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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 830147, 7 pages
On the Geometry of the Movements of Particles in a Hamilton Space
1Faculty of Art and Sciences, Department of Mathematics, Suleyman Demirel University, 32260 Isparta, Turkey
2Faculty of Education, Department of Mathematics Education, Pamukkale University, 20070 Denizli, Turkey
Received 31 December 2012; Accepted 15 February 2013
Academic Editor: Abdelghani Bellouquid
Copyright © 2013 A. Ceylan Coken and Ismet Ayhan. 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 studied on the differential geometry of the Hamilton space including trajectories of the motion of particles exposed to gravitational fields and the cotangent bundle.
As known, a Hamilton space is constructed as a differentiable manifold and a real valuable function defined on its cotangent bundle. The second order partial differentiation (Hessian) of this real valuable function with respect to momentum coordinate determines a metric tensor on the cotangent bundle. However, Hessian of the Hamiltonian with respect to momentum coordinate determines a metric tensor on the manifold. This metric tensor is considered by Miron . Recently, many studies have been done on the metrics defined on the cotangent bundles, and most of these studies are on two distinguished metrics. One of these metrics is the Riemann extension of the torsion-free affine connection [2–4] and the other one is the diagonal lift in cotangent bundle [1, 5]. Willmore  showed that a torsion-free affine connection on a manifold determines canonically a pseudo-Riemannian metric on the cotangent bundle. Furthermore, he expressed this pseudo-Riemannian metric as the Riemann extension of the affine connection. Akbulut et al.  defined a diagonal lift of a Riemannian metric of a manifold to its cotangent bundle, and they studied the differential geometry of the cotangent bundle with respect to this Riemann metric. Oproiu  studied the differential geometry of tangent bundle of a Lagrange manifold when this tangent bundle is endowed with pseudo-Riemannian metric obtained from fundamental tensor field by a method similar to the obtaining of the complete lift of a pseudo-Riemannian metric on a differentiable manifold. Ayhan [7, 8] obtained the images on the cotangent bundle of the some tensor fields (i.e., functions, vector fields, and 1-forms, and tensor fields with types (1,1), (0,2) and (2,0)) on the tangent bundle of a Lagrange manifold which are obtained by vertical, complete, and horizontal lifts under the Legendre transformation.
In this paper, it is proved that the trajectories of particles exposed to gravitational fields are geodesics and the Hamilton function as represented of the total energy of system is constant along these trajectories. We studied the differential geometry of the cotangent bundle of the Hamilton space including the trajectories of particles exposed to gravitational fields. We obtained that the pseudo-Riemannian metric on corresponds to pseudo-Riemannian metric on with respect to Legendre transformation, and we showed that is the Riemann extension of the Levi-Civita connection. Moreover we considered an almost product structure is defined on . By means of and , an almost symplectic structure on is defined. Finally we obtained that the coefficients of the Levi-Civita connection and Riemann curvature tensor of and we found the condition under which is locally flat.
In this study, all the manifolds and the geometric objects are assumed to be , and we use the Einstein summation convention.
2. The Movement in a Hamilton Space
The fundamental physical concept is that a gravitational field is identical to geometry of the Hamilton space. This geometry is determined by Hamiltonian where is a tensor with type given by . is local components of a (pseudo)Riemann metric tensor . At the same time, the second order partial differentiation (Hessian) of Hamiltonian given by (1) with respect to momentum coordinate is equal to the following tensor type of :
The Hamilton space , called a Hamilton mechanic system by mechanists, consists of n-dimensional differentiable manifold and regular Hamiltonian given by (2) providing [1, 10]. The motion of every particle in the Hamilton space depending on time is represented as a curve . For any time , the position coordinates of every particle in the Hamilton space are given by , , or briefly , , and, respectively, the velocity and momentum coordinates are given by , , and , .
