an integrable hénon heiles system on spaces with constant...
TRANSCRIPT
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Universidad de Burgos
Departamento de Fısica
An integrable Henon–Heiles system onspaces with constant curvature
Angel Ballesteros
A. Blasco, F.J. Herranz, F. Musso
XXIV IFWGP CUD, Zaragoza, September 2015
1 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The problem: curvature as an integrable deformation
Given a certain Liouville integrable Hamiltonian system on thetwo-dimensional (2D) Euclidean space with qi , pj = δij
H = T + V =1
2(p2
1 + p22) + V(q1, q2),
and whose integral of the motion is given by I(p1, p2, q1, q2),
find a one-parameter integrable generalization Hκ of this system of the form
Hκ = Tκ(p1, p2, q1, q2) + Vκ(q1, q2),
with integral of the motion given by Iκ(p1, p2, q1, q2) and such that:
Tκ is the kinetic energy of a particle on a 2D space with constantcurvature κ: the sphere S2 (κ > 0) or the hyperbolic space H2 (κ < 0).
The Euclidean system H has to be smoothly recovered in thezero-curvature limit κ→ 0, namely
H = limκ→0Hκ, I = lim
κ→0Iκ.
2 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The problem: curvature as an integrable deformation
Given a certain Liouville integrable Hamiltonian system on thetwo-dimensional (2D) Euclidean space with qi , pj = δij
H = T + V =1
2(p2
1 + p22) + V(q1, q2),
and whose integral of the motion is given by I(p1, p2, q1, q2),
find a one-parameter integrable generalization Hκ of this system of the form
Hκ = Tκ(p1, p2, q1, q2) + Vκ(q1, q2),
with integral of the motion given by Iκ(p1, p2, q1, q2) and such that:
Tκ is the kinetic energy of a particle on a 2D space with constantcurvature κ: the sphere S2 (κ > 0) or the hyperbolic space H2 (κ < 0).
The Euclidean system H has to be smoothly recovered in thezero-curvature limit κ→ 0, namely
H = limκ→0Hκ, I = lim
κ→0Iκ.
2 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The problem: curvature as an integrable deformation
Given a certain Liouville integrable Hamiltonian system on thetwo-dimensional (2D) Euclidean space with qi , pj = δij
H = T + V =1
2(p2
1 + p22) + V(q1, q2),
and whose integral of the motion is given by I(p1, p2, q1, q2),
find a one-parameter integrable generalization Hκ of this system of the form
Hκ = Tκ(p1, p2, q1, q2) + Vκ(q1, q2),
with integral of the motion given by Iκ(p1, p2, q1, q2) and such that:
Tκ is the kinetic energy of a particle on a 2D space with constantcurvature κ: the sphere S2 (κ > 0) or the hyperbolic space H2 (κ < 0).
The Euclidean system H has to be smoothly recovered in thezero-curvature limit κ→ 0, namely
H = limκ→0Hκ, I = lim
κ→0Iκ.
2 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The problem: curvature as an integrable deformation
If these two conditions are fulfilled, we shall say that Hκ is a curved H systemon the sphere and the hyperbolic space.
The uniqueness of this construction is not guaranteed, since differentVκ integrable potentials (and their associated Iκ integrals) having thesame κ→ 0 limit could be found.
Example: the construction of integrable curved analogues of theanisotropic oscillator [A.B., A. Blasco, F. J. Herranz, F. Musso,(2013–2014)] on S2 (κ > 0) and H2 (κ < 0).
If the Hamiltonian H is superintegrable (i.e. if another globally definedand functionally independent integral of the motion K(p1, p2, q1, q2) doesexist) then we could further impose the existence of the curved (andfunctionally independent) analogue Kκ of the second integral. If wesucceed in finding such second integral, we would obtain asuperintegrable curved generalization of H.
3 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The problem: curvature as an integrable deformation
If these two conditions are fulfilled, we shall say that Hκ is a curved H systemon the sphere and the hyperbolic space.
The uniqueness of this construction is not guaranteed, since differentVκ integrable potentials (and their associated Iκ integrals) having thesame κ→ 0 limit could be found.
Example: the construction of integrable curved analogues of theanisotropic oscillator [A.B., A. Blasco, F. J. Herranz, F. Musso,(2013–2014)] on S2 (κ > 0) and H2 (κ < 0).
If the Hamiltonian H is superintegrable (i.e. if another globally definedand functionally independent integral of the motion K(p1, p2, q1, q2) doesexist) then we could further impose the existence of the curved (andfunctionally independent) analogue Kκ of the second integral. If wesucceed in finding such second integral, we would obtain asuperintegrable curved generalization of H.
3 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The problem: curvature as an integrable deformation
If these two conditions are fulfilled, we shall say that Hκ is a curved H systemon the sphere and the hyperbolic space.
The uniqueness of this construction is not guaranteed, since differentVκ integrable potentials (and their associated Iκ integrals) having thesame κ→ 0 limit could be found.
Example: the construction of integrable curved analogues of theanisotropic oscillator [A.B., A. Blasco, F. J. Herranz, F. Musso,(2013–2014)] on S2 (κ > 0) and H2 (κ < 0).
If the Hamiltonian H is superintegrable (i.e. if another globally definedand functionally independent integral of the motion K(p1, p2, q1, q2) doesexist) then we could further impose the existence of the curved (andfunctionally independent) analogue Kκ of the second integral. If wesucceed in finding such second integral, we would obtain asuperintegrable curved generalization of H.
3 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The problem: curvature as an integrable deformation
In this framework Gaussian curvature κ of the space enters as adeformation parameter, and the curved systems Hκ can be thought ofas integrable perturbations of the flat ones H in terms of the curvatureparameter κ.
In this way, integrable Hamiltonian systems on S2 (κ > 0), H2 (κ < 0)and E2 (κ = 0) can be simultaneously constructed and analysed.
This approach has been followed so far in order to construct analoguesof the oscillator and Kepler–Coulomb systems on spaces with constantcurvature (see the previous works by Ranada, Santander, Carinena, Diacuand our group).
