Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric

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, an N 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.

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 Section 2, 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 (𝑀^𝑁,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 𝐷_𝑋𝑌=𝑋^𝑗𝜕𝑌^𝑖𝜕𝑥^𝑗+Γ^𝑖_𝑗𝑘𝑌^𝑘𝜕𝜕𝑥_𝑖,∀𝑋,𝑌∈𝔛(𝑀). (2.1) The coefficients Γ^𝑘_𝑖𝑗 of the connection 𝐷 can be represented in terms of the metric tensor 𝑔 by Γ^𝑘_𝑖𝑗=12𝑔^𝑘𝑙𝜕_𝑗𝑔_𝑖𝑙+𝜕_𝑖𝑔_𝑗𝑙−𝜕_𝑙𝑔_𝑖𝑗, (2.2) where 𝜕_𝑖 means 𝜕/𝜕𝑥^𝑖, and the Einstein summation convention is used. For any 𝑋,𝑌,𝑍∈𝔛(𝑀), the curvature tensor is defined by 𝑅(𝑋,𝑌)𝑍=𝐷_𝑋𝐷_𝑌𝑍−𝐷_𝑌𝐷_𝑋𝑍−𝐷_[𝑋,𝑌]𝑍, (2.3) 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 𝑅^𝑙_𝑘𝑖𝑗=𝜕_𝑖Γ^𝑙_𝑘𝑗−𝜕_𝑗Γ^𝑙_𝑘𝑖+Γ^ℎ_𝑘𝑗Γ^𝑙_ℎ𝑖−Γ^ℎ_𝑘𝑖Γ^𝑙_ℎ𝑗. (2.4) The Ricci curvature is defined by 𝑅_𝑖𝑗=𝑔^𝑘𝑙𝑅_{𝑖𝑘𝑙𝑗}, (2.5) where 𝑅_{𝑖𝑘𝑙𝑗}=𝑔_𝑘𝑠𝑅^𝑠_𝑖𝑙𝑗. The scalar curvature is defined by ℛ=𝑔^𝑖𝑗𝑅_𝑖𝑗. (2.6) 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, 𝐷_̇𝜉(𝑡)𝜉̇(𝑡)=0. (2.7) In local coordinate 𝜉(𝑡)=(𝑥¹(𝑡),𝑥²(𝑡),…,𝑥^𝑁(𝑡)), the geodesic equation satisfies d²𝑥^𝑘d𝑡²+Γ^𝑘_𝑖𝑗d𝑥^𝑖d𝑡d𝑥^𝑗d𝑡=0. (2.8) In addition, the equation for the well-known Jacobi field 𝐽=𝐽^𝑘𝜕_𝑘 satisfies 𝐷_̇𝜉(𝑡)𝐷_̇𝜉(𝑡)𝐽=𝑅𝜉̇(𝑡),𝐽𝜉̇(𝑡), (2.9) 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 metric 𝑔 defined on 𝑀 is said to be conformally equivalent to 𝑔, if there exists a function 𝑓∈𝐶^∞(𝑀) such that 𝑔=𝑒^𝑓𝑔. (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 𝑔_𝑖𝑗=𝑒^𝑓𝑔_𝑖𝑗. (2.11)

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 (𝑀^𝑁,𝑔₀) and evolves its metric by the equation 𝜕𝑔_𝑡𝜕𝑡_𝑖𝑗=−2𝑅_𝑖𝑗, (2.12) where 𝑔_𝑡|_𝑡=0=𝑔₀ and 𝑅_𝑖𝑗 denote the Ricci curvatures with 𝑔_𝑡.

In this paper, we study a similar flow, which is an evolution equation started from a smooth manifold 𝑀^𝑁 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(𝑞^𝑖,𝑞̇^𝑖,𝑡), where 𝑞=(𝑞¹,…,𝑞^𝑁) and 𝑞̇=(𝑞̇¹,…,𝑞̇^𝑁) are called the generalized coordinate and generalized velocity, respectively. While a Hamiltonian function is represented as 𝐻(𝑞^𝑖,𝑝_𝑖,𝑡), where 𝑝=(𝜕𝐿/𝜕𝑞¹,…,𝜕𝐿/𝜕𝑞^𝑁) 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_𝑎𝐿𝑞,𝑞,̇𝑡d𝑡. (2.13) The Hamiltonian variational principle states 𝛿𝑆=0. (2.14) From the calculus of variation, (2.14) equates to the Euler-Lagrange equations dd𝑡𝜕𝐿𝜕𝑞̇^𝑖−𝜕𝐿𝜕𝑞^𝑖=0,𝑖=1,…,𝑁. (2.15)

To pass to the Hamiltonian formalism, we introduce the generalized momenta as 𝑝_𝑖=𝜕𝐿𝜕𝑞̇^𝑖,𝑖=1,…,𝑁, (2.16) make the change (𝑞^𝑖,𝑞̇^𝑖)↦(𝑞^𝑖,𝑝_𝑖), and introduce the Hamiltonian 𝐻𝑞^𝑖,𝑝_𝑖,𝑡=𝑝_𝑗𝑞̇^𝑗−𝐿𝑞^𝑖,𝑞̇^𝑖,𝑡. (2.17) Then the Euler-Lagrange equations are equivalent to the Hamiltonian equations, which are given by d𝑞^𝑖=d𝑡𝜕𝐻𝜕𝑝_𝑖,d𝑝_𝑖d𝑡=−𝜕𝐻𝜕𝑞^𝑖,𝑖=1,…,𝑁. (2.18)

