i-Ci i a, Sulle as o mazioni delle equazioni dinamiche,
Ann. Ma . (2) 24 (1896), 255–300.
[7] A. V. Amino a, Pseudo-Riemannian mani olds wi h gene al geo-
desics, Uspekhi Ma . Nauk 48(2) (1993), 107–164 (in Russian);
ansla ion in Russian Ma h. Su eys 48(2) (1993), 105–160.
[8] N. S. Sinyuko , To he heo y o geodesical mappings, So ie
Ma h. Docl. 169(4) (1966), 770–772.
[9] S. Tabachniko , P ojec i ely equi alen me ics, exac ans e se
line field and he geodesic flow on he ellipsoid, Commen . Ma h.
Hel . 74(2) (1999), 306–321.
[10] A. V. B ailo , Cons uc ion o comple ely in eg able geodesic
flows on compac symme ic spaces, Ma h. USSR Iz es iya 29(1)
(1987), 19–31.
[11] A. V. Bolsino and A. T. Fomenko, O bi al classifica ion o
geodesic flows o wo-dimensional ellipsoids. The Jacobi p oblem is
o bi ally equi alen o he in eg able Eule case in he dynamics
o a igid body, Funk sional. Anal. i P ilozhen 29(3) (1995), 1–15,
96 (in Russian); ansla ion in Func . Anal. Appl. 29(3) (1995),
149–160.
[12] S. P. No iko and I. Shmel’ se , Pe iodic solu ions o Ki chhoff
equa ions o he ee mo ion o a igid body in a fluid and he
ex ended Lyus e nik-Shni el’man-Mo se heo y, Funk ional. Anal. i
P ilozhen. 15(3) (1981), 54–66 (in Russian).
276 P. Topalo
[13] V. V. Kozlo , Two in eg able p oblems o classical dynamics,
Ves nik Mosko . Uni . Se . I Ma . Mekh. 4(1981), 80–83, 87 (in
Russian).
[14] A. V. Bolsino , A. T. Fomenko and V. V. Kozlo , The de
Maupe uis p inciple and geodesic flows on a sphe e ha a ise om
in eg able cases o he dynamics o a igid body, Uspekhi Ma . Nauk
50(3) (1995), 3–32 (in Russian); ansla ion in Russian Ma h. Su -
eys 50(3) (1995), 473–501.
Depa men o Diffe en ial Equa ions
Ins i u o Ma hema ics and In o ma ics
Bulga ian Academy o Sciences
Acad. G. Bonche S ., bl. 8
Sofia 1113
Bulga ia
E-mail add ess:[email p o ec ed]
P ime a e si´o ebuda el 30 d’ab il de 1999,
da e a e si´o ebuda el 23 de se emb e de 1999.