Here we face the construction of the first, to the best of our knowledge,example of a curved integrable Henon–Heiles system on S2 and H2
and of the full series of RDG integrable potentials.
Details in arXiv:1411.2033 (Nonlinearity) and arXiv:1503.09187 (JPCS).
4 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The problem: curvature as an integrable deformation
In this framework Gaussian curvature κ of the space enters as adeformation parameter, and the curved systems Hκ can be thought ofas integrable perturbations of the flat ones H in terms of the curvatureparameter κ.
In this way, integrable Hamiltonian systems on S2 (κ > 0), H2 (κ < 0)and E2 (κ = 0) can be simultaneously constructed and analysed.
This approach has been followed so far in order to construct analoguesof the oscillator and Kepler–Coulomb systems on spaces with constantcurvature (see the previous works by Ranada, Santander, Carinena, Diacuand our group).
Here we face the construction of the first, to the best of our knowledge,example of a curved integrable Henon–Heiles system on S2 and H2
and of the full series of RDG integrable potentials.
Details in arXiv:1411.2033 (Nonlinearity) and arXiv:1503.09187 (JPCS).
4 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The problem: curvature as an integrable deformation
In this framework Gaussian curvature κ of the space enters as adeformation parameter, and the curved systems Hκ can be thought ofas integrable perturbations of the flat ones H in terms of the curvatureparameter κ.
In this way, integrable Hamiltonian systems on S2 (κ > 0), H2 (κ < 0)and E2 (κ = 0) can be simultaneously constructed and analysed.
This approach has been followed so far in order to construct analoguesof the oscillator and Kepler–Coulomb systems on spaces with constantcurvature (see the previous works by Ranada, Santander, Carinena, Diacuand our group).
Here we face the construction of the first, to the best of our knowledge,example of a curved integrable Henon–Heiles system on S2 and H2
and of the full series of RDG integrable potentials.
Details in arXiv:1411.2033 (Nonlinearity) and arXiv:1503.09187 (JPCS).
4 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
2. Integrable Henon-Heiles Hamiltonians andRDG potentials
5 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The Henon-Heiles Hamiltonian
The Henon-Heiles Hamiltonian system, [M. Henon, C. Heiles (1964)]
H =1
2(p2
1 + p22) +
1
2(q2
1 + q22) + α
„q2
1q2 −1
3q3
2
«was introduced in order to model a Newtonian axially-symmetric galacticsystem, and it was soon considered as the paradigm of a two-dimensional(2D) system that exhibits chaotic behaviour.
Multiparameter generalization
H =1
2(p2
1 + p22) + Ω1q
21 + Ω2q
22 + α
“q2
1q2 + βq32
”.
The only Liouville-integrable cases were shown to be the so–called
Sawada–Kotera system: β = 1/3 and Ω1 = Ω2.
Korteweg–de Vries (KdV) system: β = 2 and Ω1,Ω2 arbitrary.
Kaup–Kupershmidt system: β = 16/3 and Ω2 = 16 Ω1.
[T. Bountis, H. Segur, F. Vivaldi (1982), J. Hietarinta (1983), A. P. Fordy(1983), S. R. Wojciechowski (1984)]
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Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The Henon-Heiles Hamiltonian
The Henon-Heiles Hamiltonian system, [M. Henon, C. Heiles (1964)]
H =1
2(p2
1 + p22) +
1
2(q2
1 + q22) + α
„q2
1q2 −1
3q3
2
«was introduced in order to model a Newtonian axially-symmetric galacticsystem, and it was soon considered as the paradigm of a two-dimensional(2D) system that exhibits chaotic behaviour.
Multiparameter generalization
H =1
2(p2
1 + p22) + Ω1q
21 + Ω2q
22 + α
“q2
1q2 + βq32
”.
The only Liouville-integrable cases were shown to be the so–called
Sawada–Kotera system: β = 1/3 and Ω1 = Ω2.
Korteweg–de Vries (KdV) system: β = 2 and Ω1,Ω2 arbitrary.
Kaup–Kupershmidt system: β = 16/3 and Ω2 = 16 Ω1.
[T. Bountis, H. Segur, F. Vivaldi (1982), J. Hietarinta (1983), A. P. Fordy(1983), S. R. Wojciechowski (1984)]
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Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The three integrable Euclidean Henon-Heiles systems
• Sawada–Kotera HH system (separable in rotated Euclidean coordinates):
H =1
2(p2
1 + p22) + Ω
“q2
1 + q22
”+ α
„q2
1q2 +1
3q3
2
«• KdV HH system (separable in parabolic coordinates):
H =1
2(p2
1 + p22) + Ω1q
21 + Ω2q
22 + α
“q2
1q2 + 2q32
”• Kaup–Kupershmidt HH system (integral quartic in the momenta):
H =1
2(p2
1 + p22) + Ω
“q2
1 + 16q22
”+ α
„q2
1q2 +16
3q3
2
«
These three Henon-Heiles systems can be thought of as cubic integrableperturbations of oscillators with different frequency relations.
7 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The three integrable Euclidean Henon-Heiles systems
• Sawada–Kotera HH system (separable in rotated Euclidean coordinates):
H =1
2(p2
1 + p22) + Ω
“q2
1 + q22
”+ α
„q2
1q2 +1
3q3
2
«
• KdV HH system (separable in parabolic coordinates):
H =1
2(p2
1 + p22) + Ω1q
21 + Ω2q
22 + α
“q2
1q2 + 2q32
”• Kaup–Kupershmidt HH system (integral quartic in the momenta):
H =1
2(p2
1 + p22) + Ω
“q2
1 + 16q22
”+ α
„q2
1q2 +16
3q3
2
«
These three Henon-Heiles systems can be thought of as cubic integrableperturbations of oscillators with different frequency relations.
