geometry of hamiltonian dynamics with conformal eisenhart metric

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 Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China School of Automation, Beijing Institute of Technology, Beijing 100081, China Correspondence should be addressed to Huafei Sun, huaf.sun@gmail.com Received 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 H´enonHeiles 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 H´enon-Heiles model are obtained And the numerical results for the H´enon-Heiles model show us the instability of the associated geodesic spreads 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, , 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 International Journal of Mathematics and Mathematical Sciences the restricted three body problem Iwai and Yamaoka 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 , nonlocal Hamiltonian operators , hydrodynamics and realization , fluid mechanics , quantum systems , reaction dynamics , 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 is devoted to the conclusion 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 Xj ∂Y i ∂xj Γijk Y k ∂ , ∂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 kl g ∂j gil ∂i gjl − ∂l gij , 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 R X, Y Z DX DY Z − DY DX Z − D X,Y Z, where ·, · is the Lie bracket on M defined as X, Y the curvature tensor can be given by Rlkij ∂i Γlkj − ∂j Γlki 2.3 XY − Y X Then the components of Γhkj Γlhi − Γhki Γlhj 2.4 International Journal of Mathematics and Mathematical Sciences The Ricci curvature is defined by g kl Riklj , Rij where Riklj 2.5 gks Rsilj The scalar curvature is defined by R g ij Rij 2.6 A curve ξ t on M is said to be geodesic if its tangent ξ˙ t is displaced parallel along the curve ξ t , that is, Dξ˙ t ξ˙ t In local coordinate ξ t 2.7 x1 t , x2 t , , xN t , the geodesic equation satisfies d2 x k dt2 Γkij dxi dxj dt dt In addition, the equation for the well-known Jacobi field J Dξ˙ t Dξ˙ t J 2.8 R ξ˙ t , J ξ˙ t , J k ∂k satisfies 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 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 International Journal of Mathematics and Mathematical Sciences 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 Poincar´e 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 −2Rij , ij 2.12 where gt |t 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 q1 , , qN and q˙ q˙ , , q˙ N are called the is usually defined by L qi , q˙ i , t , where q generalized coordinate and generalized velocity, respectively While a Hamiltonian function ∂L/∂q1 , , ∂L/∂qN is generalized momenta The is represented as H qi , pi , t , where p space M of the generalized coordinate is called the configuration space The Hamiltonian action within t ∈ a, b is b S a L q, q, ˙ t dt 2.13 The Hamiltonian variational principle states δS 2.14 From the calculus of variation, 2.14 equates to the Euler-Lagrange equations d ∂L dt ∂q˙ i − ∂L ∂qi 0, i 1, , N 2.15 To pass to the Hamiltonian formalism, we introduce the