# Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric

Authors: Linyu Peng; Huafei Sun; Xiao Sun
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

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, anN degrees of freedom linear Hamiltonian system and the Hénon-Heiles
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.


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## Body

## 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 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 Section2, the foundation of the Hamiltonian mechanics, the Eisenhart metric of a manifold M×ℝ2, 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 𝔛(M), and D is the affine connection, which is defined as (2.1)DXY=Xj(∂Yi∂xj+ΓjkiYk)∂∂xi,∀X,Y∈X(M).
The coefficients Γijk of the connection D can be represented in terms of the metric tensor g by (2.2)Γijk=12gkl(∂jgil+∂igjl-∂lgij),
where ∂i means ∂/∂xi, and the Einstein summation convention is used. For any X,Y,Z∈𝔛(M), the curvature tensor is defined by (2.3)R(X,Y)Z=DXDYZ-DYDXZ-D[X,Y]Z,
where [·,·] is the Lie bracket on 𝔛(M) defined as [X,Y]=XY-YX. Then the components of the curvature tensor can be given by (2.4)Rkijl=∂iΓkjl-∂jΓkil+ΓkjhΓhil-ΓkihΓhjl.
The Ricci curvature is defined by (2.5)Rij=gklRiklj,
where Riklj=gksRiljs. The scalar curvature is defined by (2.6)R=gijRij.
A curve ξ(t) on M is said to be geodesic if its tangent ξ̇(t) is displaced parallel along the curve ξ(t), that is, (2.7)Dξ̇(t)ξ̇(t)=0.
In local coordinate ξ(t)=(x1(t),x2(t),…,xN(t)), the geodesic equation satisfies (2.8)d2xkdt2+Γijkdxidtdxjdt=0.
In addition, the equation for the well-known Jacobi field J=Jk∂k satisfies (2.9)Dξ̇(t)Dξ̇(t)J=R(ξ̇(t),J)ξ̇(t),
where ξ(t) 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 metricg̃ defined on M is said to be conformally equivalent to g, if there exists a function f∈C∞(M) such that (2.10)g̃=efg.
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 (2.11)g̃ij=efgij.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 (2.12)(∂gt∂t)ij=-2Rij,
where gt|t=0=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 manifoldMN 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 L(qi,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 H(qi,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 withint∈[a,b] is (2.13)S=∫abL(q,q̇,t)dt.
The Hamiltonian variational principle states (2.14)δS=0.
From the calculus of variation, (2.14) equates to the Euler-Lagrange equations (2.15)ddt(∂L∂q̇i)-∂L∂qi=0,i=1,…,N.To pass to the Hamiltonian formalism, we introduce the generalized momenta as(2.16)pi=∂L∂q̇i,i=1,…,N,
make the change (qi,q̇i)↦(qi,pi), and introduce the Hamiltonian (2.17)H(qi,pi,t)=pjq̇j-L(qi,q̇i,t).
Then the Euler-Lagrange equations are equivalent to the Hamiltonian equations, which are given by (2.18)dqidt=∂H∂pi,dpidt=-∂H∂qi,i=1,…,N.In this paper, we will only consider the classical Hamiltonian dynamics, whose Hamiltonian is of the form(2.19)H=12∑i=1Npi2+V(q),
where V(q) is the potential energy.The energyE of the classical Hamiltonian dynamics is a constant equal to Hamiltonian H. Considering the manifold M×ℝ2 defined by (2.20)M×R2={q∣q=(q0,q1,q2,…,qN,qN+1)},
in which q0=t and qN+1=(C12/2)t+C22-∫0tLdτ, 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 symbolsα,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×ℝ2 is defined as (2.21)(gαβ)=(-2V0⋯0100⋮IN×N⋮0010⋯00),
where IN×N is the identity matrix, and we know that it is not degenerate as det(gαβ)=(-1)N+3.The inverse matrix of(gαβ) is (2.22)(gαβ)=(00⋯0100⋮IN×N⋮0010⋯02V).
