Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 710274, 26 pages
doi:10.1155/2011/710274
Research Article
Geometry of Hamiltonian Dynamics with
Conformal Eisenhart Metric
Linyu Peng,1 Huafei Sun,2 and Xiao Sun3
1
Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK
Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
3
School of Automation, Beijing Institute of Technology, Beijing 100081, China
2
Correspondence should be addressed to Huafei Sun, huaf.sun@gmail.com
Received 3 December 2010; Revised 19 March 2011; Accepted 12 April 2011
Academic Editor: Heinrich Begehr
Copyright q 2011 Linyu Peng et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After
investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the
geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the
conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow
constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the
conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the
flow of the famous models, an N degrees of freedom linear Hamiltonian system and the HénonHeiles model are given, and in a special case, numerical solutions of the conformal geodesics,
the generalized momenta, and the Jacobi field along the geodesics of the Hénon-Heiles model are
obtained. And the numerical results for the Hénon-Heiles model show us the instability of the
associated geodesic spreads.
1. Introduction
As the development of differential geometry, symplectic geometry, and Riemannian
geometry, the field of dynamics has been studied from the point of geometry, and many
significant results have been reached, especially in Hamiltonian dynamics. In order to
consider the stability of the dynamical systems, it is usually necessary and efficient for one to
study the geometric structure of them.
In 1, 2, the applications of methods used in classical differential geometry are
concerned to study the chaotic dynamics of Hamiltonian systems. After getting the geodesics
in the configuration space which is equipped with a suitable metric, the geometry theory
of chaotic dynamics is investigated and applied into studying the Kepler problem and
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the restricted three body problem. Iwai and Yamaoka 3 proposed a problem, that is how
a many-body system behaves in a neighborhood of a collinear configuration and dealt with
the behavior of boundaries for three bodies in space. From the viewpoint of Riemannian
geometry, similarly to the method for obtaining Jacobi equation for geodesic deviations, the
equations of the variational vector obtained along the solution of the equations of motion, are
used to study the boundary behavior, and small vibrations near an equilibrium of dynamical
systems. Moreover, geometry is also applied to other fields of dynamics, such as Hamiltonian
and gradient control system 4, nonlocal Hamiltonian operators 5, hydrodynamics and
realization 6, fluid mechanics 7, quantum systems 8, reaction dynamics 9, and other
dynamics systems 10–12.
In the present paper, in Section 2, the foundation of the Hamiltonian mechanics,
the Eisenhart metric of a manifold M × Ê2 , where M is the configuration space, and the
corresponding geometric structure are introduced. We study the conformal Eisenhart metric
in Section 3, and the curvatures, the geodesic equations, the equations of Jacobi field along
the geodesics, and the equations of a certain flow for the classical Hamiltonian dynamics are
obtained. Moreover, in Section 4, the conformal geometric structures of two models are given,
and numerical simulations for one of the models are shown. The final Section 5 is devoted to
the conclusion.
2. Geometry and Dynamics
2.1. Geometry and Flow
To make the paper readable, we recall some useful background of Riemannian geometry. The
readers could refer to 13–15 for more information. Let MN , g, D be a smooth manifold
with dimension N, where g is a nondegenerate metric defined on the vector field M, and
D is the affine connection, which is defined as
DX Y X
j
∂Y i
Γijk Y k
∂xj
∂
,
∂xi
∀X, Y ∈ M.
2.1
The coefficients Γkij of the connection D can be represented in terms of the metric tensor g by
Γkij 1 kl g ∂j gil ∂i gjl − ∂l gij ,
2
2.2
where ∂i means ∂/∂xi , and the Einstein summation convention is used. For any X, Y, Z ∈
M, the curvature tensor is defined by
RX, Y Z DX DY Z − DY DX Z − DX,Y Z,
2.3
where ·, · is the Lie bracket on M defined as X, Y XY − Y X. Then the components of
the curvature tensor can be given by
Rlkij ∂i Γlkj − ∂j Γlki Γhkj Γlhi − Γhki Γlhj .
2.4
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3
The Ricci curvature is defined by
Rij g kl Riklj ,
2.5
where Riklj gks Rsilj . The scalar curvature is defined by
R g ij Rij .
2.6
A curve ξt on M is said to be geodesic if its tangent ξ̇t is displaced parallel along the curve
ξt, that is,
Dξ̇t ξ̇t 0.
2.7
In local coordinate ξt x1 t, x2 t, . . . , xN t, the geodesic equation satisfies
i
j
d2 x k
k dx dx
0.