In this paper, we will only consider the classical Hamiltonian dynamics, whose Hamiltonian is of the form 1𝐻=2^𝑁_𝑖=1𝑝²_𝑖+𝑉𝑞, (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×ℝ² defined by 𝑀×ℝ²=𝑞∣𝑞=𝑞⁰,𝑞¹,𝑞²,…,𝑞^𝑁,𝑞^𝑁+1, (2.20) in which 𝑞⁰=𝑡 and 𝑞^𝑁+1=(𝐶²₁/2)𝑡+𝐶²₂−∫^𝑡₀𝐿d𝜏, 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×ℝ² is defined as 𝑔_𝛼𝛽=−2𝑉0⋯0100⋮𝐼_𝑁×𝑁⋮0010⋯00, (2.21) where 𝐼_𝑁×𝑁 is the identity matrix, and we know that it is not degenerate as det(𝑔_𝛼𝛽)=(−1)^𝑁+3.

The inverse matrix of (𝑔_𝛼𝛽) is 𝑔^𝛼𝛽=00⋯0100⋮𝐼_𝑁×𝑁⋮0010⋯02𝑉. (2.22) The nonzero coefficients of the Riemanian connection are Γ^𝑘₀₀=−Γ^𝑁+1_0𝑘=𝜕_𝑘𝑉, (2.23) where 𝜕_𝑘 means 𝜕/𝜕𝑞^𝑘.

The nonzero components of the curvature tensor are 𝑅_0𝑗0𝑙=−𝜕_𝑗𝜕_𝑙𝑉. (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 d²𝑞⁰d𝑡²d=0,₂𝑞^𝑖d𝑡²=−𝜕𝑉𝜕𝑞^𝑖d,𝑖=1,2,…𝑁,₂𝑞^𝑁+1d𝑡²=−d𝐿,d𝑡 (2.26) 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×ℝ², in which 𝑀 is the configuration space, we investigate the conformal Eisenhart metric which states as 𝑔=𝑒^𝑓𝑔, (3.1) where 𝑔 is the Eisenhart metric and 𝑓∶𝑀×ℝ²→ℝ is a smooth function. Then the arc length parameter 𝑠̃ is shown as d𝑠²̃=𝑔_𝛼𝛽d𝑞^𝛼d𝑞^𝛽=𝐶²₁𝑒^𝑓d𝑡²=𝑒^𝑓d𝑡². (3.2) And the volume element d𝑣 of the manifold under the conformal Eisenhart metric is given by d𝑣=det𝑔d𝑞⁰∧d𝑞¹∧⋯∧d𝑞^𝑁+1=−𝑒^{((𝑁/2)+1)𝑓}d𝑞⁰∧d𝑞¹∧⋯∧d𝑞^𝑁+1. (3.3)