The nonzero coefficients of the Riemanian connection are (2.23)Γ00k=-Γ0kN+1=∂kV,
where ∂k means ∂/∂qk.The nonzero components of the curvature tensor are(2.24)R0j0l=-∂j∂lV.
The nonzero Ricci curvature and the scalar curvature are, respectively, given by (2.25)R00=ΔV,R=0,
where Δ is the Laplacian operator in the Euclidean space.The geodesic equations read(2.26)d2q0dt2=0,d2qidt2=-∂V∂qi,i=1,2,…N,d2qN+1dt2=-dLdt,
in which, the first and third equations are identical, and the second ones are the equations of motion for the associated 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 𝔛(M), and D is the affine connection, which is defined as (2.1)DXY=Xj(∂Yi∂xj+ΓjkiYk)∂∂xi,∀X,Y∈X(M).
The coefficients Γijk of the connection D can be represented in terms of the metric tensor g by (2.2)Γijk=12gkl(∂jgil+∂igjl-∂lgij),
where ∂i means ∂/∂xi, and the Einstein summation convention is used. For any X,Y,Z∈𝔛(M), the curvature tensor is defined by (2.3)R(X,Y)Z=DXDYZ-DYDXZ-D[X,Y]Z,
where [·,·] is the Lie bracket on 𝔛(M) defined as [X,Y]=XY-YX. Then the components of the curvature tensor can be given by (2.4)Rkijl=∂iΓkjl-∂jΓkil+ΓkjhΓhil-ΓkihΓhjl.
The Ricci curvature is defined by (2.5)Rij=gklRiklj,
where Riklj=gksRiljs. The scalar curvature is defined by (2.6)R=gijRij.
A curve ξ(t) on M is said to be geodesic if its tangent ξ̇(t) is displaced parallel along the curve ξ(t), that is, (2.7)Dξ̇(t)ξ̇(t)=0.
In local coordinate ξ(t)=(x1(t),x2(t),…,xN(t)), the geodesic equation satisfies (2.8)d2xkdt2+Γijkdxidtdxjdt=0.
In addition, the equation for the well-known Jacobi field J=Jk∂k satisfies (2.9)Dξ̇(t)Dξ̇(t)J=R(ξ̇(t),J)ξ̇(t),
where ξ(t) 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 metricg̃ defined on M is said to be conformally equivalent to g, if there exists a function f∈C∞(M) such that (2.10)g̃=efg.
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 (2.11)g̃ij=efgij.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 (2.12)(∂gt∂t)ij=-2Rij,
where gt|t=0=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 manifoldMN 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 L(qi,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 H(qi,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 withint∈[a,b] is (2.13)S=∫abL(q,q̇,t)dt.
The Hamiltonian variational principle states (2.14)δS=0.
From the calculus of variation, (2.14) equates to the Euler-Lagrange equations (2.15)ddt(∂L∂q̇i)-∂L∂qi=0,i=1,…,N.To pass to the Hamiltonian formalism, we introduce the generalized momenta as(2.16)pi=∂L∂q̇i,i=1,…,N,
make the change (qi,q̇i)↦(qi,pi), and introduce the Hamiltonian (2.17)H(qi,pi,t)=pjq̇j-L(qi,q̇i,t).
Then the Euler-Lagrange equations are equivalent to the Hamiltonian equations, which are given by (2.18)dqidt=∂H∂pi,dpidt=-∂H∂qi,i=1,…,N.In this paper, we will only consider the classical Hamiltonian dynamics, whose Hamiltonian is of the form(2.19)H=12∑i=1Npi2+V(q),
where V(q) is the potential energy.The energyE of the classical Hamiltonian dynamics is a constant equal to Hamiltonian H. Considering the manifold M×ℝ2 defined by (2.20)M×R2={q∣q=(q0,q1,q2,…,qN,qN+1)},
in which q0=t and qN+1=(C12/2)t+C22-∫0tLdτ, 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 symbolsα,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×ℝ2 is defined as (2.21)(gαβ)=(-2V0⋯0100⋮IN×N⋮0010⋯00),
where IN×N is the identity matrix, and we know that it is not degenerate as det(gαβ)=(-1)N+3.The inverse matrix of(gαβ) is (2.22)(gαβ)=(00⋯0100⋮IN×N⋮0010⋯02V).