The movement equation of any particle in the Hamilton space from the position to is determined by the canonic Hamilton equation which is defined by The solution curves of the differential equation system in (3) are one-parameter group of diffeomorphisms of the Hamilton space . The Hamiltonian is fixed on family of one-parameter curves, which defines a conservation law. As any particle is moving on any curve in the Hamilton space, the total energy of the system is the same on every point of curve. In other words, the Hamiltonian is not changed with respect to variable and the total energy must be constant as the particles move. Since the tangent vector field of a curve , , satisfies the canonic Hamilton equation in (3), this tangent vector field of is called the Hamilton vector field. Integral curves of the Hamilton vector field correspond to geodesics in the Hamilton space . In Section 3, we proved that the integral curves of the Hamilton vector field on correspond to geodesics on and also the value of Hamiltonian does not change on geodesics of for the Hamiltonian of the gravitational fields given by (1).
The Hamilton space is an -dimensional differentiable manifold with , , the local chart and is -dimensional its cotangent bundle with , , the local chart, where is canonical projection, and are the vector space coordinates of an element from with respect to the local frame of defined by the local chart . In classical mechanics, and are called momentum phase space and velocity phase space, respectively. The tangent bundle of has an integrable vector subbundle called the vertical distribution on . A nonlinear connection on is defined by the horizontal distribution by and is complementary to in . Thus . The system of the local vector fields is a local frame in and the system of the local vector fields is a local frame in .
The Legendre transformation is a diffeomorphism between the open set of and the open set of . Let , be an adapted frame (coframe) on and , be an adapted frame (coframe) on . Then the differential geometric objects on can be expressed in terms of those of by using the Legendre transformation as follows:
3. The Integral Curves and Metrics
In this section, we studied the relation between the integral curves of the Hamilton vector field on and the geodesics on . Then we obtained the pseudo-Riemann metric on by using two different methods. In addition, we defined an almost symplectic structure on by using and an almost product structure . Finally, the fact that the total energy is constant for each stage of the system as the system with -particles moves with the effect of the gravitational field is reexpressed in terms of differential geometric objects on the cotangent bundle of the Hamilton space .
Theorem 1. Let be the Hamiltonian given by (1). Let be a curve in , be a projection of to ; that is, , and let be a 1-form associated with the tangent vector of curve .(i)If the curve is an integral curve of the Hamilton vector field , the curve is geodesic.(ii)The Hamiltonian of the gravitational field is constant along the geodesic of the Hamilton space.
Proof. (i) Let be the Hamilton vector field. has local expression with respect to induced coordinate system on
The tangent vector field of in has local coordinate expression
with respect to induced coordinate system of . If the curve is an integral curve of the Hamilton vector field , the equation holds. From this equation, we obtain the following canonic Hamilton equations:
The right part of the above equations is expressed by
Using the composite function differentiation, we get
Next, transvecting by , we get
We get a nonlinear connection on defined by
Then, we obtain
and since the following equation is satisfied:
Subsequently we obtained that and . If we substitute the above equation into (12), we get
and transvecting by , we get
Then we get
Since is a 1-form associated with the tangent vector, of curves , and Riemann connection satisfies the following property:
Therefore it can be seen that straightforward the curve is a geodesic curve.
(ii) It is sufficient to show the Hamiltonian is not changed with respect to variable in order to prove the theorem. We calculate If we take into account (3), we obtain . Thus is not dependent on value .
Therefore, we obtain that the trajectories of particles exposed to gravitational fields are geodesics and the Hamilton function represented of the total energy of system is constant along these trajectories.
We consider differential geometric objects on the cotangent bundle of the Hamilton space. Let us start by obtaining a metric on . A pseudo-Riemann metric on the cotangent bundle of the Hamilton space is obtained by using two different ways. Firstly, the pseudo-Riemann metric on the cotangent bundle is obtained as we were inspired by the paper of Willmore  as follows.
Theorem 2. The Levi-Civita connection on determines canonically a pseudo-Riemannian metric on .
Proof. Let be a point on such that . Let be a tangent vector to at . The image of the curve under the bundle projection map is a curve on , passing through . The curve can be regarded as a field of covariant vectors defined along the curve . The covariant derivative is a covector at which can be evaluated on the projected tangent vector . This defines a quadratic differential form on . From this we obtain a bilinear form on at by the usual formula where and are tangent vectors to at . We shall consider that corresponds to Riemann extension of on . We choose a local coordinate system , , valid in some neighborhood around . Then a local coordinate system for is , where . The curve may be expressed locally by and the corresponding curve is . The vector at is given by and its projection by . Then, at when, we have where are the connection coefficients of . We evaluate this covector on to get the number If the equation which is obtained above taken into account in (26), we get for the following: Therefore has local expression with respect to the induced coordinates in . Since the adapted dual frame on is , where we get .