Γ
ij
dt dt
dt2
2.8
In addition, the equation for the well-known Jacobi field J J k ∂k satisfies
Dξ̇t Dξ̇t J R ξ̇t, J ξ̇t,
2.9
where ξt is a geodesic on manifold M. It is also called the geodesic derivation equation, as
its close connection with the behavior or completeness of the geodesics. Usually Jacobi field is
used to study the stability of the geodesic spreads, that is, the behavior of the geodesics with
the time parameter t changing, of dynamical systems. In 16, the instability of the geodesic
spreads of the entropic dynamical models is obtained via the study of the Jacobi field. The
readers could also refer to 17 for more about its applications to physical systems.
A metric g defined on M is said to be conformally equivalent to g, if there exists a
function f ∈ C∞ M such that
g ef g.
2.10
Then the equivalent relation between metrics and the equivalence class of g is called its
conformal class denoted by g. In local coordinate, 2.10 becomes
gij ef gij .
2.11
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Hamilton 18 introduced the Ricci flow in 1982, which ultimately led to the proof,
by Perelman, of the Thurston geometrization conjecture and the solution of the Poincaré
conjecture. It is a geometric evolution equation in which one starts with a smooth Riemannian
manifold MN , g0 and evolves its metric by the equation
∂gt
∂t
ij
−2Rij ,
2.12
where gt |t0 g0 and Rij denote the Ricci curvatures with gt .
In this paper, we study a similar flow, which is an evolution equation started from a
smooth manifold MN with nondegenerate metric g, which is not needed to be Riemannian.
2.2. Geometry in Hamiltonian Dynamics
In mechanics 19, a Lagrangian function L of a dynamical system of N degrees of freedom
is usually defined by Lqi , q̇i , t, where q q1 , . . . , qN and q̇ q̇1 , . . . , q̇N are called the
generalized coordinate and generalized velocity, respectively. While a Hamiltonian function
is represented as Hqi , pi , t, where p ∂L/∂q1 , . . . , ∂L/∂qN is generalized momenta. The
space M of the generalized coordinate is called the configuration space.
The Hamiltonian action within t ∈ a, b is
S
b
a
L q, q̇, t dt.
2.13
The Hamiltonian variational principle states
δS 0.
2.14
From the calculus of variation, 2.14 equates to the Euler-Lagrange equations
d ∂L
∂L
− i 0,
dt ∂q̇i
∂q
i 1, . . . , N.
2.15
To pass to the Hamiltonian formalism, we introduce the generalized momenta as
pi ∂L
,
∂q̇i
i 1, . . . , N,
2.16
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5
make the change qi , q̇i → qi , pi , and introduce the Hamiltonian
H qi , pi , t pj q̇j − L qi , q̇i , t .
2.17
Then the Euler-Lagrange equations are equivalent to the Hamiltonian equations, which are
given by
dqi ∂H
,
dt
∂pi
dpi
∂H
− i,
dt
∂q
i 1, . . . , N.
2.18
In this paper, we will only consider the classical Hamiltonian dynamics, whose
Hamiltonian is of the form
N
1
pi2 V q ,
2 i1
H
2.19
where V q is the potential energy.
The energy E of the classical Hamiltonian dynamics is a constant equal to Hamiltonian
H. Considering the manifold M × Ê2 defined by
M × Ê2 q | q q0 , q1 , q2 , . . . , qN , qN1 ,
2.20
t
in which q0 t and qN1 C12 /2t C22 − 0 Ldτ, where C1 , C2 are real numbers, and in the
following, C1 is assumed to be equal to 1.
In this paper, we assume that, the Greek symbols α, β, γ, . . . are from 0 to N 1 and the
Latin symbols i, j, k, . . . are from 1 to N.
The Eisenhart metric cf. 20 of the manifold M × Ê2 is defined as
⎛
−2V 0
⎜
⎜
⎜
⎜
⎜
gαβ ⎜
⎜
⎜
⎜
⎝
···
0
..
.
IN×N
0
1
0
···
⎞
0 1
⎟
0⎟
⎟
⎟
.. ⎟,
.⎟
⎟
⎟
0⎟
⎠
2.21
0 0
where IN×N is the identity matrix, and we know that it is not degenerate as detgαβ −1N3 .
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The inverse matrix of gαβ is
⎛
0 0
⎜
⎜0
⎜
⎜
⎜
αβ
g
⎜ ...
⎜
⎜
⎜0
⎝
···
0 1
⎟
0 ⎟
⎟
⎟
.. ⎟.
. ⎟
⎟
⎟
0 ⎟
⎠
IN×N
1 0
···
⎞
2.22
0 2V
The nonzero coefficients of the Riemanian connection are
∂k V,
Γk00 −ΓN1
0k
2.23
where ∂k means ∂/∂qk .