The nonzero coefficients of the Riemanian connection are (2.23)Γ00k=-Γ0kN+1=∂kV,
where ∂k means ∂/∂qk.The nonzero components of the curvature tensor are(2.24)R0j0l=-∂j∂lV.
The nonzero Ricci curvature and the scalar curvature are, respectively, given by (2.25)R00=ΔV,R=0,
where Δ is the Laplacian operator in the Euclidean space.The geodesic equations read(2.26)d2q0dt2=0,d2qidt2=-∂V∂qi,i=1,2,…N,d2qN+1dt2=-dLdt,
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 manifoldM×ℝ2, in which M is the configuration space, we investigate the conformal Eisenhart metric which states as (3.1)g̃=efg,
where g is the Eisenhart metric and f:M×ℝ2→ℝ is a smooth function. Then the arc length parameter s̃ is shown as (3.2)ds̃2=g̃αβdqαdqβ=C12efdt2=efdt2.
And the volume element dṽ of the manifold under the conformal Eisenhart metric is given by (3.3)dṽ=det⁡(g̃)dq0∧dq1∧⋯∧dqN+1=-e((N/2)+1)fdq0∧dq1∧⋯∧dqN+1.Proposition 3.1.
The conformal and independent components of the curvature tensor are given by(3.4)R̃0j0l=ef(-∂j∂lV-V∂j∂lf+12V∂jf∂lf)+δjlef4(2∂0∂0f-(∂0f)2-4V∂0f∂N+1f-4V2(∂N+1f)2-2(∇f⋅∇V)-2V|∇f|2),R̃0j0N+1=ef(-12∂0∂jf-V∂j∂N+1f+14∂0f∂jf+12V∂jf∂N+1f-12∂jV∂N+1f),R̃0jkl=δjlef(12∂0∂kf+12∂kV∂N+1f-14∂0f∂kf)-δjkef(12∂0∂lf+12∂lV∂N+1f-14∂0f∂lf),R̃0jkN+1=ef(-12∂j∂kf+14∂jf∂kf)-δjkef(12∂0∂N+1f+12V(∂N+1f)2+14∂0f∂N+1f+14|∇f|2),R̃0N+10N+1=-ef(∂0∂N+1f-V∂N+1∂N+1f+14|∇f|2),R̃0N+1kl=0,R̃0N+1kN+1=ef(-12∂k∂N+1f+14∂kf∂N+1f),R̃ijkl=δjlef(12∂i∂kf-14∂if∂kf)-δilef(12∂j∂kf-14∂jf∂kf)+δikef(12∂j∂lf-14∂jf∂lf)-δjkef(12∂i∂lf-14∂if∂lf)+(δikδjl-δilδjk)ef(12∂0f∂N+1f+12V(∂N+1f)2+14|∇f|2),R̃ijkN+1=δikef(12∂j∂N+1f-14∂jf∂N+1f)-δjkef(12∂i∂N+1f-14∂if∂N+1f),R̃iN+1kN+1=δikef(12∂N+1∂N+1f-14(∂N+1f)2),
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
(3.5)Γ̃000=∂0f+V∂N+1f,Γ̃00k=∂kV+V∂kf,Γ̃00N+1=V∂0f+2V2∂N+1f,Γ̃0j0=12∂jf,Γ̃0jk=12δjk∂0f,Γ̃0jN+1=-∂jV,Γ̃0N+10=0,Γ̃0N+1k=-12∂kf,Γ̃0N+1N+1=-V∂N+1f,Γ̃ij0=-12δij∂N+1f,Γ̃ijk=12δik∂jf+12δjk∂if-12δij∂kf,Γ̃ijN+1=-12δij∂0f-δijV∂N+1f,Γ̃iN+10=0,Γ̃iN+1k=12δik∂N+1f,Γ̃iN+1N+1=12∂if,Γ̃N+1N+10=0,Γ̃N+1N+1k=0,Γ̃N+1N+1N+1=∂N+1f.