generalized momenta as pi ∂L , ∂q˙ i i 1, , N, 2.16 International Journal of Mathematics and Mathematical Sciences make the change qi , q˙ i → qi , pi , and introduce the Hamiltonian H q i , pi , t pj q˙ j − L qi , q˙ i , t 2.17 Then the Euler-Lagrange equations are equivalent to the Hamiltonian equations, which are given by dqi dt ∂H , ∂pi dpi dt − ∂H , ∂qi i 1, , N 2.18 In this paper, we will only consider the classical Hamiltonian dynamics, whose Hamiltonian is of the form 1N p 2i i H V q , 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 q0 , q1 , q2 , , qN , qN q|q , 2.20 t in which q0 t and qN C12 /2 t C22 − Ldτ, where C1 , C2 are real numbers, and in the following, C1 is assumed to be equal to In this paper, we assume that, the Greek symbols α, β, γ, are from to N and the Latin symbols i, j, k, are from to N The Eisenhart metric cf 20 of the manifold M × Ê2 is defined as ⎛ −2V gαβ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ··· IN×N ··· ⎞ ⎟ 0⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ 0⎟ ⎠ 2.21 0 where IN×N is the identity matrix, and we know that it is not degenerate as det gαβ −1 N International Journal of Mathematics and Mathematical Sciences The inverse matrix of gαβ is ⎛ ··· 0 g αβ ⎜ ⎜0 ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜0 ⎝ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ IN×N ··· ⎞ 2.22 2V The nonzero coefficients of the Riemanian connection are Γk00 −ΓN 0k ∂k V, 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 2.25 0, where Δ is the Laplacian operator in the Euclidean space The geodesic equations read d2 q dt2 d2 q i dt2 − ∂V , ∂qi d2 q N dt2 0, i 1, 2, N, − 2.26 dL , dt in which, the first and third equations are identical, and the second ones are the equations of motion for the associated dynamics 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 International Journal of Mathematics and Mathematical Sciences where g is the Eisenhart metric and f : M × Ê2 → parameter s is shown as Ê is a smooth function Then the arc length gαβ dqα dqβ ds2 C12 ef dt2 3.2 ef dt2 And the volume element dv of the manifold under the conformal Eisenhart metric is given by det g dq0 ∧ dq1 ∧ · · · ∧ dqN dv −e N/2 f dq ∧ dq ∧ · · · ∧ dq 1 N 3.3 Proposition 3.1 The conformal and independent components of the curvature tensor are given by R0j0l δjl ef 2∂0 ∂0 f − ∂0 f −2V ∇f R0j0N R0jkl 2 δjl ef ∂0 ∂k f − δjk ef − ∇f · ∇V ∂0 f∂j f 1 V ∂j f∂N f − ∂j V ∂N f , 2 1 ∂k V ∂N f − ∂0 f∂k f ∂0 ∂l f ef − ∂j ∂k f − 4V ∂0 f∂N f − 4V ∂N f , ef − ∂0 ∂j f − V ∂j ∂N f − δjk ef R0jkN V ∂j f∂l f ef −∂j ∂l V − V ∂j ∂l f 1 ∂l V ∂N f − ∂0 f∂l f , ∂j f∂k f ∂0 ∂N f V ∂N f 2 ∂0 f∂N f ∇f , International Journal of Mathematics and Mathematical Sciences R0N 10N R0N R0N 1kl 1kN Rijkl ef − ∂k ∂N f , ∂k f∂N f , 1 ∂i ∂k f − ∂i f∂k f δjl ef 1 ∂j ∂k f − ∂j f∂k f − δil ef 1 ∂j ∂l f − ∂j f∂l f 1 ∂i ∂l f − ∂i f∂l f − δjk ef ∂0 f∂N f δik δjl − δil δjk ef δik ef 1 ∂j ∂N f − ∂j f∂N f 1kN δik ef 1 ∂N ∂N f − ∂N f RijkN 0, δik ef RiN ∇f −ef ∂0 ∂N f − V ∂N ∂N f V ∂N f ∇f , 1 ∂i ∂N f − ∂i f∂N f , − δjk ef 2 , 