The nonzero components of the curvature tensor are
R0j0l −∂j ∂l V.
2.24
The nonzero Ricci curvature and the scalar curvature are, respectively, given by
R00 ΔV,
R 0,
2.25
where Δ is the Laplacian operator in the Euclidean space.
The geodesic equations read
d2 q 0
0,
dt2
d2 q i
∂V
− i,
2
dt
∂q
i 1, 2, . . . N,
2.26
d2 qN1
dL
− ,
2
dt
dt
in which, the first and third equations are identical, and the second ones are the equations of
motion for the associated dynamics.
3. Conformal Structure
From now on, in order to study the conformal structure of the manifold M × Ê2 , in which M
is the configuration space, we investigate the conformal Eisenhart metric which states as
g ef g,
3.1
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where g is the Eisenhart metric and f : M × Ê2 →
parameter s is shown as
7
Ê is a smooth function. Then the arc length
d
s2 gαβ dqα dqβ
C12 ef dt2
3.2
ef dt2 .
And the volume element d
v of the manifold under the conformal Eisenhart metric is given
by
d
v
det g dq0 ∧ dq1 ∧ · · · ∧ dqN1
−e
N/21f
dq ∧ dq ∧ · · · ∧ dq
0
1
N1
3.3
.
Proposition 3.1. The conformal and independent components of the curvature tensor are given by
0j0l ef −∂j ∂l V − V ∂j ∂l f 1 V ∂j f∂l f
R
2
δjl
2
2
ef 2∂0 ∂0 f − ∂0 f − 4V ∂0 f∂N1 f − 4V 2 ∂N1 f − 2 ∇f · ∇V
4
2 −2V ∇f ,
0j0N1 ef − 1 ∂0 ∂j f − V ∂j ∂N1 f 1 ∂0 f∂j f 1 V ∂j f∂N1 f − 1 ∂j V ∂N1 f ,
R
2
4
2
2
0jkl δjl ef 1 ∂0 ∂k f 1 ∂k V ∂N1 f − 1 ∂0 f∂k f
R
2
2
4
1
1
1
∂0 ∂l f ∂l V ∂N1 f − ∂0 f∂l f ,
− δjk ef
2
2
4
0jkN1 ef − 1 ∂j ∂k f 1 ∂j f∂k f
R
2
4
2 1
1 1 2
f 1
∂0 ∂N1 f V ∂N1 f ∂0 f∂N1 f ∇f
,
− δjk e
2
2
4
4
8
0N10N1
R
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1 2
−ef ∂0 ∂N1 f − V ∂N1 ∂N1 f ∇f ,
4
0N1kl 0,
R
0N1kN1 ef − 1 ∂k ∂N1 f 1 ∂k f∂N1 f ,
R
2
4
1
1
f 1
f 1
∂i ∂k f − ∂i f∂k f − δil e
∂j ∂k f − ∂j f∂k f
Rijkl δjl e
2
4
2
4
1
1
1
1
∂j ∂l f − ∂j f∂l f − δjk ef
∂i ∂l f − ∂i f∂l f
δik ef
2
4
2
4
2 1 2
1
1 ∂0 f∂N1 f V ∂N1 f ∇f ,
δik δjl − δil δjk ef
2
2
4
ijkN1 δik ef 1 ∂j ∂N1 f − 1 ∂j f∂N1 f − δjk ef 1 ∂i ∂N1 f − 1 ∂i f∂N1 f ,
R
2
4
2
4
iN1kN1 δik ef 1 ∂N1 ∂N1 f − 1 ∂N1 f 2 ,
R
2
4
3.4
where ∇ and , ·, are the gradient operator and the inner product in the Euclidean space, respectively.
Proof. From 2.2, one can get the independent conformal coefficients
0 ∂0 f V ∂N1 f,
Γ
00
k ∂k V V ∂k f,
Γ
00
N1 V ∂0 f 2V 2 ∂N1 f,
Γ
00
k 1 δjk ∂0 f,
Γ
0j
2
0
Γ
0N1 0,
0 1 ∂j f,
Γ
0j
2
N1 −∂j V,
Γ
0j
1
k
Γ
0N1 − ∂k f,
2
0 − 1 δij ∂N1 f,
Γ
ij
2
k 1 δik ∂j f 1 δjk ∂i f − 1 δij ∂k f,
Γ
ij
2
2
2
N1 − 1 δij ∂0 f − δij V ∂N1 f,
Γ
ij
2
1
k
Γ
iN1 δik ∂N1 f,
2
k
Γ
N1N1 0,
N1 −V ∂N1 f,
Γ
0N1
0
Γ
iN1 0,
N1 1 ∂i f,
Γ
iN1
2
0
Γ
N1N1 0,
N1
Γ
N1N1 ∂N1 f.