Then from (2.4) and R̃αβγη=g̃βλR̃αγηλ, one can get the conclusion, immediately.Proposition 3.2.
The conformal Ricci curvatures ofM×ℝ2 are given by
(3.6)R̃00=ΔV+VΔf-N2∂0∂0f+N4(∂0f)2+NV∂0f∂N+1f+NV2(∂N+1f)2+N2(∇f⋅∇V)+N2V|∇f|2+2V∂0∂N+1f+2V2∂N+1∂N+1f,R̃0k=-N(12∂0∂kf+12∂kV∂N+1f-14∂0f∂kf),R̃0N+1=-12Δf-N+22∂0∂N+1f-V∂N+1∂N+1f-N(12V(∂N+1f)2+14∂0f∂N+1f+14|∇f|2),R̃ik=-N2∂i∂kf+N4∂if∂kf-δik(∂0∂N+1f+N2∂0f∂N+1f+N2V(∂N+1f)2+N4|∇f|2+V∂N+1∂N+1f+12Δf),R̃iN+1=-N2∂i∂N+1f+N4∂if∂N+1f,R̃N+1N+1=-N2∂N+1∂N+1f+N4(∂N+1f)2.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(3.7)R̃=-e-f((N+1)Δf+N(N+1)4|∇f|2+2(N+1)∂0∂N+1f)-e-f(N(N+1)2∂0f∂N+1f+2(N+1)V∂N+1∂N+1f+N(N+1)2(∂N+1f)2).Proof.
From (2.6), and the conclusion in Proposition 3.2, we can obtain the conclusion of Theorem 3.3, immediately.Remark 3.4.
WhenV is a smooth function on the configuration space and let f=V, we can get
(3.8)R̃00=(V+1)(ΔV+N2|∇V|2),R̃0k=0,R̃0N+1=-12ΔV-N4|∇V|2,R̃ik=-N2∂i∂kV+N4∂iV∂kV-δik(12ΔV+N4|∇V|2),R̃iN+1=0,R̃N+1N+1=0,R̃=-e-V((N+1)ΔV+N(N+1)4|∇V|2).Theorem 3.5.
The conformal geodesic equations are(3.9)d2qkdt2+6E-4V-12(1-2L)∂N+1fdqkdt-12∂kf(dqkdt)2+(E-V+12)∂kf+∂kV=0,
and f satisfies
(3.10)∂0f=L∂N+1f,(12-L)dfdt=2(E-V-14)∂N+1f,
where
(3.11)dfdt=∂ifdqidt,i=1,…,N,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 assumingf:M→ℝ, the geodesic equations are
(3.12)d2qkdt2-12∂kf(dqkdt)2+(E-V+12)∂kf+∂kV=0,
and f satisfies that
(3.13)dfdt=0.Lemma 3.7 (Hopf).
Let(M,g) be a compact manifold, f∈C∞(M). Then ∫MΔgf=0.Lemma 3.8.
Letf,h∈C∞(M), g̃=efg, then
(3.14)∇g̃h=e-f∇gh,Δg̃h=e-fΔgh+N-22g̃(∇g̃f,∇g̃h).Proof.
From the definition of gradient and Laplacian ofh∈C∞(M), one can obtain
(3.15)∇g̃h=∂ihg̃ij∂j=e-f∂ihgij∂j=e-f∇gh,Δg̃h=1G̃∂i(G̃g̃ij∂jh)=e-(N/2)f1G∂i(e((N-2)/2)fGgij∂jh)=e-f1G∂i(Ggij∂jh)+N-22e-fgij∂jh∂if=e-fΔgh+N-22g̃(∇g̃f,∇g̃h),
where G=det(gij).Theorem 3.9.