Proposition 3.1. The conformal and independent components of the curvature tensor are given by 𝑅_0𝑗0𝑙=𝑒^𝑓−𝜕_𝑗𝜕_𝑙𝑉−𝑉𝜕_𝑗𝜕_𝑙1𝑓+2𝑉𝜕_𝑗𝑓𝜕_𝑙𝑓+𝛿_𝑗𝑙𝑒^𝑓42𝜕₀𝜕₀𝑓−𝜕₀𝑓²−4𝑉𝜕₀𝑓𝜕_𝑁+1𝑓−4𝑉²𝜕_𝑁+1𝑓²−2∇𝑓⋅∇𝑉−2𝑉∇𝑓²,𝑅_0𝑗0𝑁+1=𝑒^𝑓−12𝜕₀𝜕_𝑗𝑓−𝑉𝜕_𝑗𝜕_𝑁+11𝑓+4𝜕₀𝑓𝜕_𝑗1𝑓+2𝑉𝜕_𝑗𝑓𝜕_𝑁+11𝑓−2𝜕_𝑗𝑉𝜕_𝑁+1𝑓,𝑅_{0𝑗𝑘𝑙}=𝛿_𝑗𝑙𝑒^𝑓12𝜕₀𝜕_𝑘1𝑓+2𝜕_𝑘𝑉𝜕_𝑁+11𝑓−4𝜕₀𝑓𝜕_𝑘𝑓−𝛿_𝑗𝑘𝑒^𝑓12𝜕₀𝜕_𝑙1𝑓+2𝜕_𝑙𝑉𝜕_𝑁+11𝑓−4𝜕₀𝑓𝜕_𝑙𝑓,𝑅_{0𝑗𝑘𝑁+1}=𝑒^𝑓−12𝜕_𝑗𝜕_𝑘1𝑓+4𝜕_𝑗𝑓𝜕_𝑘𝑓−𝛿_𝑗𝑘𝑒^𝑓12𝜕₀𝜕_𝑁+11𝑓+2𝑉𝜕_𝑁+1𝑓²+14𝜕₀𝑓𝜕_𝑁+11𝑓+4∇𝑓²,𝑅_{0𝑁+10𝑁+1}=−𝑒^𝑓𝜕₀𝜕_𝑁+1𝑓−𝑉𝜕_𝑁+1𝜕_𝑁+11𝑓+4∇𝑓²,𝑅_{0𝑁+1𝑘𝑙}𝑅=0,_{0𝑁+1𝑘𝑁+1}=𝑒^𝑓−12𝜕_𝑘𝜕_𝑁+11𝑓+4𝜕_𝑘𝑓𝜕_𝑁+1𝑓,𝑅_{𝑖𝑗𝑘𝑙}=𝛿_𝑗𝑙𝑒^𝑓12𝜕_𝑖𝜕_𝑘1𝑓−4𝜕_𝑖𝑓𝜕_𝑘𝑓−𝛿_𝑖𝑙𝑒^𝑓12𝜕_𝑗𝜕_𝑘1𝑓−4𝜕_𝑗𝑓𝜕_𝑘𝑓+𝛿_𝑖𝑘𝑒^𝑓12𝜕_𝑗𝜕_𝑙1𝑓−4𝜕_𝑗𝑓𝜕_𝑙𝑓−𝛿_𝑗𝑘𝑒^𝑓12𝜕_𝑖𝜕_𝑙1𝑓−4𝜕_𝑖𝑓𝜕_𝑙𝑓+𝛿_𝑖𝑘𝛿_𝑗𝑙−𝛿_𝑖𝑙𝛿_𝑗𝑘𝑒^𝑓12𝜕₀𝑓𝜕_𝑁+11𝑓+2𝑉𝜕_𝑁+1𝑓²+14∇𝑓²,𝑅_{𝑖𝑗𝑘𝑁+1}=𝛿_𝑖𝑘𝑒^𝑓12𝜕_𝑗𝜕_𝑁+11𝑓−4𝜕_𝑗𝑓𝜕_𝑁+1𝑓−𝛿_𝑗𝑘𝑒^𝑓12𝜕_𝑖𝜕_𝑁+11𝑓−4𝜕_𝑖𝑓𝜕_𝑁+1𝑓,𝑅_{𝑖𝑁+1𝑘𝑁+1}=𝛿_𝑖𝑘𝑒^𝑓12𝜕_𝑁+1𝜕_𝑁+11𝑓−4𝜕_𝑁+1𝑓², (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 Γ⁰₀₀=𝜕₀𝑓+𝑉𝜕_𝑁+1𝑓,Γ^𝑘₀₀=𝜕_𝑘𝑉+𝑉𝜕_𝑘Γ𝑓,_𝑁+100=𝑉𝜕₀𝑓+2𝑉²𝜕_𝑁+1𝑓,Γ⁰_0𝑗=12𝜕_𝑗Γ𝑓,_𝑘0𝑗=12𝛿_𝑗𝑘𝜕₀𝑓,Γ^𝑁+1_0𝑗=−𝜕_𝑗Γ𝑉,_00𝑁+1=0,Γ^𝑘_0𝑁+11=−2𝜕_𝑘𝑓,Γ^𝑁+1_0𝑁+1=−𝑉𝜕_𝑁+1Γ𝑓,_0𝑖𝑗1=−2𝛿_𝑖𝑗𝜕_𝑁+1𝑓,Γ^𝑘_𝑖𝑗=12𝛿_𝑖𝑘𝜕_𝑗1𝑓+2𝛿_𝑗𝑘𝜕_𝑖1𝑓−2𝛿_𝑖𝑗𝜕_𝑘Γ𝑓,_{𝑁+1𝑖𝑗}1=−2𝛿_𝑖𝑗𝜕₀𝑓−𝛿_𝑖𝑗𝑉𝜕_𝑁+1𝑓,Γ⁰_𝑖𝑁+1Γ=0,_{𝑘𝑖𝑁+1}=12𝛿_𝑖𝑘𝜕_𝑁+1𝑓,Γ^𝑁+1_𝑖𝑁+1=12𝜕_𝑖𝑓,Γ⁰_𝑁+1𝑁+1Γ=0,_{𝑘𝑁+1𝑁+1}=0,Γ^𝑁+1_𝑁+1𝑁+1=𝜕_𝑁+1𝑓. (3.5) Then from (2.4) and 𝑅_{𝛼𝛽𝛾𝜂}=𝑔_𝛽𝜆𝑅^𝜆_𝛼𝛾𝜂, one can get the conclusion, immediately.