Secondly, the pseudo-Riemann metric on the cotangent bundle is obtained as follows.
Theorem 3. Let be a manifold with a (pseudo)Riemann metric . Then the pseudo-Riemannian metric on corresponds to the pseudo-Riemannian metric on .
Proof. Let be a (pseudo)Riemannian metric on then given by is a pseudo-Riemann metric on . By using the equalities in (4), we get which gives a pseudo-Riemann metric on .
In order to understand the relation between the pseudo- Riemann manifold and the symplectic manifold , we need to define an almost symplectic structure on the cotangent bundle of the Hamilton space and an almost product structure on . The definition of and was obtained as we were inspired by the studies of Miron  for the Hamilton space.
Definition 4. Let be globally defined as 1-form on . The exterior differential dw of the 1-form w is called an almost symplectic structure on the cotangent bundle of the Hamilton space given by
Definition 5. Let be a 2n-dimensional manifold. A mixed tensor field defines an endomorphism on each tangent space of . If there exists a mixed tensor field which satisfies
we say that the field gives an almost product structure to .
We can consider the tensor field with type (1, 1) on :
Theorem 6. is an almost product structure on .
Proof. We have from which .
Theorem 7. Let be a pseudo-Riemann metric defined as Riemann extension of in and let be an almost product structure on . is an almost symplectic structure associated with (). Nondegenerate skew-symmetric 2-form on is given by following equation:
Proof. By using (33), the value of the vector fields , on under is On the other hand, the value of is Therefore, it is seen forward the accuracy of the claim of the theorem.
Theorem 8. Let be Riemann manifold and let be Hamiltonian. For any vector field on ,
Proof. is a 1-form on with local expression
with respect to adapted local dual frame, and has local expression
From (40) and (41), we get
Since is a Hamilton vector field, has local expression with respect to adapted frame on :
where . Thus we have
If we substitute the above equation into (3), we get
From (43) and (46) it is easily seen that .
By using the differential geometric objects , and on the cotangent bundle of the Hamilton space considered in this section, we obtain where is the pseudo-Riemannian metric defining the Riemann extension of the Levi-Civita connection on . If we put instead of in the above equation, we obtain Since , the Hamilton function which gives the total energy of each stage of the system is constant.
4. The Differential Geometry of
In this section, we obtained that the coefficients of the Levi-Civita connection and Riemannian curvature tensor of and we found the condition under which is locally flat.
Theorem 9. The Lie brackets of the horizontal base vector fields and vertical base vector fields on are given by(i),(ii),(iii), where . for .
Theorem 10. Let be Hamilton space, the cotangent bundle of , a pseudo-Riemann metric defined as Riemann extension of Levi-Civita connection in , and the Levi-Civita connection on . Then the connection coefficients of the Levi-Civita connection of the pseudo-Riemannian metric on are given by where
Proof. Let , , and be vector fields on . According to the Koszul formula, we get We put , , and instead of , , and in (51); then we get By the equality , we find and we put , , and instead of , , and in (51). So we get Since the Levi-Civita connection which is defined on is torsion-free, we have . Subsequently we find Thus we get and the rest of the equalities can be obtained similarly.
Theorem 11. Let be a Hamilton space, the cotangent bundle of , a pseudo-Riemann metric defined as Riemann extension of Levi-Civita connection in , the Levi-Civita connection on , and the Riemann curvature tensor on . Then the components of the Riemann curvature tensor on are given by
Proof. Let , , and be vector fields on . Then If we put , , and instead of , , and in (58), we get By the equality , we obtain The other coefficients of the curvature tensor can be obtained similarly.
Theorem 12. The pseudo-Riemann manifold is flat if and only if the Riemann manifold is Euclidean.
Proof. If the Riemann manifold is Euclidean, the Christoffel symbols must be zero. Thus, Riemann curvature tensor on and on must be zero.
5. Concluding Remarks
The projected curves in the Hamilton space of the integral curves of the Hamilton vector field are geodesics. Furthermore, the total energy of each stage of the system is constant. The cotangent bundle of the Hamilton space is flat if and only if the Hamilton space is Euclidean.
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