Letf:M→ℝ, M×U1×U2 be compact, where U1 and U2 are compact subsets of ℝ. Then from Theorem 3.3, Lemmas 3.7 and 3.8, one can get that
(3.16)∫M×U1×U2R̃=-∫M×U1×U2e-f((N+1)Δf+N(N+1)4|∇f|2)=-∫M×U1×U2((N+1)Δg̃f-(N+1)(N-4)4e-f|∇f|2)=(N+1)(N-4)4∫M×U1×U2(e-f|∇f|2),
and moreover,(1)
whenN>4, ∫M×U1×U2R̃≥0,(2)
whenN=4, ∫M×U1×U2R̃≡0,(3)
whenN<4, ∫M×U1×U2R̃≤0.In addition, when N≠4 and the equality of (3.16) holds, |∇g̃f|2=e-2f|∇f|2=0, so that f is a constant function on the configuration space M.Theorem 3.10.
Letf:M→ℝ 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
(3.17)d2J0dt2+∂ifdJidt+A1J0+∂i∂jfdqjdtJi=0,d2Jidt2+A2dJ0dt+A3dJjdt-∂ifdJN+1dt+A4Jj=0,d2JN+1dt2-2dVdtdJ0dt+A5dJjdt+A6J0+A7Ji+A8JN+1=0,
where
(3.18)A1=12∂ifd2qidt2+12(∇f⋅∇V)-2L+18|∇f|2,A2=2∂iV+(E-12)∂if,A3=∂jfdqidt-∂ifdqjdt,A4=12(δik∂jf+δij∂kf-δkj∂if)d2qkdt2+∂i∂jV-14δij|∇f|2+12(∂iV∂jf+∂jV∂if-∂i∂jf+δij(∇f⋅∇V)+∂j∂sfdqsdtdqjdt),A5=-2∂jV+(12-L)∂jf,A6=-∂jVd2qjdt2+12(∇f⋅∇V)-|∇V|2,A7=-2∂i∂jVdqjdt+(12-L)∂i∂jfdqjdt,A8=12∂jfd2qjdt2+12(∇f⋅∇V)-14|∇f|2.Proof.
Firstly, from the Jacobi equation (2.9), we can get the conformal Jacobi equations represented in components as
(3.19)d2Jαdt2+2Γ̃βσαdqβdtdJσdt+(Γ̃βσαd2qβdt2+Γ̃βσγΓ̃γλαdqβdtdqλdt+∂λΓ̃βσαdqβdtdqλdt-R̃βλσαdqβdtdqλdt)Jσ=0,
where R̃βλσα are given by (2.4). From (2.4), (3.19) turns into
(3.20)d2Jαdt2+2Γ̃βσαdqβdtdJσdt+(Γ̃βσαd2qβdt2+Γ̃βλγΓ̃γσαdqβdtdqλdt+∂σΓ̃βλαdqβdtdqλdt)Jσ=0.
Furthermore, substituting (3.5) into (3.20), in which f should be constrained in C∞(M) but not C∞(M×ℝ2), we complete the proof of Theorem 3.10.Now letfu:M×[a,b]→ℝ be a smooth function with fu|u=0=0, and gu=efug, that is, gu|u=0=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
(3.21)efu∂fu∂u=-Δfu-N2|∇fu|2,ΔV+N2(∇fu⋅∇V)=0,2∂i∂kfu-∂ifu∂kfu=0,i≠k,Δfu-N∂i∂ifu+N2(∂ifu)2+N-12|∇fu|2=0,i=1,2,…,N.Proof.