7 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The three integrable Euclidean Henon-Heiles systems
• Sawada–Kotera HH system (separable in rotated Euclidean coordinates):
H =1
2(p2
1 + p22) + Ω
“q2
1 + q22
”+ α
„q2
1q2 +1
3q3
2
«• KdV HH system (separable in parabolic coordinates):
H =1
2(p2
1 + p22) + Ω1q
21 + Ω2q
22 + α
“q2
1q2 + 2q32
”
• Kaup–Kupershmidt HH system (integral quartic in the momenta):
H =1
2(p2
1 + p22) + Ω
“q2
1 + 16q22
”+ α
„q2
1q2 +16
3q3
2
«
These three Henon-Heiles systems can be thought of as cubic integrableperturbations of oscillators with different frequency relations.
7 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The three integrable Euclidean Henon-Heiles systems
• Sawada–Kotera HH system (separable in rotated Euclidean coordinates):
H =1
2(p2
1 + p22) + Ω
“q2
1 + q22
”+ α
„q2
1q2 +1
3q3
2
«• KdV HH system (separable in parabolic coordinates):
H =1
2(p2
1 + p22) + Ω1q
21 + Ω2q
22 + α
“q2
1q2 + 2q32
”• Kaup–Kupershmidt HH system (integral quartic in the momenta):
H =1
2(p2
1 + p22) + Ω
“q2
1 + 16q22
”+ α
„q2
1q2 +16
3q3
2
«
These three Henon-Heiles systems can be thought of as cubic integrableperturbations of oscillators with different frequency relations.
7 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The three integrable Euclidean Henon-Heiles systems
• Sawada–Kotera HH system (separable in rotated Euclidean coordinates):
H =1
2(p2
1 + p22) + Ω
“q2
1 + q22
”+ α
„q2
1q2 +1
3q3
2
«• KdV HH system (separable in parabolic coordinates):
H =1
2(p2
1 + p22) + Ω1q
21 + Ω2q
22 + α
“q2
1q2 + 2q32
”• Kaup–Kupershmidt HH system (integral quartic in the momenta):
H =1
2(p2
1 + p22) + Ω
“q2
1 + 16q22
”+ α
„q2
1q2 +16
3q3
2
«
These three Henon-Heiles systems can be thought of as cubic integrableperturbations of oscillators with different frequency relations.
7 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The Euclidean 1:2 KdV Henon-Heiles Hamiltonian
The 1:2 KdV Henon-Heiles Hamiltonian can be regarded as an integrablecubic perturbation of the anisotropic 1:2 oscillator with the followingfrequency relations
Ω1 = Ω, Ω2 = 4Ω.
HKdV1:2 =
1
2(p2
1 + p22) + Ω(q2
1 + 4q22) + α
“q2
1q2 + 2q32
”.
This system is endowed with a constant of motion quadratic in the momenta:
IKdV1:2 = p1(q1p2 − q2p1) + q2
1
h2Ωq2 +
α
4(q2
1 + 4q22)i.
The 1:2 KdV Henon-Heiles potential is just a superposition of two of theso-called Ramani, Dorizzi, Gammaticos (RDG) potentials:
V2(q1, q2) = q21 + 4q2
2 , V3(q1, q2) = 4q21q2 + 8q3
2 .
This fact will be essential for the construction of the curved Hamiltoinan.
8 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The Euclidean 1:2 KdV Henon-Heiles Hamiltonian
The 1:2 KdV Henon-Heiles Hamiltonian can be regarded as an integrablecubic perturbation of the anisotropic 1:2 oscillator with the followingfrequency relations
Ω1 = Ω, Ω2 = 4Ω.
HKdV1:2 =
1
2(p2
1 + p22) + Ω(q2
1 + 4q22) + α
“q2
1q2 + 2q32
”.
This system is endowed with a constant of motion quadratic in the momenta:
IKdV1:2 = p1(q1p2 − q2p1) + q2
1
h2Ωq2 +
α
4(q2
1 + 4q22)i.
The 1:2 KdV Henon-Heiles potential is just a superposition of two of theso-called Ramani, Dorizzi, Gammaticos (RDG) potentials:
V2(q1, q2) = q21 + 4q2
2 , V3(q1, q2) = 4q21q2 + 8q3
2 .
This fact will be essential for the construction of the curved Hamiltoinan.
8 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The Euclidean 1:2 KdV Henon-Heiles Hamiltonian
The 1:2 KdV Henon-Heiles Hamiltonian can be regarded as an integrablecubic perturbation of the anisotropic 1:2 oscillator with the followingfrequency relations
Ω1 = Ω, Ω2 = 4Ω.
HKdV1:2 =
1
2(p2
1 + p22) + Ω(q2
1 + 4q22) + α
“q2
1q2 + 2q32
”.
This system is endowed with a constant of motion quadratic in the momenta:
IKdV1:2 = p1(q1p2 − q2p1) + q2
1
h2Ωq2 +
α
4(q2
1 + 4q22)i.
The 1:2 KdV Henon-Heiles potential is just a superposition of two of theso-called Ramani, Dorizzi, Gammaticos (RDG) potentials:
V2(q1, q2) = q21 + 4q2
2 , V3(q1, q2) = 4q21q2 + 8q3
2 .
This fact will be essential for the construction of the curved Hamiltoinan.
8 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The Euclidean RDG potentials
The RDG potentials Vn(q1, q2) [A. Ramani, B. Dorizzi, B. Grammaticos(1982)], are integrable homogeneous polynomial potentials with degree nthat are separable in parabolic coordinates:
Vn(q1, q2) =
[ n2
]Xi=0
2n−2i
n − i
i
!q2i
1 qn−2i2 , n = 1, 2, . . .
V1(q1, q2) = 2q2,
V2(q1, q2) = q21 + 4q2
2 ,
V3(q1, q2) = 4q21q2 + 8q3
2 ,
V4(q1, q2) = q41 + 12q2
1q22 + 16q4
2 .