Proposition 3.2. The conformal Ricci curvatures of 𝑀×ℝ² are given by 𝑅₀₀𝑁=Δ𝑉+𝑉Δ𝑓−2𝜕₀𝜕₀𝑁𝑓+4𝜕₀𝑓²+𝑁𝑉𝜕₀𝑓𝜕_𝑁+1𝑓+𝑁𝑉²𝜕_𝑁+1𝑓²+𝑁2+𝑁∇𝑓⋅∇𝑉2𝑉∇𝑓²+2𝑉𝜕₀𝜕_𝑁+1𝑓+2𝑉²𝜕_𝑁+1𝜕_𝑁+1𝑅𝑓,_0𝑘1=−𝑁2𝜕₀𝜕_𝑘1𝑓+2𝜕_𝑘𝑉𝜕_𝑁+11𝑓−4𝜕₀𝑓𝜕_𝑘𝑓,𝑅_0𝑁+11=−2Δ𝑓−𝑁+22𝜕₀𝜕_𝑁+1𝑓−𝑉𝜕_𝑁+1𝜕_𝑁+1𝑓1−𝑁2𝑉𝜕_𝑁+1𝑓²+14𝜕₀𝑓𝜕_𝑁+11𝑓+4∇𝑓²,𝑅_𝑖𝑘𝑁=−2𝜕_𝑖𝜕_𝑘𝑁𝑓+4𝜕_𝑖𝑓𝜕_𝑘𝑓−𝛿_𝑖𝑘𝜕₀𝜕_𝑁+1𝑁𝑓+2𝜕₀𝑓𝜕_𝑁+1𝑁𝑓+2𝑉𝜕_𝑁+1𝑓²+𝑁4∇𝑓²+𝑉𝜕_𝑁+1𝜕_𝑁+11𝑓+2,𝑅Δ𝑓_𝑖𝑁+1𝑁=−2𝜕_𝑖𝜕_𝑁+1𝑁𝑓+4𝜕_𝑖𝑓𝜕_𝑁+1𝑅𝑓,_𝑁+1𝑁+1𝑁=−2𝜕_𝑁+1𝜕_𝑁+1𝑁𝑓+4𝜕_𝑁+1𝑓². (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 ℛ=−𝑒^−𝑓()𝑁()𝑁+1Δ𝑓+𝑁+14∇𝑓²+2(𝑁+1)𝜕₀𝜕_𝑁+1𝑓−𝑒^−𝑓𝑁(𝑁+1)2𝜕₀𝑓𝜕_𝑁+1𝑓+2(𝑁+1)𝑉𝜕_𝑁+1𝜕_𝑁+1𝑁()𝑓+𝑁+12𝜕_𝑁+1𝑓². (3.7)

Proof. From (2.6), and the conclusion in Proposition 3.2, we can obtain the conclusion of Theorem 3.3, immediately.

Remark 3.4. When 𝑉 is a smooth function on the configuration space and let 𝑓=𝑉, we can get 𝑅₀₀=()𝑁𝑉+1Δ𝑉+2|∇𝑉|²,𝑅_0𝑘=0,𝑅_0𝑁+11=−2𝑁Δ𝑉−4|∇𝑉|²,𝑅_𝑖𝑘𝑁=−2𝜕_𝑖𝜕_𝑘𝑁𝑉+4𝜕_𝑖𝑉𝜕_𝑘𝑉−𝛿_𝑖𝑘12𝑁Δ𝑉+4|∇𝑉|²,𝑅_𝑖𝑁+1=0,𝑅_𝑁+1𝑁+1=0,ℛ=−𝑒^−𝑉()𝑁()𝑁+1Δ𝑉+𝑁+14|∇𝑉|². (3.8)

Theorem 3.5. The conformal geodesic equations are d²𝑞^𝑘d𝑡²+6𝐸−4𝑉−12()𝜕1−2𝐿^𝑁+1𝑓d𝑞^𝑘−1d𝑡2𝜕_𝑘𝑓d𝑞^𝑘d𝑡²+1𝐸−𝑉+2𝜕_𝑘𝑓+𝜕_𝑘𝑉=0, (3.9) and 𝑓 satisfies 𝜕₀𝑓=𝐿𝜕_𝑁+11𝑓,2−𝐿d𝑓1d𝑡=2𝐸−𝑉−4𝜕_𝑁+1𝑓, (3.10) where d𝑓d𝑡=𝜕_𝑖𝑓d𝑞^𝑖d𝑡,𝑖=1,…,𝑁,𝐿=𝐸−2𝑉. (3.11)

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→ℝ, the geodesic equations are d²𝑞^𝑘d𝑡²−12𝜕_𝑘𝑓d𝑞^𝑘d𝑡²+1𝐸−𝑉+2𝜕_𝑘𝑓+𝜕_𝑘𝑉=0, (3.12) and 𝑓 satisfies that d𝑓d𝑡=0. (3.13)

Lemma 3.7 (Hopf). Let (𝑀,𝑔) be a compact manifold, 𝑓∈𝐶^∞(𝑀). Then ∫_𝑀Δ_𝑔𝑓=0.