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 ΓN 00 ∂0 f Γk0j Γ00N ΓN ij ΓkiN ΓkN V ∂0 f 2V ∂N f, δjk ∂0 f, ΓN 0j Γk0N 0, 1 − δij ∂N f, Γ0ij Γk00 V ∂N f, ∂k V −∂j V, 1 − ∂k f, Γkij 1N 1 δik ∂N f, 0, ΓN N ∂j f, Γ00j ΓN 0N 1 δik ∂j f − δij ∂0 f − δij V ∂N f, V ∂k f, ΓN iN 1 1N Γ0iN ∂i f, −V ∂N f, 1 δjk ∂i f − δij ∂k f, 2 0, Γ0N 1N 0, ∂N f 3.5 Then from 2.4 and Rαβγ η gβλ Rλαγ η , one can get the conclusion, immediately International Journal of Mathematics and Mathematical Sciences Proposition 3.2 The conformal Ricci curvatures of M × Ê2 are given by R00 ΔV V Δf − N ∂0 ∂0 f N V ∇f N ∇f · ∇V R0k R0N −N RN NV ∂0 f∂N f 2V ∂0 ∂N f NV ∂N f 2V ∂N ∂N f, 1 ∂k V ∂N f − ∂0 f∂k f , − V ∂N f N ∂i ∂k f 2 ∂0 f∂N f ∇f , N ∂i f∂k f − δik ∂0 ∂N f − N ∂i ∂N f 1N − N ∂N ∂N f RiN 2 N ∂0 ∂N f − V ∂N ∂N f − Δf − 2 −N Rik ∂0 ∂k f N ∂0 f N ∂0 f∂N f N V ∂N f 2 N ∇f Δf , V ∂N ∂N f N ∂i f∂N f, N ∂N f 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 −e −f N Δf N N N N ∂0 f∂N f ∇f N 2 N ∂0 ∂N f V ∂N ∂N f N N 3.7 ∂N f Proof From 2.6 , and the conclusion in Proposition 3.2, we can obtain the conclusion of Theorem 3.3, immediately 10 International Journal of Mathematics and Mathematical Sciences Remark 3.4 When V is a smooth function on the configuration space and let f get R00 V Rik ΔV − N |∇V |2 , N ∂i ∂k V R0N 0, 1 N ∂i V ∂k V − δik ΔV RiN R R0k −e−V RN 0, N 1N N N ΔV V , we can N − ΔV − |∇V |2 , N |∇V |2 , 3.8 0, |∇V |2 Theorem 3.5 The conformal geodesic equations are d2 q k dt dqk 6E − 4V − ∂N f − ∂k f − 2L dt dqk dt ∂k f E−V ∂k V 0, 3.9 and f satisfies ∂0 f df −L dt L∂N f, E−V − 3.10 ∂N f, where df dt ∂i f dqi , dt L i 1, , N, 3.11 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 Therefore, we can get the following Remark 3.6 When assuming f : M → d2 q k − ∂k f dt2 Ê, the geodesic equations are dqk dt E−V ∂k f ∂k V 0, 3.12 International Journal of Mathematics and Mathematical Sciences 13 where Rαβλσ are given by 2.4 From 2.4 , 3.19 turns into d2 J α dt2 2Γαβσ dqβ dJ σ dt dt Γαβσ d2 q β dt2 γ Γβλ Γαγσ dqβ dqλ dt dt ∂σ Γαβλ dqβ dqλ dt dt Jσ 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 |u 0, and gu is, gu |u g, where g is the Eisenhart metric Then we can get the following efu g, that Theorem 3.11 The equations of the flow defined in 2.12 are efu ∂fu ∂u ΔV −Δfu − N ∇fu , N ∇fu · ∇V 2∂i ∂k fu − ∂i fu ∂k fu Δfu − N∂i ∂i fu N ∂i fu 2 0, N−1 ∇fu 0, 3.21 i / k, 0, i −2R0N 1, 2, , N Proof From 2.12 , one can obtain −2R00 −2Rik ∂fu 2V, ∂u −2R0i ∂fu δik , e ∂u −2RiN −efu fu 0, 0, −2RN efu 1N ∂fu , ∂u 3.22 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 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 1N p 2i i V q1 , q2 , , qN , 4.1 14 International Journal of Mathematics and Mathematical Sciences in which N−1 i ω q 2i i V − λ2 N q 2 , 4.2 where ωi and λ are constants This system consists of N − uncoupled linear oscillators, with the remaining uncoupled degree of freedom consisting of a parabolic barrier From 4.2 , one can get ω12 q1 , ω22 q2 , , ωN−1 qN−1 , −λ2 qN , ∇V N−1 ΔV i Ê2 The coordinate of the manifold