The integral of the motion Ln for the Vn RDG Hamiltonian contains theprecedent Vn−1 potential (we take V0(q1, q2) := 1)
Hn =1
2(p2
1 + p22) + αnVn, Ln = p1(q1p2 − q2p1) + αnq
21Vn−1
Hn,Ln = 0.
9 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The Euclidean RDG potentials
The RDG potentials Vn(q1, q2) [A. Ramani, B. Dorizzi, B. Grammaticos(1982)], are integrable homogeneous polynomial potentials with degree nthat are separable in parabolic coordinates:
Vn(q1, q2) =
[ n2
]Xi=0
2n−2i
n − i
i
!q2i
1 qn−2i2 , n = 1, 2, . . .
V1(q1, q2) = 2q2,
V2(q1, q2) = q21 + 4q2
2 ,
V3(q1, q2) = 4q21q2 + 8q3
2 ,
V4(q1, q2) = q41 + 12q2
1q22 + 16q4
2 .
The integral of the motion Ln for the Vn RDG Hamiltonian contains theprecedent Vn−1 potential (we take V0(q1, q2) := 1)
Hn =1
2(p2
1 + p22) + αnVn, Ln = p1(q1p2 − q2p1) + αnq
21Vn−1
Hn,Ln = 0.
9 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The Euclidean RDG potentials and the KdV HH system
The RDG potentials can be superposed by preserving integrability
HM =1
2
“p2
1 + p22
”+
MXn=1
αnVn, LM = p1(q1p2 − q2p1) + q21
MXn=1
αnVn−1.
The 1:2 KdV HH Hamiltonian as an RDG system
By setting
M = 3, α1 = 0, α2 = Ω, α3 =α
4.
we get an integrable cubic perturbation of the 1 : 2 anisotropic oscillator
HKdV1:2 =
1
2(p2
1 + p22) + α2V2 + α3V3,
LKdV1:2 = p1(q1p2 − q2p1) + q2
1 (α2V1 + α3V2) ,
such that LKdV1:2 := IKdV
1:2 , and the integral involves also V1(q1, q2) = 2q2.
10 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The Euclidean RDG potentials and the KdV HH system
The RDG potentials can be superposed by preserving integrability
HM =1
2
“p2
1 + p22
”+
MXn=1
αnVn, LM = p1(q1p2 − q2p1) + q21
MXn=1
αnVn−1.
The 1:2 KdV HH Hamiltonian as an RDG system
By setting
M = 3, α1 = 0, α2 = Ω, α3 =α
4.
we get an integrable cubic perturbation of the 1 : 2 anisotropic oscillator
HKdV1:2 =
1
2(p2
1 + p22) + α2V2 + α3V3,
LKdV1:2 = p1(q1p2 − q2p1) + q2
1 (α2V1 + α3V2) ,
such that LKdV1:2 := IKdV
1:2 , and the integral involves also V1(q1, q2) = 2q2.
10 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Integrable rational perturbations of the Euclidean RDG potentials
Proposition. [A.B., A. Blasco (2010)] The multiparametric family ofperturbations of the RDG system given by
H(M,R) =1
2
“p2
1 + p22
”+
MXn=1
αnVn + α0V0 +RX
n=1
λnVn−1
q2n1
=1
2
“p2
1 + p22
”+
MXn=1
[ n2
]Xi=0
αn2n−2i
n − i
i
!q2i
1 qn−2i2 + α0
+RX
n=1
[ n−12
]Xi=0
λn2n−1−2i
n − 1− i
i
!qn−1−2i
2
q2(n−i)1
,
where αn, α0 and λn are arbitrary real constants, is integrable for any indicesM,R ∈ N+. The corresponding integral of the motion reads
L(M,R) = p1(q1p2 − q2p1) + q21
MXn=1
αnVn−1 + α0V0 −RX
n=1
λnVn
q2n1
,
where Vn are the RDG potentials and V0(q1, q2) := 1.
11 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Integrable rational perturbations of the Euclidean RDG potentials
Example. If we set M = 3 and R = 4 we obtain the following integrableperturbation of the KdV Henon–Heiles Hamiltonian:
H(3,4) =1
2
“p2
1 + p22
”+ α1V1 + α2V2 + α3V3 + α0V0 + λ1
V0
q21
+ λ2V1
q41
+ λ3V2
q61
+ λ4V3
q81
=1
2
“p2
1 + p22
”+ α1 (2q2) + α2
“q2
1 + 4q22
”+ α3
“4q2
1q2 + 8q32
”+ α0
+λ11
q21
+ λ22q2
q41
+ λ3q2
1 + 4q22
q61
+ λ44q2
1q2 + 8q32
q81
,
which Poisson-commutes with
L(3,4) = p1(q1p2 − q2p1) + q21 (α1V0 + α2V1 + α3V2) + α0V0
−„λ1V1
q21
+ λ2V2
q41
+ λ3V3
q61
+ λ4V4
q81
«.
Recall that the λ1-potential behaves as a centrifugal barrier on E2 when λ1 > 0.
12 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The method
• Use projective coordinates in order to describe the free motionon S2 and H2.
• Construct an integrable curved anisotropic 1:2 oscillator andtake it as the curved V2 RDG potential.
• Construct the full family of curved RDG potentials on S2 andH2.
• Show that the curved RDG potentials can be superposed bypreserving integrability.
• Obtain the curved 1:2 KdV Henon-Heiles system as aparticular case of the later system.
13 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The method
• Use projective coordinates in order to describe the free motionon S2 and H2.
• Construct an integrable curved anisotropic 1:2 oscillator andtake it as the curved V2 RDG potential.
• Construct the full family of curved RDG potentials on S2 andH2.
• Show that the curved RDG potentials can be superposed bypreserving integrability.
• Obtain the curved 1:2 KdV Henon-Heiles system as aparticular case of the later system.
13 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The method
• Use projective coordinates in order to describe the free motionon S2 and H2.
• Construct an integrable curved anisotropic 1:2 oscillator andtake it as the curved V2 RDG potential.
• Construct the full family of curved RDG potentials on S2 andH2.