Lemma 3.8. Let 𝑓,ℎ∈𝐶^∞(𝑀), 𝑔=𝑒^𝑓𝑔, then ∇_𝑔ℎ=𝑒^−𝑓∇_𝑔Δℎ,_𝑔ℎ=𝑒^−𝑓Δ_𝑔ℎ+𝑁−22𝑔∇_𝑔𝑓,∇_𝑔ℎ. (3.14)

Proof. From the definition of gradient and Laplacian of ℎ∈𝐶^∞(𝑀), one can obtain ∇_𝑔ℎ=𝜕_𝑖ℎ𝑔^𝑖𝑗𝜕_𝑗=𝑒^−𝑓𝜕_𝑖ℎ𝑔^𝑖𝑗𝜕_𝑗=𝑒^−𝑓∇_𝑔Δℎ,_𝑔1ℎ=𝐺𝜕_𝑖𝐺𝑔^𝑖𝑗𝜕_𝑗ℎ=𝑒^{−(𝑁/2)𝑓}1√𝐺𝜕_𝑖𝑒^{((𝑁−2)/2)𝑓}√𝐺𝑔^𝑖𝑗𝜕_𝑗ℎ=𝑒^−𝑓1√𝐺𝜕_𝑖√𝐺𝑔^𝑖𝑗𝜕_𝑗ℎ+𝑁−22𝑒^−𝑓𝑔^𝑖𝑗𝜕_𝑗ℎ𝜕_𝑖𝑓=𝑒^−𝑓Δ_𝑔ℎ+𝑁−22𝑔∇_𝑔𝑓,∇_𝑔ℎ, (3.15) where 𝐺=det(𝑔_𝑖𝑗).

Theorem 3.9. Let 𝑓∶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 _{𝑀×𝑈1×𝑈2}𝑅=−_{𝑀×𝑈1×𝑈2}𝑒^−𝑓()𝑁()𝑁+1Δ𝑓+𝑁+14∇𝑓²=−_{𝑀×𝑈1×𝑈2}(𝑁+1)Δ_𝑔()𝑓−𝑁+1)(𝑁−44𝑒^−𝑓∇𝑓²=(𝑁+1)(𝑁−4)4_{𝑀×𝑈1×𝑈2}𝑒^−𝑓∇𝑓², (3.16) and moreover,(1)when 𝑁>4, ∫_{𝑀×𝑈1×𝑈2}𝑅≥0,(2)when 𝑁=4, ∫_{𝑀×𝑈1×𝑈2}𝑅≡0,(3)when 𝑁<4, ∫_{𝑀×𝑈1×𝑈2}𝑅≤0.In addition, when 𝑁≠4 and the equality of (3.16) holds, |∇_𝑔𝑓|²=𝑒^−2𝑓|∇𝑓|²=0, so that 𝑓 is a constant function on the configuration space 𝑀.

Theorem 3.10. Let 𝑓∶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 d²𝐽⁰d𝑡²+𝜕_𝑖𝑓d𝐽^𝑖d𝑡+𝐴₁𝐽⁰+𝜕_𝑖𝜕_𝑗𝑓d𝑞^𝑗𝐽d𝑡^𝑖d=0,₂𝐽^𝑖d𝑡²+𝐴₂d𝐽⁰d𝑡+𝐴₃d𝐽^𝑗d𝑡−𝜕_𝑖𝑓d𝐽^𝑁+1d𝑡+𝐴₄𝐽^𝑗d=0,₂𝐽^𝑁+1d𝑡²−2d𝑉d𝑡d𝐽⁰d𝑡+𝐴₅d𝐽^𝑗d𝑡+𝐴₆𝐽⁰+𝐴₇𝐽^𝑖+𝐴₈𝐽^𝑁+1=0, (3.17) where 𝐴₁=12𝜕_𝑖𝑓d²𝑞^𝑖d𝑡²+12∇𝑓⋅∇𝑉−2𝐿+18∇𝑓²,𝐴₂=2𝜕_𝑖1𝑉+𝐸−2𝜕_𝑖𝐴𝑓,₃=𝜕_𝑗𝑓d𝑞^𝑖d𝑡−𝜕_𝑖𝑓d𝑞^𝑗,𝐴d𝑡₄=12𝛿_𝑖𝑘𝜕_𝑗𝑓+𝛿_𝑖𝑗𝜕_𝑘𝑓−𝛿_𝑘𝑗𝜕_𝑖𝑓d²𝑞^𝑘d𝑡²+𝜕_𝑖𝜕_𝑗1𝑉−4𝛿_𝑖𝑗∇𝑓²+12𝜕_𝑖𝑉𝜕_𝑗𝑓+𝜕_𝑗𝑉𝜕_𝑖𝑓−𝜕_𝑖𝜕_𝑗𝑓+𝛿_𝑖𝑗∇𝑓⋅∇𝑉+𝜕_𝑗𝜕_𝑠𝑓d𝑞^𝑠d𝑡d𝑞^𝑗,𝐴d𝑡₅=−2𝜕_𝑗1𝑉+2−𝐿𝜕_𝑗𝐴𝑓,₆=−𝜕_𝑗𝑉d²𝑞^𝑗d𝑡²+12∇𝑓⋅∇𝑉−|∇𝑉|²,𝐴₇=−2𝜕_𝑖𝜕_𝑗𝑉d𝑞^𝑗+1d𝑡2−𝐿𝜕_𝑖𝜕_𝑗𝑓d𝑞^𝑗,𝐴d𝑡₈=12𝜕_𝑗𝑓d²𝑞^𝑗d𝑡²+12−1∇𝑓⋅∇𝑉4∇𝑓². (3.18)

