Erratum for "Non singular Hamiltonian systems and geodesic flows on surfaces with negative curvature"

Publicacions Ma em`a iques, Vol. 44 (2000), 355–357
ERRATUM FOR “NON SINGULAR HAMILTONIAN
SYSTEMS AND GEODESIC FLOWS ON SURFACES
WITH NEGATIVE CURVATURE”
E. A. Lacomba and J. G. Reyes
A mis ake was ound in he easoning leading o a Lag angian which
we conside ed as equi alen om he o mula o he ac ion S(γ) below
he classical mechanical p oblem (3) on [1, page 271]. I u ns ou ha
he new Lag angian, co esponding o he con o mal me ic gha he
bo om o he page, is equi alen o he classical mechanical Lag angian,
bu only o he case o one deg ee o eedom. This means ha P opo-
si ion 2 is no alid o wo o mo e deg ees o eedom. Howe e , he
esul ing geodesic flow has many quali a i e ea u es simila o hose o
he o iginal mechanical p oblem. Indeed, compu ing he Eule -Lag ange
equa ions o
L=1
√2
δij ˙xi˙xj
√h−U,
subjec o he cons ain 1
2δij ˙xi˙xj−U(x)=h, we find, a e a long
compu a ion, he sys em o diffe en ial equa ions
2(h−U)[¨
x+∇U]=(˙
x·˙
x)∇U−˙
x(∇U·˙
x).
Since he igh hand side is o hogonal o he eloci y ˙
x, we e i y ha
indeed he ene gy is a fi s in eg al. Fo n=1 deg ee o eedom, he abo e
equa ion becomes 2(h−U)[¨x+U]= ˙x2U−˙xU˙x= 0 o ¨x+U=0,
he same han o he o iginal classical mechanical Lag angian. Bu i
n= 1 he igh hand side is no ze o in gene al, as can be seen wi h he
example n= 2 and U=1
2[(x1)2+(x2)2]. Then he wo componen s o
he igh hand side o he equa ions o mo ion become ˙x2(x1˙x2−x2˙x1)
and −˙x1(x1˙x2−x2˙x1) espec i ely; hese exp essions can be ze o only
i he eloci y is ze o o i i is o hogonal o he posi ion ec o .
Fo now, we a e le wi h esul s abou ei he one o he wo sys ems,
as ollows. The emaining pa o Sec ion 2 is alid o he geodesic
con o mal sys em (in (Mh),g
h,U). These a e mo e p ecisely
Theo em 2, classi ying he sign o Gaussian cu a u e in e ms o he
g adien and he Laplacian o he po en ial, as well as Co olla y 1.
356 E. A. Lacomba, J. G. Reyes
Lemma 1, s a ing ha geodesics escaping o infini y ha e infini e
leng h.
Lemma 2, s a ing ha any geodesic going o he cu e Γ = {U(x, y)=
0}has fini e leng h. Also Co olla y 2, s a ing ha he geodesic sys em
is no geodesically comple e.
All he esul s in Sec ion 3 a e s a ed o he o iginal classical mechan-
ical p oblem. Hence, hey a e all ue, excep o he commen abou
examples in Sec ion 5 on he final line o Sec ion 3.
Sec ion 4 e e s o he geodesic con o mal sys em (in (Mh),g
h,U), bu
no all he esul s emain alid since i was pa ly based on Sec ion 3.
We do no know i he ollowing esul s s ill hold:
Theo em 6. Gi en any di ec ion z= a c an(α)in he Hill’s egion
and any poin P∈in (Mh), he e exis s a unique geodesic γP h ough P
ha ing posi i ely asymp o ic di ec ion z. Analogously o he nega i ely
asymp o ic case.
Co olla y 5. I P∈in (Mh)and z∈Mh(∞), hen he e exis s a
unique geodesic γ⊂Mhdefined o any big enough posi i e ime, such
ha γ(0) = Pand γ(+∞)=z+. Simila ly o he case γ(0) = Pand
γ(−∞)=z−.
Theo em 7. Gi en wo poin s z1,z2in Mh(∞), he e exis s a unique
geodesic γ( )such ha γ⊂z−
1and γ⊂z+
2.
Co olla y 7. Any geodesic γ⊂Mhcan be w i en in he o m γ=
z−
1z+
2as he in e sec ion o wo classes o geodesics.
The las h ee heo ems in he pape also emain open, since hey
ela e he o iginal mechanical sys em wi h he geodesic flow. They a e:
Theo em 9. The se o physical cu es in he configu a ion space Mh
is ela ed, as a one-dimensional olia ion, o he geodesic flow o a space
homeomo phic o he uni disk, associa ed o a con o mal quasi-comple e
me ic wi h nega i e cu a u e.
Theo em 10 (Geome y o he Repulsi e Isosceles P oblem).I he
masses sa is y he ela ion µ≥2m, we ha e ha he se o physical
cu es o his p oblem is ela ed, as a one-dimensional olia ion, o he
geodesic flow in a space homeomo phic o he uni disk, wi h espec o
a con o mally Euclidean quasi-comple e me ic wi h nega i e cu a u e.
E a um o “Non singula Hamil onian sys ems...” 357
Theo em 11 (Geome y o he Repulsi e Rhomboidal P oblem).I
he masses sa is y he ela ion m1
2≤m3≤2m1, we ha e ha o a
fixed ene gy le el h>0, he se o physical cu es is ela ed, as a one-
dimensional olia ion, o he geodesic flow in a space homeomo phic o
he uni disk wi h espec o a quasi-comple e con o mal me ic wi h neg-
a i e cu a u e.
We belie e ha all he o he esul s in Sec ion 4 emain alid o he
geodesic sys em.
Re e ence
[1] E. A. Lacomba and J. G. Reyes, Non singula Hamil onian sys-
ems and geodesic flows on su aces wi h nega i e cu a u e, Publ.
Ma . 42 (1998), 267–299.
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Rebu el 20 de se emb e de 1999.