3.5
αβγ η gβλ R
λαγ η , one can get the conclusion, immediately.
Then from 2.4 and R
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9
Proposition 3.2. The conformal Ricci curvatures of M × Ê2 are given by
00 ΔV V Δf − N ∂0 ∂0 f N ∂0 f 2 NV ∂0 f∂N1 f NV 2 ∂N1 f 2
R
2
4
N 2
N
∇f · ∇V V ∇f 2V ∂0 ∂N1 f 2V 2 ∂N1 ∂N1 f,
2
2
1
1
1
∂0 ∂k f ∂k V ∂N1 f − ∂0 f∂k f ,
−N
2
2
4
0k
R
0N1 − 1 Δf − N 2 ∂0 ∂N1 f − V ∂N1 ∂N1 f
R
2
2
2 1
1 1 2
V ∂N1 f ∂0 f∂N1 f ∇f ,
−N
2
4
4
ik − N ∂i ∂k f N ∂i f∂k f
R
2
4
2 N 2
N
N 1
− δik ∂0 ∂N1 f ∂0 f∂N1 f V ∂N1 f ∇f V ∂N1 ∂N1 f Δf ,
2
2
4
2
iN1 − N ∂i ∂N1 f N ∂i f∂N1 f,
R
2
4
N1N1 − N ∂N1 ∂N1 f N ∂N1 f 2 .
R
2
4
3.6
Proof. From 2.5, and the conformal curvature tensors obtained in Proposition 3.1, we can
get the conclusion, immediately.
Theorem 3.3. The conformal scalar curvature is given by
−e−f N 1Δf NN 1 ∇f 2 2N 1∂0 ∂N1 f
R
4
2
NN 1
NN 1 − e−f
∂0 f∂N1 f 2N 1V ∂N1 ∂N1 f ∂N1 f
.
2
2
3.7
Proof. From 2.6, and the conclusion in Proposition 3.2, we can obtain the conclusion of
Theorem 3.3, immediately.
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Remark 3.4. When V is a smooth function on the configuration space and let f V , we can
get
00 V 1 ΔV N |∇V |2 ,
R
2
0N1 − 1 ΔV − N |∇V |2 ,
R
2
4
ik − N ∂i ∂k V N ∂i V ∂k V − δik 1 ΔV N |∇V |2 ,
R
2
4
2
4
0k 0,
R
3.8
N1N1 0,
iN1 0,
R
R
−e−V N 1ΔV NN 1 |∇V |2 .
R
4
Theorem 3.5. The conformal geodesic equations are
d2 q k
dqk 1
6E − 4V − 1
∂
− ∂k f
f
N1
21 − 2L
dt
2
d t2
dqk
dt
2
1
∂k f ∂k V 0,
E−V 2
3.9
and f satisfies
∂0 f L∂N1 f,
df
1
1
−L
2 E−V −
∂N1 f,
2
dt
4
3.10
where
df
dqi
∂i f
,
dt
dt
i 1, . . . , N,
3.11
L E − 2V.
Proof. From the geodesic equation 2.8, and the coefficients of Riemannian connection in
3.5, we can get the equations of Theorem 3.5.
For the classical Hamiltonian system, L is usually not a constant. Then if f is
constrained in C∞ M, the second equation in 3.10 can be reduced into df/dt 0.
Therefore, we can get the following.
Remark 3.6. When assuming f : M →
d2 q k 1
− ∂k f
2
dt2
Ê, the geodesic equations are
dqk
dt
2
1
∂k f ∂k V 0,
E−V 2
3.12
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11
and f satisfies that
df
0.
dt
Lemma 3.7 Hopf. Let M, g be a compact manifold, f ∈ C∞ M. Then
3.13
M
Δg f 0.
Lemma 3.8. Let f, h ∈ C∞ M, g ef g, then
∇g h e−f ∇g h,
Δg h e−f Δg h N−2 g ∇g f, ∇g h .
2
3.14
Proof. From the definition of gradient and Laplacian of h ∈ C∞ M, one can obtain
∇g h ∂i h
g ij ∂j e−f ∂i hg ij ∂j
e−f ∇g h,
√
1
g ij ∂j h e−N/2f √1 ∂i eN−2/2f Gg ij ∂j h
Δg h ∂i
G
G
G
N −2
1 √
e−f √ ∂i Gg ij ∂j h e−f g ij ∂j h∂i f
2
G
e−f Δg h 3.15
N −2 g ∇g f, ∇g h ,
2
where G detgij .