Proof. Firstly, from the Jacobi equation (2.9), we can get the conformal Jacobi equations represented in components as d²𝐽^𝛼d𝑡²+2Γ^𝛼_𝛽𝜎d𝑞^𝛽d𝑡d𝐽^𝜎+Γd𝑡^𝛼𝛽𝜎d²𝑞^𝛽d𝑡²+Γ^𝛾_𝛽𝜎Γ^𝛼_𝛾𝜆d𝑞^𝛽d𝑡d𝑞^𝜆d𝑡+𝜕_𝜆Γ^𝛼_𝛽𝜎d𝑞^𝛽d𝑡d𝑞^𝜆−𝑅d𝑡^{𝛼𝛽𝜆𝜎}d𝑞^𝛽d𝑡d𝑞^𝜆𝐽d𝑡^𝜎=0, (3.19) where 𝑅^𝛼_𝛽𝜆𝜎 are given by (2.4). From (2.4), (3.19) turns into d²𝐽^𝛼d𝑡²+2Γ^𝛼_𝛽𝜎d𝑞^𝛽d𝑡d𝐽^𝜎+Γd𝑡^𝛼𝛽𝜎d²𝑞^𝛽d𝑡²+Γ^𝛾_𝛽𝜆Γ^𝛼_𝛾𝜎d𝑞^𝛽d𝑡d𝑞^𝜆d𝑡+𝜕_𝜎Γ^𝛼_𝛽𝜆d𝑞^𝛽d𝑡d𝑞^𝜆𝐽d𝑡^𝜎=0. (3.20) 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 let 𝑓_𝑢∶𝑀×[𝑎,𝑏]→ℝ be a smooth function with 𝑓_𝑢|_𝑢=0=0, and 𝑔_𝑢=𝑒^𝑓𝑢𝑔, that is, 𝑔_𝑢|_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 𝑒^𝑓𝑢𝜕𝑓_𝑢𝜕𝑢=−Δ𝑓_𝑢−𝑁2∇𝑓_𝑢²,𝑁Δ𝑉+2∇𝑓_𝑢⋅∇𝑉=0,2𝜕_𝑖𝜕_𝑘𝑓_𝑢−𝜕_𝑖𝑓_𝑢𝜕_𝑘𝑓_𝑢=0,𝑖≠𝑘,Δ𝑓_𝑢−𝑁𝜕_𝑖𝜕_𝑖𝑓_𝑢+𝑁2𝜕_𝑖𝑓_𝑢²+𝑁−12∇𝑓_𝑢²=0,𝑖=1,2,…,𝑁. (3.21)

Proof. From (2.12), one can obtain −2𝑅₀₀=−𝑒^𝑓𝑢𝜕𝑓_𝑢𝜕𝑢2𝑉,−2𝑅_0𝑖=0,−2𝑅_0𝑁+1=𝑒^𝑓𝑢𝜕𝑓_𝑢,𝜕𝑢−2𝑅_𝑖𝑘=𝑒^𝑓𝑢𝜕𝑓_𝑢𝛿𝜕𝑢^𝑖𝑘,−2𝑅_𝑖𝑁+1=0,−2𝑅_𝑁+1𝑁+1=0, (3.22) where 𝑅_𝛼𝛽 are the Ricci curvatures with the metric 𝑔_𝑢.

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→ℝ ia a smooth function and the conformal metric 𝑔=𝑒^𝑓𝑔, 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 1𝐻=2^𝑁_𝑖=1𝑝²_𝑖+𝑉𝑞¹,𝑞²,…,𝑞^𝑁, (4.1) in which 1𝑉=2^𝑁−1_𝑖=1𝜔²_𝑖𝑞^𝑖2−𝜆²2𝑞^𝑁2, (4.2) where 𝜔_𝑖 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 ∇𝑉=𝜔²₁𝑞¹,𝜔²₂𝑞²,…,𝜔²_𝑁−1𝑞^𝑁−1,−𝜆²𝑞^𝑁,Δ𝑉=^𝑁−1_𝑖=1𝜔²_𝑖−𝜆². (4.3) The coordinate of the manifold 𝑀×ℝ² is 𝑞=(𝑞⁰=𝑡,𝑞¹,𝑞²,…,𝑞^𝑁,𝑞^𝑁+1) where 𝑞^𝑁+1=(𝑡/2)+𝐶²₂−∫^𝑡₀𝐿d𝜏.