• Show that the curved RDG potentials can be superposed bypreserving integrability.
• Obtain the curved 1:2 KdV Henon-Heiles system as aparticular case of the later system.
13 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The method
• Use projective coordinates in order to describe the free motionon S2 and H2.
• Construct an integrable curved anisotropic 1:2 oscillator andtake it as the curved V2 RDG potential.
• Construct the full family of curved RDG potentials on S2 andH2.
• Show that the curved RDG potentials can be superposed bypreserving integrability.
• Obtain the curved 1:2 KdV Henon-Heiles system as aparticular case of the later system.
13 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The method
• Use projective coordinates in order to describe the free motionon S2 and H2.
• Construct an integrable curved anisotropic 1:2 oscillator andtake it as the curved V2 RDG potential.
• Construct the full family of curved RDG potentials on S2 andH2.
• Show that the curved RDG potentials can be superposed bypreserving integrability.
• Obtain the curved 1:2 KdV Henon-Heiles system as aparticular case of the later system.
13 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
3. Projective geometry and dynamicson S2 and H2
14 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Geometry and geodesic dynamics on S2 and H2
Let us consider the one-parameter family of 3D real Lie algebras soκ(3) withcommutation relations and Casimir invariant given by
[J12, J01] = J02, [J12, J02] = −J01, [J01, J02] = κJ12,
C = J201 + J2
02 + κJ212,
where κ is a real parameter.
The 2D homogeneous spaces SOκ(3)/SO(2), where SOκ(3) is the Lie groupof soκ(3) and SO(2) = 〈J12〉, have constant Gaussian curvature given by κ.
The three constant curvature spaces
κ > 0 : Sphere κ = 0 : Euclidean plane κ < 0 : Hyperbolic space
S2 = SO(3)/SO(2) E2 = ISO(2)/SO(2) H2 = SO(2, 1)/SO(2)
15 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Geometry and geodesic dynamics on S2 and H2
Let us consider the one-parameter family of 3D real Lie algebras soκ(3) withcommutation relations and Casimir invariant given by
[J12, J01] = J02, [J12, J02] = −J01, [J01, J02] = κJ12,
C = J201 + J2
02 + κJ212,
where κ is a real parameter.
The 2D homogeneous spaces SOκ(3)/SO(2), where SOκ(3) is the Lie groupof soκ(3) and SO(2) = 〈J12〉, have constant Gaussian curvature given by κ.
The three constant curvature spaces
κ > 0 : Sphere κ = 0 : Euclidean plane κ < 0 : Hyperbolic space
S2 = SO(3)/SO(2) E2 = ISO(2)/SO(2) H2 = SO(2, 1)/SO(2)
15 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
These 2D spaces can be embedded in R3 provided their ambient or Weierstrasscoordinates (x0, x1, x2) satisfy
x20 + κ(x2
1 + x22 ) = 1.
If we consider a central projection with pole (0, 0, 0) ∈ R3 that maps apoint of the 2D space (x0, x1, x2) ∈ R3 to the 2D plane given by(1, q1, q2) ∈ R3, we obtain that the Beltrami projective coordinatesq = (q1, q2) ∈ R2 are defined by the relations
x0 =1p
1 + κq2, x =
qp1 + κq2
, q =x
x0,
where x = (x1, x2) and q2 = q21 + q2
2 .
The Beltrami conjugate momenta will be by p = (p1, p2) and we alsodenote
p2 = p21 + p2
2 , q · p = q1p1 + q2p2.
These canonical momenta can be computed by writing in Beltramicoordinates the Lagrangian for a free particle that moves on the 2Dcurved space.
16 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
These 2D spaces can be embedded in R3 provided their ambient or Weierstrasscoordinates (x0, x1, x2) satisfy
x20 + κ(x2
1 + x22 ) = 1.
If we consider a central projection with pole (0, 0, 0) ∈ R3 that maps apoint of the 2D space (x0, x1, x2) ∈ R3 to the 2D plane given by(1, q1, q2) ∈ R3, we obtain that the Beltrami projective coordinatesq = (q1, q2) ∈ R2 are defined by the relations
x0 =1p
1 + κq2, x =
qp1 + κq2
, q =x
x0,
where x = (x1, x2) and q2 = q21 + q2
2 .
The Beltrami conjugate momenta will be by p = (p1, p2) and we alsodenote
p2 = p21 + p2
2 , q · p = q1p1 + q2p2.
These canonical momenta can be computed by writing in Beltramicoordinates the Lagrangian for a free particle that moves on the 2Dcurved space.
16 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
These 2D spaces can be embedded in R3 provided their ambient or Weierstrasscoordinates (x0, x1, x2) satisfy
x20 + κ(x2
1 + x22 ) = 1.
If we consider a central projection with pole (0, 0, 0) ∈ R3 that maps apoint of the 2D space (x0, x1, x2) ∈ R3 to the 2D plane given by(1, q1, q2) ∈ R3, we obtain that the Beltrami projective coordinatesq = (q1, q2) ∈ R2 are defined by the relations
x0 =1p
1 + κq2, x =
qp1 + κq2
, q =x
x0,
where x = (x1, x2) and q2 = q21 + q2
2 .
The Beltrami conjugate momenta will be by p = (p1, p2) and we alsodenote
p2 = p21 + p2
2 , q · p = q1p1 + q2p2.
These canonical momenta can be computed by writing in Beltramicoordinates the Lagrangian for a free particle that moves on the 2Dcurved space.
16 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The central (gnomonic) projective plane for the hemisphere
17 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The central projection for (a half of) the hyperbolic space
18 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The kinetic energy in Beltrami coordinates
A symplectic realization of the Poisson soκ(3) algebra
J12, J01 = J02, J12, J02 = −J01, J01, J02 = κJ12,
in terms of the Beltrami canonical variables (q, p) turns out to be
J0i = pi + κ(q · p)qi , i = 1, 2; J12 = q1p2 − q2p1.