Theorem 3.9. Let f : M → Ê, M × U1 × U2 be compact, where U1 and U2 are compact subsets of
Ê. Then from Theorem 3.3, Lemmas 3.7 and 3.8, one can get that
M×U1 ×U2
NN 1 2
∇f
e−f N 1Δf 4
M×U1 ×U2
N 1N − 4 −f 2
e ∇f
−
N 1Δg f −
4
M×U1 ×U2
2 N 1N − 4
e−f ∇f ,
4
M×U1 ×U2
−
R
and moreover,
1 when N > 4,
2 when N 4,
3 when N < 4,
M×U1 ×U2
≥ 0,
R
M×U1 ×U2
≡ 0,
R
M×U1 ×U2
≤ 0.
R
3.16
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In addition, when N /
4 and the equality of 3.16 holds, |∇g f|2 e−2f |∇f|2 0, so that f is a
constant function on the configuration space M.
Theorem 3.10. Let f : M → Ê be a smooth function. With the conformal Eisenhart metric, the
equations for Jacobi field along the geodesics, which is the solution of 3.12, are given by
dqj i
d2 J 0
dJ i
0
A
J 0,
∂
f
J
∂
∂
f
i
1
i
j
dt
dt
dt2
dJ 0
dJ j
d2 J i
dJ N1
A
−
∂
A4 J j 0,
A
f
2
3
i
dt
dt
dt
dt2
3.17
dJ j
d2 J N1
dV dJ 0
A
A6 J 0 A7 J i A8 J N1 0,
−
2
5
dt dt
dt
dt2
where
2L 1 2
d2 q i 1 1
∇f ,
∂i f 2 ∇f · ∇V −
2
2
8
dt
1
∂i f,
A2 2∂i V E −
2
A1 A3 ∂j f
dqi
dqj
− ∂i f
,
dt
dt
d2 q k
1
1 2
δik ∂j f δij ∂k f − δkj ∂i f
∂i ∂j V − δij ∇f 2
2
4
dt
dqs dqj
1
∂i V ∂j f ∂j V ∂i f − ∂i ∂j f δij ∇f · ∇V ∂j ∂s f
,
2
dt dt
1
− L ∂j f,
A5 −2∂j V 2
A4 3.18
d2 q j 1 ∇f · ∇V − |∇V |2 ,
2
2
dt
j
dq
dqj
1
− L ∂i ∂j f
,
A7 −2∂i ∂j V
dt
2
dt
A6 −∂j V
A8 d2 q j 1 1
∂j f 2 ∇f · ∇V −
2
2
dt
1 2
∇f .
4
Proof. Firstly, from the Jacobi equation 2.9, we can get the conformal Jacobi equations
represented in components as
β
σ
d2 J α
α dq dJ 2
Γ
βσ
dt dt
dt2
2 β
β
λ
β
λ
β
λ
α dq dq
γ Γ
α dq dq ∂λ Γ
α dq dq − R
α d q Γ
Γ
βσ
γ
λ
βσ
βλσ
βσ
dt dt
dt dt
dt dt
dt2
J σ 0,
3.19
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13
α are given by 2.4. From 2.4, 3.19 turns into
where R
βλσ
β
σ
d2 J α
α dq dJ 2Γ
βσ dt dt
2
dt
α
Γ
βσ
β
λ
d2 qβ γ α dqβ dqλ
α dq dq
∂σ Γ
Γβλ Γγ σ
βλ
2
dt dt
dt dt
dt
J σ 0.
3.20
Furthermore, substituting 3.5 into 3.20, in which f should be constrained in C∞ M but
not C∞ M × Ê2 , we complete the proof of Theorem 3.10.
Now let fu : M × a, b → Ê be a smooth function with fu |u0 0, and gu efu g, that
is, gu |u0 g, where g is the Eisenhart metric. Then we can get the following.
Theorem 3.11. The equations of the flow defined in 2.12 are
efu
2
∂fu
N
−Δfu − ∇fu ,
∂u
2
ΔV N
∇fu · ∇V 0,
2
2∂i ∂k fu − ∂i fu ∂k fu 0,
Δfu − N∂i ∂i fu 3.21
i
/ k,
2 N − 1 N
∇fu 2 0,
∂i fu 2
2
i 1, 2, . . . , N.
Proof. From 2.12, one can obtain
−2R00 −efu
∂fu
2V,
∂u
∂fu
−2Rik e
δik ,
∂u
fu
−2R0i 0,
−2RiN1 0,
−2R0N1 efu
∂fu
,
∂u
3.22
−2RN1N1 0,
where Rαβ are the Ricci curvatures with the metric gu .