M × t t/2 C22 − Ldτ 4.3 ωi2 − λ2 is q t, q1 , q2 , , qN , qN q0 where qN 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 R0N ωi2 − λ2 N ∇f · ∇V V Δf N ∇f , − Δf − N − ∂i ∂k f Rik N Δf N N R0k RN 0, N ∂i f∂k f − δik Δf −e−f R RiN N V ∇f , ∇f 0, 1N N ∇f 0, 4.4 , , where i, k are from to N Moreover, from 3.12 , we can get the geodesic equations as q0 d2 q k − ∂k f dt2 dqk dt d2 q N − ∂N f dt2 qN where L ∂k f E−V dqN dt t, ωk2 qk E − 2V , and f satisfies df/dt C22 − k 1, 2, , N − 1, ∂k f − λ2 qN E−V t 0, t L dτ, 4.5 0, International Journal of Mathematics and Mathematical Sciences 15 Remark 4.3 When f is a constant function, 4.5 yields that t, q0 d2 qk dt2 ωk2 qk k 0, 1, 2, , N − 1, 4.6 d2 q N − λ2 q N dt2 qN t 0, t C22 − L dτ 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 efu N−1 i ∂fu ∂u −Δfu − ωi2 qi ∂i fu − λ2 qN ∂N fu where {fu |fu |u N ∂i fu N−1 N 2∂i ∂k fu − ∂i fu ∂k fu Δfu − N∂i ∂i fu N ∇fu , i 0, 4.7 i / k, 0, N−1 ∇fu 2 ωi2 − λ2 0, i 1, 2, , N, 0} is a family of smooth functions Example 4.5 For the H´enon-Heiles Model, the Hamiltonian is H p p22 V q1 , q2 , 4.8 in which the potential V is given by V q1 q2 q1 q2 − q 3 4.9 Next we are going to study the geometry of the H´enon-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 R00 ΔV V Δf R03 1 − Δf − ∇f , 2 Rik −∂i ∂k f q1 V ∇f , Ri3 R0k 0, R33 1 ∂i f∂k f − δik Δf 2 ∇f ∇f −e−f 3Δf R where i, k are from to 2, ∇V ∇f · ∇V 2q1 q2 , q2 q1 2 − q2 0, 0, 4.10 , , and ΔV From the Hamiltonian equations, one can get that E−V H −V p p22 ⎛ ⎝ dq1 dt dq2 dt 4.11 ⎞ ⎠ Then the geodesic equations are given by q0 d2 q 1 − ∂1 f dt2 d2 q − ∂2 f dt2 dq2 dt dq1 dt where df/dt ∂1 f E−V E−V q3 t, t ∂2 f C22 − 0, and E − V is given by 4.11 t q2 L dτ, q1 q1 2q1 q2 − q2 0, 4.12 0, International Journal of Mathematics and Mathematical Sciences 17 4.5 q0 3.5 2.5 1.5 1 1.5 2.5 3.5 4.5 t Figure 1: Component q0 of the geodesic 0.8 0.6 0.4 q1 0.2 −0.2 −0.4 −0.6 −0.8 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 d2 q1 dt2 d2 q dt2 q3 q2 q0 t, q1 2q1 q2 q1 t 2 C22 − − q2 t 0, L dτ 4.13 0, 18 International Journal of Mathematics and Mathematical Sciences 150 q2 100 50 0 t Figure 3: Component q2 of the geodesic ×104 0.5 −0.5 −1 q3 −1.5 −2 −2.5 −3 −3.5 −4 −4.5 t Figure 4: Component q3 of the geodesic Moreover, for the above geodesic equations of the H´enon-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 simulation, firstly we introduce two parameters p1 , p2 , which are the generalized momenta, and pi dqi /dt, i 1, International Journal of Mathematics and Mathematical Sciences 19 p1 −1 −2 t Figure 5: Generalized momentum p1 Then the second-order differential equations in 4.13 become into the following first-order differential equations q0 dp1 dt dp2 dt q2 q3 where L dq1 dt p1 , q1 2q1 q2 dq2 dt p2 , q1 t t, 4.14 − q2 C22 − t 0, 0, L dτ, E − 2V Let the initial vector q1 , q2 , q3 , p1 , p2 be 0, 0, 1, 1, , and the interval for time t be 0, , by use of the ode45 method in Matlab, that is, Runge-Kutta 4, method, we can get the following numerical results of geodesics