Proposition 4.2. The conformal Ricci curvatures and scalar curvature of the 𝑁 degrees of freedom linear Hamiltonian system are given by 𝑅₀₀=^𝑁−1_𝑖=1𝜔²_𝑖−𝜆²𝑁+𝑉Δ𝑓+2+𝑁∇𝑓⋅∇𝑉2𝑉∇𝑓²,𝑅_0𝑘𝑅=0,_0𝑁+11=−2𝑁Δ𝑓−4∇𝑓²,𝑅_𝑖𝑁+1=0,𝑅_𝑁+1𝑁+1𝑅=0,_𝑖𝑘𝑁=−2𝜕_𝑖𝜕_𝑘𝑁𝑓+4𝜕_𝑖𝑓𝜕_𝑘𝑓−𝛿_𝑖𝑘12𝑁Δ𝑓+4∇𝑓²,ℛ=−𝑒^−𝑓)𝑁()(𝑁+1Δ𝑓+𝑁+14∇𝑓², (4.4) where 𝑖,𝑘 are from 1 to 𝑁.

Moreover, from (3.12), we can get the geodesic equations as 𝑞⁰d=𝑡,₂𝑞^𝑘d𝑡²−12𝜕_𝑘𝑓d𝑞^𝑘d𝑡²+1𝐸−𝑉+2𝜕_𝑘𝑓+𝜔²_𝑘𝑞^𝑘d=0,𝑘=1,2,…,𝑁−1,₂𝑞^𝑁d𝑡²−12𝜕_𝑁𝑓d𝑞^𝑁d𝑡²+1𝐸−𝑉+2𝜕_𝑘𝑓−𝜆²𝑞^𝑁𝑞=0,_𝑁+1=𝑡2+𝐶²₂−^𝑡₀𝐿d𝜏, (4.5) where 𝐿=𝐸−2𝑉, and 𝑓 satisfies d𝑓/d𝑡=0.

Remark 4.3. When 𝑓 is a constant function, (4.5) yields that 𝑞⁰d=𝑡,₂𝑞^𝑘d𝑡²+𝜔²_𝑘𝑞^𝑘d=0,𝑘=1,2,…,𝑁−1,₂𝑞^𝑁d𝑡²−𝜆²𝑞^𝑁𝑞=0,_𝑁+1=𝑡2+𝐶²₂−^𝑡₀𝐿d𝜏. (4.6) 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 𝑒^𝑓𝑢𝜕𝑓_𝑢𝜕𝑢=−Δ𝑓_𝑢−𝑁2∇𝑓_𝑢²,_{𝑁−1𝑖=1}𝜔²_𝑖𝑞^𝑖𝜕_𝑖𝑓_𝑢−𝜆²𝑞^𝑁𝜕_𝑁𝑓_𝑢+2𝑁^𝑁−1_𝑖=1𝜔²_𝑖−𝜆²=0,2𝜕_𝑖𝜕_𝑘𝑓_𝑢−𝜕_𝑖𝑓_𝑢𝜕_𝑘𝑓_𝑢=0,𝑖≠𝑘,Δ𝑓_𝑢−𝑁𝜕_𝑖𝜕_𝑖𝑓_𝑢+𝑁2𝜕_𝑖𝑓_𝑢²+𝑁−12∇𝑓_𝑢²=0,𝑖=1,2,…,𝑁, (4.7) where {𝑓_𝑢|𝑓_𝑢|_𝑢=0=0} is a family of smooth functions.

Example 4.5. For the Hénon-Heiles Model, the Hamiltonian is 1𝐻=2𝑝²₁+𝑝²₂+𝑉𝑞¹,𝑞², (4.8) in which the potential 𝑉 is given by 1𝑉=2𝑞¹²+𝑞²²+𝑞¹²𝑞₂−13𝑞²³. (4.9) 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 𝑅₀₀=Δ𝑉+𝑉Δ𝑓+∇𝑓⋅∇𝑉+𝑉∇𝑓²,𝑅_0𝑘𝑅=0,₀₃1=−21Δ𝑓−2∇𝑓²,𝑅_𝑖3=0,𝑅₃₃𝑅=0,_𝑖𝑘=−𝜕_𝑖𝜕_𝑘1𝑓+2𝜕_𝑖𝑓𝜕_𝑘𝑓−𝛿_𝑖𝑘121Δ𝑓+2∇𝑓²,ℛ=−𝑒^−𝑓33Δ𝑓+2∇𝑓², (4.10) 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 =1𝐸−𝑉=𝐻−𝑉2𝑝²₁+𝑝²₂=12d𝑞¹d𝑡²+d𝑞²d𝑡². (4.11) Then the geodesic equations are given by 𝑞⁰d=𝑡,₂𝑞¹d𝑡²−12𝜕₁𝑓d𝑞¹d𝑡²+1𝐸−𝑉+2𝜕₁𝑓+𝑞¹+2𝑞¹𝑞²d=0,₂𝑞²d𝑡²−12𝜕₂𝑓d𝑞²d𝑡²+1𝐸−𝑉+2𝜕₂𝑓+𝑞²+𝑞¹²−𝑞²²𝑞=0,₃=𝑡2+𝐶²₂−^𝑡₀𝐿d𝜏, (4.12) where d𝑓/d𝑡=0, and 𝐸−𝑉 is given by (4.11).