The kinetic energy Tκ for a particle moving on these curved paces can bewritten as the symplectic realization of the Casimir function, namely
Tκ ≡1
2C =
1
2(J2
01 + J202 + κJ2
12) =1
2
“1 + κq2
”“p2 + κ(q · p)2
”.
Notice that the flat/Euclidean limit κ→ 0 of the above expressions leads to
x0 = 1, x = q, J0i = pi , J12 = q1p2 − q2p1, T =1
2p2.
19 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The kinetic energy in Beltrami coordinates
A symplectic realization of the Poisson soκ(3) algebra
J12, J01 = J02, J12, J02 = −J01, J01, J02 = κJ12,
in terms of the Beltrami canonical variables (q, p) turns out to be
J0i = pi + κ(q · p)qi , i = 1, 2; J12 = q1p2 − q2p1.
The kinetic energy Tκ for a particle moving on these curved paces can bewritten as the symplectic realization of the Casimir function, namely
Tκ ≡1
2C =
1
2(J2
01 + J202 + κJ2
12) =1
2
“1 + κq2
”“p2 + κ(q · p)2
”.
Notice that the flat/Euclidean limit κ→ 0 of the above expressions leads to
x0 = 1, x = q, J0i = pi , J12 = q1p2 − q2p1, T =1
2p2.
19 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
4. Curved RDG and Henon–Heiles potentials
20 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Starting point: a curved anisotropic oscillator
Proposition. [A.B., A. Blasco, F. J. Herranz, F. Musso, (2014)] Let Hκ be thefollowing Hamiltonian written in terms of Beltrami variables as
Hκ =
Tκz | 1
2
“1 + κq2
”“p2 + κ(q · p)2
”+
VΩ1,Ω2κz |
Ω1q2
1
1 + κq22
+ Ω2q2
2(1 + κq2)
(1 + κq22)(1− κq2
2)2,
For any value of the real constants κ, Ω1 and Ω2, the Hamiltonian is endowedwith a constant of motion given by
Iκ =1
2
“J2
01 + κJ212
”+ Ω1
q21(1 + κq2
2)
(1− κq22)2
+ κ (Ω2 − 4Ω1)q2
1q22
(1 + κq22)(1− κq2
2)2.
The function VΩ1,Ω2κ (q1, q2), is an integrable curved anisotropic oscillator
potential with arbitrary frequencies.
In the Euclidean space with κ = 0 these expressions reduce to:
H0 = T0 + VΩ1,Ω20 =
1
2p2 + Ω1q
21 + Ω2q
22 , I0 =
1
2p2
1 + Ω1q21 .
21 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Starting point: a curved anisotropic oscillator
Proposition. [A.B., A. Blasco, F. J. Herranz, F. Musso, (2014)] Let Hκ be thefollowing Hamiltonian written in terms of Beltrami variables as
Hκ =
Tκz | 1
2
“1 + κq2
”“p2 + κ(q · p)2
”+
VΩ1,Ω2κz |
Ω1q2
1
1 + κq22
+ Ω2q2
2(1 + κq2)
(1 + κq22)(1− κq2
2)2,
For any value of the real constants κ, Ω1 and Ω2, the Hamiltonian is endowedwith a constant of motion given by
Iκ =1
2
“J2
01 + κJ212
”+ Ω1
q21(1 + κq2
2)
(1− κq22)2
+ κ (Ω2 − 4Ω1)q2
1q22
(1 + κq22)(1− κq2
2)2.
The function VΩ1,Ω2κ (q1, q2), is an integrable curved anisotropic oscillator
potential with arbitrary frequencies.
In the Euclidean space with κ = 0 these expressions reduce to:
H0 = T0 + VΩ1,Ω20 =
1
2p2 + Ω1q
21 + Ω2q
22 , I0 =
1
2p2
1 + Ω1q21 .
21 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
A curved 1:2 anisotropic oscillator
We need a suitable curved analogue of the 1:2 anisotropic oscillator
H2 =1
2(p2
1 + p22) + Ω(q2
1 + 4q22),
which is given by the previous Proposition with Ω2 = 4Ω1 = 4Ω:
Proposition. The Hamiltonian
H1:2κ =
1
2
“1 + κq2
”“p2 + κ(q · p)2
”+ Ω
q21(1 + κq2
2) + 4q22
(1− κq22)2
defines a superintegrable system for any value of κ and Ω, and itscorresponding two functionally independent integrals of motion are given by
I1:2κ =
1
2
“J2
01 + κJ212
”+ Ω
q21(1 + κq2
2)
(1− κq22)2
, L1:2κ = J01J12 + 2Ω
q21q2
(1− κq22)2
.
This system is just the well-known superintegrable curved 1:2 oscillatorpreviously introduced in [M. Ranada and M. Santander, (1999)].
22 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
A curved 1:2 anisotropic oscillator
We need a suitable curved analogue of the 1:2 anisotropic oscillator
H2 =1
2(p2
1 + p22) + Ω(q2
1 + 4q22),
which is given by the previous Proposition with Ω2 = 4Ω1 = 4Ω:
Proposition. The Hamiltonian
H1:2κ =
1
2
“1 + κq2
”“p2 + κ(q · p)2
”+ Ω
q21(1 + κq2
2) + 4q22
(1− κq22)2
defines a superintegrable system for any value of κ and Ω, and itscorresponding two functionally independent integrals of motion are given by
I1:2κ =
1
2
“J2
01 + κJ212
”+ Ω
q21(1 + κq2
2)
(1− κq22)2
, L1:2κ = J01J12 + 2Ω
q21q2
(1− κq22)2
.
This system is just the well-known superintegrable curved 1:2 oscillatorpreviously introduced in [M. Ranada and M. Santander, (1999)].
22 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
A curved 1:2 anisotropic oscillator
Why this curved 1:2 oscillator?