Then from the conformal Ricci curvatures obtained in Proposition 3.2, and through a
direct computation, one can get the conclusion.
4. Examples
In this section, the conformal geometric structures of two famous classical Hamiltonian
systems are shown. Assuming that f : M → Ê ia a smooth function and the conformal
metric g ef g, where M is the configuration space and g is the Eisenhart metric. For the
following examples, the total energy E is assumed to be a constant.
Example 4.1. For an N degrees of freedom linear Hamiltonian system, the Hamiltonian is
given by
H
N
1
pi2 V q1 , q2 , . . . , qN ,
2 i1
4.1
14
International Journal of Mathematics and Mathematical Sciences
in which
V 2 λ 2 2
1 N−1
qN ,
ω2 qi −
2 i1 i
2
4.2
where ωi and λ are constants.
This system consists of N − 1 uncoupled linear oscillators, with the remaining
uncoupled degree of freedom consisting of a parabolic barrier.
From 4.2, one can get
2
∇V ω12 q1 , ω22 q2 , . . . , ωN−1
qN−1 , −λ2 qN ,
ΔV N−1
i1
The coordinate of the manifold M ×
t
t/2 C22 − 0 Ldτ.
Ê2
4.3
ωi2 − λ2 .
is q q0 t, q1 , q2 , . . . , qN , qN1 where qN1 Proposition 4.2. The conformal Ricci curvatures and scalar curvature of the N degrees of freedom
linear Hamiltonian system are given by
00 R
N−1
i1
ωi2 − λ2 V Δf N 2
N
∇f · ∇V V ∇f ,
2
2
0N1 − 1 Δf − N ∇f 2 ,
R
2
4
iN1 0,
R
0k 0,
R
N1N1 0,
R
ik − N ∂i ∂k f N ∂i f∂k f − δik 1 Δf N ∇f 2 ,
R
2
4
2
4
−e−f N 1Δf NN 1 ∇f 2 ,
R
4
4.4
where i, k are from 1 to N.
Moreover, from 3.12, we can get the geodesic equations as
q0 t,
d2 q k 1
− ∂k f
2
dt2
dqk
dt
2
1
∂k f ωk2 qk 0,
E−V 2
d2 q N 1
− ∂N f
2
dt2
dqN
dt
2
qN1 k 1, 2, . . . , N − 1,
1
∂k f − λ2 qN 0,
E−V 2
t
C22 −
2
where L E − 2V , and f satisfies df/dt 0.
t
0
L dτ,
4.5
International Journal of Mathematics and Mathematical Sciences
15
Remark 4.3. When f is a constant function, 4.5 yields that
q0 t,
d2 qk
ωk2 qk 0,
dt2
k 1, 2, . . . , N − 1,
4.6
d2 q N
− λ2 qN 0,
dt2
t
t
N1
2
C2 − L dτ.
q
2
0
For the N degrees of freedom linear Hamiltonian system, from the equations of flows in
3.21, we can get the following.
Proposition 4.4. The equations of flow are given by
2
∂fu
N
−Δfu − ∇fu ,
∂u
2
N−1
2 N−1
2 i
2 N
2
2
0,
ωi q ∂i fu − λ q ∂N fu ωi − λ
N i1
i1
efu
2∂i ∂k fu − ∂i fu ∂k fu 0,
Δfu − N∂i ∂i fu 4.7
i/
k,
2 N − 1 N
∇fu 2 0,
∂i fu 2
2
i 1, 2, . . . , N,
where {fu |fu |u0 0} is a family of smooth functions.
Example 4.5. For the Hénon-Heiles Model, the Hamiltonian is
H
1
2
p1 p22 V q1 , q2 ,
2
4.8
in which the potential V is given by
2
1 1 2 2 2
1 2 3
q
q .
V q1 q2 −
q
2
3
4.9
Next we are going to study the geometry of the Hénon-Heiles Model and obtain the
numerical solutions.
16
International Journal of Mathematics and Mathematical Sciences
Proposition 4.6. The conformal Ricci curvatures and scalar curvature are given by
00 ΔV V Δf ∇f · ∇V V ∇f 2 ,
R
03 − 1 Δf − 1 ∇f 2 ,
R
2
2
i3 0,
R
0k 0,
R
33 0,
R
ik −∂i ∂k f 1 ∂i f∂k f − δik 1 Δf 1 ∇f 2 ,
R
2
2
2
−e−f 3Δf 3 ∇f 2 ,
R
2
4.10
where i, k are from 1 to 2, ∇V q1 2q1 q2 , q2 q1 2 − q2 2 and ΔV 2.