Remark 4.7. When 𝑓 is a constant function on 𝑀, the geodesic equations are given by 𝑞⁰d=𝑡,₂𝑞¹d𝑡²+𝑞¹+2𝑞¹𝑞²d=0,₂𝑞²d𝑡²+𝑞²+𝑞¹²−𝑞²²𝑞=0,₃=𝑡2+𝐶²₂−^𝑡₀𝐿d𝜏. (4.13) 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 𝑝_𝑖=d𝑞^𝑖/d𝑡, 𝑖=1,2. Then the second-order differential equations in (4.13) become into the following first-order differential equations 𝑞⁰=𝑡,d𝑞¹d𝑡=𝑝₁,d𝑝₁d𝑡+𝑞¹+2𝑞¹𝑞²=0,d𝑞²d𝑡=𝑝₂,d𝑝₂d𝑡+𝑞²+𝑞¹²−𝑞²²𝑞=0,₃=𝑡2+𝐶²₂−^𝑡₀𝐿d𝜏, (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).

Figure 1 

Component 𝑞0 of the geodesic.

Figures 2 and 3 above give the motion of the Hénon-Heiles model. Figures 5 and 6 describe the forms of generalized momenta.

Figure 2 

Component 𝑞1 of the geodesic.

Figure 3 

Component 𝑞2 of the geodesic.

Figure 4 

Component 𝑞3 of the geodesic.

Figure 5 

Generalized momentum 𝑝1.

Figure 6 

Generalized momentum 𝑝2.

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 d²𝐽⁰d𝑡²d=0,₂𝐽¹d𝑡²+2𝑞¹+2𝑞¹𝑞²d𝐽⁰+d𝑡1+2𝑞²𝐽¹+2𝑞¹𝐽²d=0,₂𝐽²d𝑡²+2𝑞²+𝑞¹²−𝑞²²d𝐽⁰d𝑡+2𝑞¹𝐽¹+1−2𝑞²𝐽²d=0,₂𝐽³d𝑡²−2d𝑉d𝑡d𝐽⁰𝑞d𝑡−2¹+2𝑞¹𝑞²d𝐽¹𝑞d𝑡−2²+𝑞¹²−𝑞²²d𝐽²−||d𝑡∇𝑉²𝐽⁰−21+2𝑞²d𝑞¹d𝑡+2𝑞¹d𝑞²𝐽d𝑡¹−22𝑞¹d𝑞¹+d𝑡1−2𝑞²d𝑞²𝐽d𝑡²=0, (4.15) 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 𝑢_𝑖=d𝐽^𝑖/d𝑡, 𝑖=0,1,…,3. Then (4.15) becomes into d𝐽⁰d𝑡=𝑢₀,d𝑢₀d𝑡=0,d𝐽¹d𝑡=𝑢₁,d𝑢₁𝑞d𝑡+2¹+2𝑞¹𝑞²𝑢₀+1+2𝑞²𝐽¹+2𝑞¹𝐽²=0,d𝐽²d𝑡=𝑢₂,d𝑢₂𝑞d𝑡+2²+𝑞¹²−𝑞²²𝑢₀+2𝑞¹𝐽¹+1−2𝑞²𝐽²=0,d𝐽³d𝑡=𝑢₃,d𝑢₃d𝑡−2d𝑉𝑢d𝑡⁰−2𝑞¹+2𝑞¹𝑞²𝑢₁−2𝑞²+𝑞¹²−𝑞²²𝑢₂−|∇𝑉|²𝐽⁰−21+2𝑞²𝑝₁+2𝑞¹𝑝₂𝐽¹−22𝑞¹𝑝₁+1−2𝑞²𝑝₂𝐽²=0, (4.16) where d𝑞^𝑖/d𝑡 and d²𝑞^𝑖/d𝑡² are replaced by 𝑝_𝑖 and d𝑝_𝑖/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.

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 Hénon-Heiles model, and [16, 17] about the geodesic spread and geodesic derivative equation and their applications (Figures 11-14).

Figure 7 

Component 𝐽0 of the Jacobi field.

Figure 8 

Component 𝐽1 of the Jacobi field.

Figure 9 

Component 𝐽2 of the Jacobi field.

Figure 10 

Component 𝐽3 of the Jacobi field.

Figure 11 

Component 𝑢0.

Figure 12 

Component 𝑢1.

Figure 13 

Component 𝑢2.

Figure 14 

Component 𝑢3.

Proposition 4.9. For the Hénon-Heiles model, from Theorem 3.11, we can obtain the equations of flow as 𝑒^𝑓𝑢𝜕𝑓_𝑢𝜕𝑢=−Δ𝑓_𝑢−∇𝑓_𝑢²,𝑞¹+2𝑞¹𝑞²𝜕₁𝑓_𝑢+𝑞²+𝑞¹²−𝑞²²𝜕₂𝑓_𝑢=−2,2𝜕_𝑖𝜕_𝑘𝑓_𝑢−𝜕_𝑖𝑓_𝑢𝜕_𝑘𝑓_𝑢=0,𝑖≠𝑘,Δ𝑓_𝑢−2𝜕_𝑖𝜕_𝑖𝑓_𝑢+𝜕_𝑖𝑓_𝑢²+12∇𝑓_𝑢²=0,𝑖=1,2. (4.17)

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 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).