The flat limit of the L1:2κ integral corresponds to the case n = 2 of the
Euclidean RDG potentials with α2 = Ω:
H1:2κ→0 ≡ H2 =
1
2(p2
1 + p22) + α2V2, V2 = q2
1 + 4q22 ,
L1:2κ→0 ≡ L2 = p1(q1p2 − q2p1) + α2q
21V1, V1 = 2q2.
Ansatz: There exists a family of curved RDG potentials Vκ,i for which thepotential in H1:2
κ is just the Vκ,2 term.
H1:2κ ≡ Hκ,2 = Tκ + α2Vκ,2, Vκ,2 =
q21(1 + κq2
2) + 4q22
(1− κq22)2
,
L1:2κ ≡ Lκ,2 = J01J12 + α2
q21
1 + κq2Vκ,1 provided Vκ,1 =
2q2(1 + κq2)
(1− κq22)2
Hκ,2,Lκ,2 = 0.
Can we say that Vκ,1 is a member of the curved RDG family?
23 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
A curved 1:2 anisotropic oscillator
Why this curved 1:2 oscillator?
The flat limit of the L1:2κ integral corresponds to the case n = 2 of the
Euclidean RDG potentials with α2 = Ω:
H1:2κ→0 ≡ H2 =
1
2(p2
1 + p22) + α2V2, V2 = q2
1 + 4q22 ,
L1:2κ→0 ≡ L2 = p1(q1p2 − q2p1) + α2q
21V1, V1 = 2q2.
Ansatz: There exists a family of curved RDG potentials Vκ,i for which thepotential in H1:2
κ is just the Vκ,2 term.
H1:2κ ≡ Hκ,2 = Tκ + α2Vκ,2, Vκ,2 =
q21(1 + κq2
2) + 4q22
(1− κq22)2
,
L1:2κ ≡ Lκ,2 = J01J12 + α2
q21
1 + κq2Vκ,1 provided Vκ,1 =
2q2(1 + κq2)
(1− κq22)2
Hκ,2,Lκ,2 = 0.
Can we say that Vκ,1 is a member of the curved RDG family?23 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The curved RDG potentials
If that is the case, we would have the following expressions for the n = 1 case:
Hκ,1 = Tκ + α1Vκ,1
Hκ,1 =1
2
“1 + κq2
”“p2 + κ(q · p)2
”+ α1
2q2(1 + κq2)
(1− κq22)2
and the constant of the motion for this potential can be again written in theRDG form as
Lκ,1 = J01J12 + α1q2
1
1 + κq2Vκ,0,
which leads us to define the curved version of the constant potential V0 = 1 as
Vκ,0 :=(1 + κq2
2)(1 + κq2)
(1− κq22)
2 ,
which, surprisingly, is a non-constant potential function.
From these results, the complete series for curved RDG potentials can beobtained.
24 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The curved RDG potentials
If that is the case, we would have the following expressions for the n = 1 case:
Hκ,1 = Tκ + α1Vκ,1
Hκ,1 =1
2
“1 + κq2
”“p2 + κ(q · p)2
”+ α1
2q2(1 + κq2)
(1− κq22)2
and the constant of the motion for this potential can be again written in theRDG form as
Lκ,1 = J01J12 + α1q2
1
1 + κq2Vκ,0,
which leads us to define the curved version of the constant potential V0 = 1 as
Vκ,0 :=(1 + κq2
2)(1 + κq2)
(1− κq22)
2 ,
which, surprisingly, is a non-constant potential function.
From these results, the complete series for curved RDG potentials can beobtained.
24 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The curved RDG potentials
If that is the case, we would have the following expressions for the n = 1 case:
Hκ,1 = Tκ + α1Vκ,1
Hκ,1 =1
2
“1 + κq2
”“p2 + κ(q · p)2
”+ α1
2q2(1 + κq2)
(1− κq22)2
and the constant of the motion for this potential can be again written in theRDG form as
Lκ,1 = J01J12 + α1q2
1
1 + κq2Vκ,0,
which leads us to define the curved version of the constant potential V0 = 1 as
Vκ,0 :=(1 + κq2
2)(1 + κq2)
(1− κq22)
2 ,
which, surprisingly, is a non-constant potential function.
From these results, the complete series for curved RDG potentials can beobtained.
24 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Proposition. [A.B., A. Blasco, F. J. Herranz, F. Musso (2015)]
The RDG potentials on the sphere S2 and the hyperbolic space H2 aredefined in Beltrami coordinates by
Vκ,n(q1, q2) =
„1 + κq2
1− κq22
«2 [ n2
]Xi=0
2n−2i
n − i
i
! q1p
1 + κq2
!2i
ׄ
1− i
n − i
»κq2
1
1 + κq2
–«„q2
1 + κq2
«n−2i
with n = 1, 2 . . . . Each RDG Hamiltonian
Hκ,n = Tκ + αnVκ,n,
is integrable, since it is endowed with a constant of motion Lκ,n which isquadratic in the momenta
Lκ,n = J01J12 + αnq2
1
1 + κq2Vκ,n−1, Hκ,n,Lκ,n = 0,
where Vκ,0 :=(1 + κq2
2)(1 + κq2)
(1− κq22)
2 .
25 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Moreover, the following result can be straightforwardly proven:
Proposition. The superposition of curved RDG potentials is integrable.
The Hamiltonian formed by the linear superposition of any number of curvedRDG potentials
Hκ,(M) = Tκ +MX
n=1
αnVκ,n , M, n ∈ N+,
Poisson-commutes with the function
Lκ,(M) = J01J12 +q2
1
1 + κq2
MXn=1
αnVκ,n−1 .
As a straightforward consequence, the curved counterpart of theHenon–Heiles KdV Hamiltonian on S2 and H2 along with its integral can beobtained from this Proposition as the particular case Hκ,(3).
26 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Moreover, the following result can be straightforwardly proven:
Proposition. The superposition of curved RDG potentials is integrable.
The Hamiltonian formed by the linear superposition of any number of curvedRDG potentials
Hκ,(M) = Tκ +MX
n=1
αnVκ,n , M, n ∈ N+,
Poisson-commutes with the function
Lκ,(M) = J01J12 +q2
1
1 + κq2
MXn=1
αnVκ,n−1 .