and the generalized momenta Figures 1, Figures and above give the motion of the H´enon-Heiles model Figures and describe the forms of generalized momenta 20 International Journal of Mathematics and Mathematical Sciences 1500 p2 1000 500 0 t Figure 6: Generalized momentum p2 5.5 4.5 J0 3.5 2.5 1.5 1 t Figure 7: Component J of the Jacobi field 20 15 10 J1 −5 −10 −15 t Figure 8: Component J of the Jacobi field International Journal of Mathematics and Mathematical Sciences 21 9000 8000 7000 6000 J2 5000 4000 3000 2000 1000 0 t Figure 9: Component J of the Jacobi field ×106 J3 −2 −4 −6 −8 −10 t Figure 10: Component J of the Jacobi field 1.8 1.6 1.4 u0 1.2 0.8 0.6 0.4 0.2 0 t Figure 11: Component u0 22 International Journal of Mathematics and Mathematical Sciences 200 100 u1 −100 −200 −300 −400 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 dt2 0, d2 J dt2 q1 2q1 q2 d2 J dt2 q2 q1 dJ dt − q2 d2 J dV dJ − q1 − dt dt dt2 −2 2q2 dq1 dt 2q2 J 1 dJ dt 2q1 q2 2q1 where dV/dt q1 2q1 q2 dq1 /dt 2 2 q − q q 2q1 J − 2q2 J 2q1 J dJ − q2 dt dq2 dt J − 2q1 q2 q1 0, q1 dq1 dt − q2 2 − q2 0, − 2q2 4.15 dJ − |∇V |2 J dt dq2 dt J2 dq2 /dt , and |∇V |2 0, q1 2q1 q2 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 −2 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, , Then 4.15 becomes into dJ dt u0 , du0 dt 0, dJ dt u1 , du1 dt q1 dJ dt u2 , du2 dt q2 dJ dt u3 , 2q1 q2 u0 q1 − q2 du3 dV −2 u0 − q dt dt −2 1 2q2 p1 2q2 J u0 2q1 J 2q1 J 2q1 q2 u1 − q2 2q1 p2 J − 2q1 p1 0, − 2q2 J q1 − q2 0, u2 − |∇V |2 J − 2q2 p2 J 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 −0.5 u3 −1 −1.5 −2 −2.5 −3 t Figure 14: Component u3 We assume that the initial vector J , J , J , J , u0 , u1 , u2 , u3 is 1, 1, 1, 1, 1, 1, 1, , and the time interval is 0, 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 H´enonHeiles model, and 16, 17 about the geodesic spread and geodesic derivative equation and their applications Figures 11-14 Proposition 4.9 For the H´enon-Heiles model, from Theorem 3.11, we can obtain the equations of flow as efu q1 ∂fu ∂u 2q1 q2 ∂1 fu −Δfu − ∇fu , q2 q1 2∂i ∂k fu − ∂i fu ∂k fu Δfu − 2∂i ∂i fu ∂i fu 2 − q2 ∂2 fu −2, 4.17 i / k, 0, ∇fu 2 0, i 1, 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 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 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no 3, part A, pp 2722–2732, 1997 Copyright of International Journal of Mathematics & Mathematical Sciences is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission However, users may print, download, or email articles for individual use ... theory of conformal metric in Riemannian geometry is applied to characterize the geometry of the classical Hamiltonian dynamics with the conformal Eisenhart metric Section We obtain the Ricci curvatures,... classical Hamiltonian dynamics, whose Hamiltonian is of the form 1N p 2i i H V q , 2.19 where V q is the potential energy The energy E of the classical Hamiltonian dynamics is a constant equal to Hamiltonian. .. 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