From the Hamiltonian equations, one can get that
E−V H −V
1
2
p1 p22
2
⎛
2 2 ⎞
1
2
dq
dq
1
⎠.
⎝
2
dt
dt
4.11
Then the geodesic equations are given by
q0 t,
d2 q 1 1
− ∂1 f
2
dt2
d2 q 2 1
− ∂2 f
2
dt2
dq2
dt
dq1
dt
2
2
1
∂1 f q1 2q1 q2 0,
E−V 2
2 2
1
∂2 f q2 q1 − q2 0,
E−V 2
q3 t
C22 −
2
where df/dt 0, and E − V is given by 4.11.
t
0
L dτ,
4.12
International Journal of Mathematics and Mathematical Sciences
17
5
4.5
4
q0
3.5
3
2.5
2
1.5
1
1
1.5
2
2.5
3
3.5
4
4.5
5
t
Figure 1: Component q0 of the geodesic.
0.8
0.6
0.4
q1
0.2
0
−0.2
−0.4
−0.6
−0.8
0
1
2
3
4
5
t
Figure 2: Component q1 of the geodesic.
Remark 4.7. When f is a constant function on M, the geodesic equations are given by
q0 t,
d2 q1
q1 2q1 q2 0,
dt2
2 2
d2 q 2
2
q
q1 − q2 0,
dt2
t
t
3
2
q C2 − L dτ.
2
0
4.13
18
International Journal of Mathematics and Mathematical Sciences
150
q2
100
50
0
0
1
2
3
4
5
t
Figure 3: Component q2 of the geodesic.
×104
0.5
0
−0.5
−1
q3
−1.5
−2
−2.5
−3
−3.5
−4
−4.5
0
1
2
3
4
5
t
Figure 4: Component q3 of the geodesic.
Moreover, for the above geodesic equations of the Hénon-Heiles model in 4.13 which can be
reduced into the following first-order differential equations in 4.14, according to the theory
of ordinary differential equations, we know that when the initial values are given, the solution
of 4.13 uniquely exists.
Next we are going to study the solution of the above geodesic equations 4.13 through
numerical simulations. In order to get the reduced equations and do simulation, firstly we
introduce two parameters p1 , p2 , which are the generalized momenta, and pi dqi /dt, i 1, 2.
International Journal of Mathematics and Mathematical Sciences
19
4
3
p1
2
1
0
−1
−2
0
1
2
3
4
5
t
Figure 5: Generalized momentum p1 .
Then the second-order differential equations in 4.13 become into the following first-order
differential equations
q0 t,
dq1
p1 ,
dt
dp1
q1 2q1 q2 0,
dt
dq2
p2 ,
dt
2 2
dp2
q2 q1 − q2 0,
dt
t
t
q3 C22 − L dτ,
2
0
4.14
where L E − 2V .
Let the initial vector q1 , q2 , q3 , p1 , p2 be 0, 0, 1, 1, 1, and the interval for time t be
0, 5, by use of the ode45 method in Matlab, that is, Runge-Kutta 4, 5 method, we can get
the following numerical results of geodesics and the generalized momenta Figures 1, 4.
Figures 2 and 3 above give the motion of the Hénon-Heiles model. Figures 5 and 6
describe the forms of generalized momenta.
20
International Journal of Mathematics and Mathematical Sciences
1500
p2
1000
500
0
0
1
2
3
4
5
t
Figure 6: Generalized momentum p2 .
6
5.5
5
4.5
J0
4
3.5
3
2.5
2
1.5
1
0
1
2
3
4
5
t
Figure 7: Component J 0 of the Jacobi field.
20
15
10
J1
5
0
−5
−10
−15
0
1
2
3
4
t
Figure 8: Component J 1 of the Jacobi field.
5
International Journal of Mathematics and Mathematical Sciences
21
9000
8000
7000
6000
J2
5000
4000
3000
2000
1000
0
0
1
2
3
4
5
t
Figure 9: Component J 2 of the Jacobi field.
×106
2
0
J3
−2
−4
−6
−8
−10
0
1
2
3
4
5
t
Figure 10: Component J 3 of the Jacobi field.
2
1.8
1.6
1.4
u0
1.2
1
0.8
0.6
0.4
0.2
0
0
1
2
3
t
Figure 11: Component u0 .
4
5
22
International Journal of Mathematics and Mathematical Sciences
200
100
u1
0
−100
−200
−300
−400
0
1
2
3
4
5
t
Figure 12: Component u1 .