As a straightforward consequence, the curved counterpart of theHenon–Heiles KdV Hamiltonian on S2 and H2 along with its integral can beobtained from this Proposition as the particular case Hκ,(3).
26 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The curved 1:2 KdV Hamiltonian
Corollary. The curved 1:2 KdV Hamiltonian is given by Hκ,(3) with
α1 = 0, α2 = Ω, and α3 =α
4.
Explicitly,
HKdVκ,1:2 = Tκ + ΩVκ,2 +
α
4Vκ,3,
where
Vκ,2 =q2
1(1 + κq22) + 4q2
2
(1− κq22)2
Vκ,3 =2q2(2 + κq2
1)(q21 + 2q2
2)
(1− κq22)2(1 + κq2)
.
The constant of motion reads
LKdVκ,1:2 = J01J12 +
q21
1 + κq2(α2Vκ,1 + α3Vκ,2) .
where
Vκ,1 =2q2(1 + κq2)
(1− κq22)2
.
27 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The curved HH potential in projective coordinates
Figure: Plot (a) represents the flat 1 : 2 (superintegrable) anharmonic oscillator potential on E2 with no
Henon–Heiles term. In case (b) the Henon–Heiles cubic term is added on E2. The (superintegrable) curved 1 : 2
anharmonic oscillator potential on H2 with no Henon–Heiles term (κ = −1, Ω = 1 and α = 0) is represented incase (c). Finally, when the integrable curved Henon–Heiles term is added (κ = −1, Ω = 1 and α = 2) the plot
(d) is obtained on H2.28 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
The potentials in the RDG series for n = 0, 1, . . . , 3 on E2 in Cartesiancoordinates q and their curved counterpart on S2 and H2 in ambient
coordinates such that x20 +κx2 = 1. Recall that x0 = 1 and x = q when κ = 0.
E2: Cartesian coordinates q S2 and H2: Ambient coordinates (x0, x)
V0 = 1 Vκ,0 =1− κx2
1
(x20 − κx2
2 )2
V1 = 2q2 Vκ,1 =2x0x2
(x20 − κx2
2 )2
V2 = q21 + 4q2
2 Vκ,2 =x2
1 (1− κx21 ) + 4x2
0 x22
(x20 − κx2
2 )2
V3 = 4q21q2 + 8q3
2 Vκ,3 =4x0x
21 x2(1− 1
2κx2
1 ) + 8x30 x3
2
(x20 − κx2
2 )2
29 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Concluding remarks
Projective coordinates are computationally very useful from theviewpoint of algebraic integrability (polynomial kinetic energy andrational functions for the potentials).
The method here presented is widely applicable:
Integrable perturbations of the KdV HH system can begeneralized to the curved case. The Vκ,0 function is essential!
The curved KdV HH system for arbitrary Ω1 and Ω2 can beconstructed.
The curved analogue of the Sawada–Kotera case is also underinvestigation. The Kaup–Kuperschdmit case is more difficult(quartic integral).
The corresponding quantum HH systems are also worth to be studied...
Any connection with a (new and hopefully integrable) deformation ofthe KdV equation?
30 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Concluding remarks
Projective coordinates are computationally very useful from theviewpoint of algebraic integrability (polynomial kinetic energy andrational functions for the potentials).
The method here presented is widely applicable:
Integrable perturbations of the KdV HH system can begeneralized to the curved case. The Vκ,0 function is essential!
The curved KdV HH system for arbitrary Ω1 and Ω2 can beconstructed.
The curved analogue of the Sawada–Kotera case is also underinvestigation. The Kaup–Kuperschdmit case is more difficult(quartic integral).
The corresponding quantum HH systems are also worth to be studied...
Any connection with a (new and hopefully integrable) deformation ofthe KdV equation?
30 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Concluding remarks
Projective coordinates are computationally very useful from theviewpoint of algebraic integrability (polynomial kinetic energy andrational functions for the potentials).
The method here presented is widely applicable:
Integrable perturbations of the KdV HH system can begeneralized to the curved case. The Vκ,0 function is essential!
The curved KdV HH system for arbitrary Ω1 and Ω2 can beconstructed.
The curved analogue of the Sawada–Kotera case is also underinvestigation. The Kaup–Kuperschdmit case is more difficult(quartic integral).
The corresponding quantum HH systems are also worth to be studied...
Any connection with a (new and hopefully integrable) deformation ofthe KdV equation?
30 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Concluding remarks
Projective coordinates are computationally very useful from theviewpoint of algebraic integrability (polynomial kinetic energy andrational functions for the potentials).
The method here presented is widely applicable:
Integrable perturbations of the KdV HH system can begeneralized to the curved case. The Vκ,0 function is essential!
The curved KdV HH system for arbitrary Ω1 and Ω2 can beconstructed.
The curved analogue of the Sawada–Kotera case is also underinvestigation. The Kaup–Kuperschdmit case is more difficult(quartic integral).
The corresponding quantum HH systems are also worth to be studied...
Any connection with a (new and hopefully integrable) deformation ofthe KdV equation?
30 / 32
Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
Concluding remarks
Projective coordinates are computationally very useful from theviewpoint of algebraic integrability (polynomial kinetic energy andrational functions for the potentials).
The method here presented is widely applicable:
Integrable perturbations of the KdV HH system can begeneralized to the curved case. The Vκ,0 function is essential!
The curved KdV HH system for arbitrary Ω1 and Ω2 can beconstructed.
The curved analogue of the Sawada–Kotera case is also underinvestigation. The Kaup–Kuperschdmit case is more difficult(quartic integral).
The corresponding quantum HH systems are also worth to be studied...
Any connection with a (new and hopefully integrable) deformation ofthe KdV equation?
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Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
THANKS FOR YOUR ATTENTION
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Introduction Henon-Heiles Hamiltonians Projective geometry and dynamics Curved RDG and HH potentials Conclusions
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