From (2.12), one can obtain
(3.22)-2R00=-efu∂fu∂u2V,-2R0i=0,-2R0N+1=efu∂fu∂u,-2Rik=efu∂fu∂uδik,-2RiN+1=0,-2RN+1N+1=0,
where Rαβ are the Ricci curvatures with the metric gu.Then from the conformal Ricci curvatures obtained in Proposition3.2, 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 thatf:M→ℝ ia a smooth function and the conformal metric g̃=efg, 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 anN degrees of freedom linear Hamiltonian system, the Hamiltonian is given by
(4.1)H=12∑i=1Npi2+V(q1,q2,…,qN),
in which
(4.2)V=12∑i=1N-1ωi2(qi)2-λ22(qN)2,
where ωi and λ are constants.This system consists ofN-1 uncoupled linear oscillators, with the remaining uncoupled degree of freedom consisting of a parabolic barrier.From (4.2), one can get (4.3)∇V=(ω12q1,ω22q2,…,ωN-12qN-1,-λ2qN),ΔV=∑i=1N-1ωi2-λ2.
The coordinate of the manifold M×ℝ2 is q=(q0=t,q1,q2,…,qN,qN+1) where qN+1=(t/2)+C22-∫0tLdτ.Proposition 4.2.
The conformal Ricci curvatures and scalar curvature of theN degrees of freedom linear Hamiltonian system are given by
(4.4)R̃00=∑i=1N-1ωi2-λ2+VΔf+N2(∇f⋅∇V)+N2V|∇f|2,R̃0k=0,R̃0N+1=-12Δf-N4|∇f|2,R̃iN+1=0,R̃N+1N+1=0,R̃ik=-N2∂i∂kf+N4∂if∂kf-δik(12Δf+N4|∇f|2),R̃=-e-f((N+1)Δf+N(N+1)4|∇f|2),
where i,k are from 1 to N.Moreover, from (3.12), we can get the geodesic equations as (4.5)q0=t,d2qkdt2-12∂kf(dqkdt)2+(E-V+12)∂kf+ωk2qk=0,k=1,2,…,N-1,d2qNdt2-12∂Nf(dqNdt)2+(E-V+12)∂kf-λ2qN=0,qN+1=t2+C22-∫0tLdτ,
where L=E-2V, and f satisfies df/dt=0.Remark 4.3.
Whenf is a constant function, (4.5) yields that
(4.6)q0=t,d2qkdt2+ωk2qk=0,k=1,2,…,N-1,d2qNdt2-λ2qN=0,qN+1=t2+C22-∫0tLdτ.
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(4.7)efu∂fu∂u=-Δfu-N2|∇fu|2,∑i=1N-1(ωi2qi∂ifu)-λ2qN∂Nfu+2N(∑i=1N-1(ωi2)-λ2)=0,2∂i∂kfu-∂ifu∂kfu=0,i≠k,Δfu-N∂i∂ifu+N2(∂ifu)2+N-12|∇fu|2=0,i=1,2,…,N,
where {fu|fu|u=0=0} is a family of smooth functions.Example 4.5.
For the Hénon-Heiles Model, the Hamiltonian is(4.8)H=12(p12+p22)+V(q1,q2),
in which the potential V is given by
(4.9)V=12((q1)2+(q2)2)+(q1)2q2-13(q2)3.
Next we are going to study the geometry of the Hénon-Heiles Model and obtain the numerical solutions.Proposition 4.6.
The conformal Ricci curvatures and scalar curvature are given by(4.10)R̃00=ΔV+VΔf+(∇f⋅∇V)+V|∇f|2,R̃0k=0,R̃03=-12Δf-12|∇f|2,R̃i3=0,R̃33=0,R̃ik=-∂i∂kf+12∂if∂kf-δik(12Δf+12|∇f|2),R̃=-e-f(3Δf+32|∇f|2),
where i,k are from 1 to 2, ∇V=(q1+2q1q2,q2+(q1)2-(q2)2) and ΔV=2.From the Hamiltonian equations, one can get that(4.11)E-V=H-V=12(p12+p22)=12((dq1dt)2+(dq2dt)2).