Proposition 4.8. When f is a constant function on M, the Jacobi field along the geodesic, which is
the solution of 4.13, is given by
d2 J 0
0,
dt2
0 d2 J 1
1
1 2 dJ
2
1
2q
J 1 2q1 J 2 0,
2
q
2q
q
dt
dt2
2 2 dJ 0
d2 J 2
2
1
2
1 1
2
2q
J 2 0,
2
q
q
−
q
J
1
−
2q
dt
dt2
4.15
2 2 dJ 2
1
d2 J 3
dV dJ 0
1
1 2 dJ
2
−
2
q
−
2
q
− |∇V |2 J 0
−
2
2q
q
q1 − q2
dt dt
dt
dt
dt2
2
1
1
2
2 dq
1 dq
1
1 dq
2 dq
− 2 1 2q
2q
J − 2 2q
1 − 2q
J 2 0,
dt
dt
dt
dt
where dV/dt q1 2q1 q2 dq1 /dt q2 q1 2 − q2 2 dq2 /dt, and |∇V |2 q1 2q1 q2 2 q2 q1 2 − q2 2 2 .
Moreover, the Jacobi equations in 4.15 are also ordinary differential equations which
can be reduced into first-order differential equations, and when the initial values are given,
the solution of 4.15 uniquely exists.
International Journal of Mathematics and Mathematical Sciences
23
×104
14
12
10
u2
8
6
4
2
0
−2
0
1
2
3
4
5
t
Figure 13: Component u2 .
For 4.15, next we are going to find the numerical solution of the Jacobi field. Using
the similar process as above, we introduce four parameters u0 , u1 , . . . , u3 and assume that
ui dJ i /dt, i 0, 1, . . . , 3. Then 4.15 becomes into
dJ 0
u0 ,
dt
du0
0,
dt
dJ 1
u1 ,
dt
du1
2 q1 2q1 q2 u0 1 2q2 J 1 2q1 J 2 0,
dt
dJ 2
u2 ,
dt
2 2 du2
u0 2q1 J 1 1 − 2q2 J 2 0,
2 q2 q1 − q2
dt
dJ 3
u3 ,
dt
2 2 du3
dV
1
1 2
2
−2
u0 − 2 q 2q q u1 − 2 q q1 − q2
u2 − |∇V |2 J 0
dt
dt
− 2 1 2q2 p1 2q1 p2 J 1 − 2 2q1 p1 1 − 2q2 p2 J 2 0,
4.16
where dqi /dt and d2 qi /dt2 are replaced by pi and dpi /dt given in 4.14, respectively.
24
International Journal of Mathematics and Mathematical Sciences
×108
0.5
0
−0.5
u3
−1
−1.5
−2
−2.5
−3
0
1
2
3
4
5
t
Figure 14: Component u3 .
We assume that the initial vector J 0 , J 1 , J 2 , J 3 , u0 , u1 , u2 , u3 is 1, 1, 1, 1, 1, 1, 1, 1, and
the time interval is 0, 5. Also using the ode45 method in Matlab, and the first-order
differential equations of geodesics and generalized momenta in 4.14, we can get the
following numerical results of Jacobi field.
Figures 7, 8, 9, and 10 show us that the Jacobi field is divergent when the time t → ∞,
which implies the instability of the geodesic spread. For more please refer to 21, 22 about
the instability of the geodesic derivation equation, that is, the Jacobi equation for the HénonHeiles model, and 16, 17 about the geodesic spread and geodesic derivative equation and
their applications Figures 11-14.
Proposition 4.9. For the Hénon-Heiles model, from Theorem 3.11, we can obtain the equations of
flow as
2
∂fu
−Δfu − ∇fu ,
∂u
2 2 q1 2q1 q2 ∂1 fu q2 q1 − q2
∂2 fu −2,
efu
2∂i ∂k fu − ∂i fu ∂k fu 0,
4.17
i/
k,
2
2 1 Δfu − 2∂i ∂i fu ∂i fu ∇fu 0,
2
i 1, 2.
5. Conclusion
The theory of conformal metric in Riemannian geometry is applied to characterize the
geometry of the classical Hamiltonian dynamics with the conformal Eisenhart metric
Section 3. We obtain the Ricci curvatures, the scalar curvatures, the geodesic equations,
the Jacobi equations, and the equation of a certain flow. The relation between the curvatures
International Journal of Mathematics and Mathematical Sciences
25
with Eisenhart metric and its conformal metric can also be shown clearly. As in the examples,
the conformal curvatures, the equation of a certain flow, and the numerical solutions of
the geodesics, generalized momenta, and Jacobi field along the geodesics are obtained
Section 4.
Acknowledgment
This paper is supported by the National Natural Science Foundations of China no. 10871218,
no. 10932002.
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