Research Article | Open Access
Linyu Peng, Huafei Sun, Xiao Sun, "Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric", International Journal of Mathematics and Mathematical Sciences, vol. 2011, Article ID 710274, 26 pages, 2011. https://doi.org/10.1155/2011/710274
Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric
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.
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  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 , where 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 be a smooth manifold with dimension , where is a nondegenerate metric defined on the vector field , and is the affine connection, which is defined as The coefficients of the connection can be represented in terms of the metric tensor by where means , and the Einstein summation convention is used. For any , the curvature tensor is defined by where is the Lie bracket on defined as . Then the components of the curvature tensor can be given by The Ricci curvature is defined by where . The scalar curvature is defined by A curve on is said to be geodesic if its tangent is displaced parallel along the curve , that is, In local coordinate , the geodesic equation satisfies In addition, the equation for the well-known Jacobi field satisfies where is a geodesic on manifold . 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 changing, of dynamical systems. In , 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  for more about its applications to physical systems.
A metric defined on is said to be conformally equivalent to , if there exists a function such that Then the equivalent relation between metrics and the equivalence class of is called its conformal class denoted by . In local coordinate, (2.10) becomes
Hamilton  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 and evolves its metric by the equation where and denote the Ricci curvatures with .
In this paper, we study a similar flow, which is an evolution equation started from a smooth manifold with nondegenerate metric , which is not needed to be Riemannian.
2.2. Geometry in Hamiltonian Dynamics
In mechanics , a Lagrangian function of a dynamical system of degrees of freedom is usually defined by , where and are called the generalized coordinate and generalized velocity, respectively. While a Hamiltonian function is represented as , where is generalized momenta. The space of the generalized coordinate is called the configuration space.
The Hamiltonian action within is The Hamiltonian variational principle states From the calculus of variation, (2.14) equates to the Euler-Lagrange equations
To pass to the Hamiltonian formalism, we introduce the generalized momenta as make the change , and introduce the Hamiltonian Then the Euler-Lagrange equations are equivalent to the Hamiltonian equations, which are given by
In this paper, we will only consider the classical Hamiltonian dynamics, whose Hamiltonian is of the form where is the potential energy.
The energy of the classical Hamiltonian dynamics is a constant equal to Hamiltonian . Considering the manifold defined by in which and , where are real numbers, and in the following, is assumed to be equal to 1.
In this paper, we assume that, the Greek symbols are from 0 to and the Latin symbols are from 1 to .
The Eisenhart metric (cf. ) of the manifold is defined as where is the identity matrix, and we know that it is not degenerate as .
The inverse matrix of is The nonzero coefficients of the Riemanian connection are where means .
The nonzero components of the curvature tensor are The nonzero Ricci curvature and the scalar curvature are, respectively, given by where is the Laplacian operator in the Euclidean space.
The geodesic equations read 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 , in which is the configuration space, we investigate the conformal Eisenhart metric which states as where is the Eisenhart metric and is a smooth function. Then the arc length parameter is shown as And the volume element of the manifold under the conformal Eisenhart metric is given by
Proposition 3.1. The conformal and independent components of the curvature tensor are given by where and are the gradient operator and the inner product in the Euclidean space, respectively.
Proposition 3.2. The conformal Ricci curvatures of are given by
Theorem 3.3. The conformal scalar curvature is given by
Remark 3.4. When is a smooth function on the configuration space and let , we can get
Theorem 3.5. The conformal geodesic equations are and satisfies where
For the classical Hamiltonian system, is usually not a constant. Then if is constrained in , the second equation in (3.10) can be reduced into . Therefore, we can get the following.
Remark 3.6. When assuming , the geodesic equations are and satisfies that
Lemma 3.7 (Hopf). Let be a compact manifold, . Then .
Lemma 3.8. Let , , then
Proof. From the definition of gradient and Laplacian of , one can obtain where .
Theorem 3.9. Let , be compact, where and are compact subsets of . Then from Theorem 3.3, Lemmas 3.7 and 3.8, one can get that and moreover,(1)when , ,(2)when , ,(3)when , .In addition, when and the equality of (3.16) holds, , so that is a constant function on the configuration space .
Theorem 3.10. Let 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 where
Proof. Firstly, from the Jacobi equation (2.9), we can get the conformal Jacobi equations represented in components as where are given by (2.4). From (2.4), (3.19) turns into Furthermore, substituting (3.5) into (3.20), in which should be constrained in but not , we complete the proof of Theorem 3.10.
Now let be a smooth function with , and , that is, , where is the Eisenhart metric. Then we can get the following.
Theorem 3.11. The equations of the flow defined in (2.12) are
Proof. From (2.12), one can obtain where are the Ricci curvatures with the metric .
Then from the conformal Ricci curvatures obtained in Proposition 3.2, and through a direct computation, one can get the conclusion.
In this section, the conformal geometric structures of two famous classical Hamiltonian systems are shown. Assuming that ia a smooth function and the conformal metric , where is the configuration space and is the Eisenhart metric. For the following examples, the total energy is assumed to be a constant.
Example 4.1. For an degrees of freedom linear Hamiltonian system, the Hamiltonian is given by in which where and are constants.
This system consists of uncoupled linear oscillators, with the remaining uncoupled degree of freedom consisting of a parabolic barrier.
From (4.2), one can get The coordinate of the manifold is where .
Proposition 4.2. The conformal Ricci curvatures and scalar curvature of the degrees of freedom linear Hamiltonian system are given by where are from 1 to .
Moreover, from (3.12), we can get the geodesic equations as where , and satisfies .
Proposition 4.4. The equations of flow are given by where is a family of smooth functions.
Example 4.5. For the Hénon-Heiles Model, the Hamiltonian is in which the potential is given by 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 where are from 1 to 2, and .
From the Hamiltonian equations, one can get that Then the geodesic equations are given by where , and is given by (4.11).
Remark 4.7. When is a constant function on , the geodesic equations are given by 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 , which are the generalized momenta, and , . Then the second-order differential equations in (4.13) become into the following first-order differential equations where .
Let the initial vector be , and the interval for time be , 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).
Proposition 4.8. When is a constant function on , the Jacobi field along the geodesic, which is the solution of (4.13), is given by where , and .
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 and assume that , . Then (4.15) becomes into where and are replaced by and given in (4.14), respectively.
We assume that the initial vector is , and the time interval is . 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 , 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).
Proposition 4.9. For the Hénon-Heiles model, from Theorem 3.11, we can obtain the equations of flow as
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).
This paper is supported by the National Natural Science Foundations of China (no. 10871218, no. 10932002).
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