Then the geodesic equations are given by (4.12)q0=t,d2q1dt2-12∂1f(dq1dt)2+(E-V+12)∂1f+q1+2q1q2=0,d2q2dt2-12∂2f(dq2dt)2+(E-V+12)∂2f+q2+(q1)2-(q2)2=0,q3=t2+C22-∫0tLdτ,
where df/dt=0, and E-V is given by (4.11).Remark 4.7.
Whenf is a constant function on M, the geodesic equations are given by
(4.13)q0=t,d2q1dt2+q1+2q1q2=0,d2q2dt2+q2+(q1)2-(q2)2=0,q3=t2+C22-∫0tLdτ.
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. Then the second-order differential equations in (4.13) become into the following first-order differential equations (4.14)q0=t,dq1dt=p1,dp1dt+q1+2q1q2=0,dq2dt=p2,dp2dt+q2+(q1)2-(q2)2=0,q3=t2+C22-∫0tLdτ,
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).Figure 1
Componentq0 of the geodesic.Figures2 and 3 above give the motion of the Hénon-Heiles model. Figures 5 and 6 describe the forms of generalized momenta.Figure 2
Componentq1 of the geodesic.Figure 3
Componentq2 of the geodesic.Figure 4
Componentq3 of the geodesic.Figure 5
Generalized momentump1.Figure 6
Generalized momentump2.Proposition 4.8.
Whenf is a constant function on M, the Jacobi field along the geodesic, which is the solution of (4.13), is given by
(4.15)d2J0dt2=0,d2J1dt2+2(q1+2q1q2)dJ0dt+(1+2q2)J1+2q1J2=0,d2J2dt2+2(q2+(q1)2-(q2)2)dJ0dt+2q1J1+(1-2q2)J2=0,d2J3dt2-2dVdtdJ0dt-2(q1+2q1q2)dJ1dt-2(q2+(q1)2-(q2)2)dJ2dt-|∇V|2J0-2((1+2q2)dq1dt+2q1dq2dt)J1-2(2q1dq1dt+(1-2q2)dq2dt)J2=0,
where dV/dt=(q1+2q1q2)(dq1/dt)+(q2+(q1)2-(q2)2)(dq2/dt), and |∇V|2=(q1+2q1q2)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.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=dJi/dt, i=0,1,…,3. Then (4.15) becomes into (4.16)dJ0dt=u0,du0dt=0,dJ1dt=u1,du1dt+2(q1+2q1q2)u0+(1+2q2)J1+2q1J2=0,dJ2dt=u2,du2dt+2(q2+(q1)2-(q2)2)u0+2q1J1+(1-2q2)J2=0,dJ3dt=u3,du3dt-2dVdtu0-2(q1+2q1q2)u1-2(q2+(q1)2-(q2)2)u2-|∇V|2J0-2((1+2q2)p1+2q1p2)J1-2(2q1p1+(1-2q2)p2)J2=0,
where dqi/dt and d2qi/dt2 are replaced by pi and dpi/dt given in (4.14), respectively.We assume that the initial vector(J0,J1,J2,J3,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.Figures7, 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 Hénon-Heiles model, and [16, 17] about the geodesic spread and geodesic derivative equation and their applications (Figures 11-14).Figure 7
ComponentJ0 of the Jacobi field.Figure 8
ComponentJ1 of the Jacobi field.Figure 9
ComponentJ2 of the Jacobi field.Figure 10
ComponentJ3 of the Jacobi field.Figure 11
Componentu0.Figure 12
Componentu1.Figure 13
Componentu2.Figure 14
Componentu3.Proposition 4.9.
For the Hénon-Heiles model, from Theorem3.11, we can obtain the equations of flow as
(4.17)efu∂fu∂u=-Δfu-|∇fu|2,(q1+2q1q2)∂1fu+(q2+(q1)2-(q2)2)∂2fu=-2,2∂i∂kfu-∂ifu∂kfu=0,i≠k,Δfu-2∂i∂ifu+(∂ifu)2+12|∇fu|2=0,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 (Section3). 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 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).



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