# On the Geometry of the Movements of Particles in a Hamilton Space

Authors: A. Ceylan Coken; Ismet Ayhan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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.


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## Body

## 1. Introduction

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(pi) 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 [1]. 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 [3] 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. [5] 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 [6] 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 bundleT*M of the Hamilton space M including the trajectories of particles exposed to gravitational fields. We obtained that the pseudo-Riemannian metric G on T*M corresponds to pseudo-Riemannian metric gC on TM with respect to Legendre transformation, and we showed that G is the Riemann extension of the Levi-Civita connection. Moreover we considered an almost product structure P is defined on T*M. By means of P and G, an almost symplectic structure θ on T*M is defined. Finally we obtained that the coefficients of the Levi-Civita connection ∇˙ and Riemann curvature tensor K of (T*M,G) and we found the condition under whichT*M is locally flat.In this study, all the manifolds and the geometric objects are assumed to beC∞, 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(1)H(xi,pi)=12gik(x)pipk,
where gik(x) is a tensor with type (2,0) given by gikgkj=δji. gkj(x) is local components of a (pseudo)Riemann metric tensor [9]. At the same time, the second order partial differentiation (Hessian) of Hamiltonian given by (1) with respect to momentum coordinate (pi) is equal to the following tensor type of (2,0):
(2)gik(x)=∂2H∂pi∂pk.The Hamilton spaceM, called a Hamilton mechanic system by mechanists, consists of n-dimensional differentiable manifold M and regular Hamiltonian H given by (2) providing det[gik]≠0 [1, 10]. The motion of every particle in the Hamilton space depending on time is represented as a curve γ:I⊂R→M. For any time t, the position coordinates of every particle in the Hamilton space are given by xi∘γ(t), i=1,…,n, or briefly xi, i=1,…,n, and, respectively, the velocity and momentum coordinates are given by yi=dxi/dt, i=1,…,n, and pi=gijyj, i=1,…,n.The movement equation of any particle in the Hamilton space from the position(x1(t1),…,xn(t1)) to (x1(t2),…,xn(t2)) is determined by the canonic Hamilton equation which is defined by
(3)dxidt=∂H∂pi;dpidt=-∂H∂xi.
The solution curves of the differential equation system in (3) are one-parameter group of diffeomorphisms of the Hamilton space [11]. The Hamiltonian H 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 H of the system is the same on every point of curve. In other words, the Hamiltonian is not changed with respect to variable t and the total energy H must be constant as the particles move. Since the tangent vector field of a curve C:t→(xi(t),pi(t)), i=1,…,n, satisfies the canonic Hamilton equation in (3), this tangent vector field of C is called the Hamilton vector field. Integral curves of the Hamilton vector field correspond to geodesics in the Hamilton space M [12]. In Section 3, we proved that the integral curves of the Hamilton vector field on T*M correspond to geodesics on M and also the value of Hamiltonian H  does not change on geodesics of M for the Hamiltonian of the gravitational fields given by (1).The Hamilton spaceM is an n-dimensional differentiable manifold with (U,xi), i=1,…,n, the local chart and T*M is 2n-dimensional its cotangent bundle with (π-1(U),xi,pi), i=1,…,n, the local chart, where π:T*M→M is canonical projection, xi=xioπ and pi are the vector space coordinates of an element from π-1(U) with respect to the local frame (dx1,…,dxn) of T*M defined by the local chart (U,xi). In classical mechanics, T*M and TM are called momentum phase space and velocity phase space, respectively. The tangent bundle of T*M has an integrable vector subbundle VT*M=Kerπ* called the vertical distribution on T*M. A nonlinear connection on T*M is defined  by the horizontal distribution by HT*M and HT*M is complementary to VT*M in TT*M. Thus TT*M=VT*M⊕HT*M. The system of the local vector fields (∂/∂p1,…,∂/∂pn) is a local frame in VT*M and the system of the local vector fields (δ/δx1,…,δ/δxn) is a local frame in HT*M.The Legendre transformationφ is a diffeomorphism between the open set of U~⊂TM and the open set of U⊂T*M. Let {δ/δxi,∂/∂yi}, {dxi,δyi} be an adapted frame (coframe) on TM and {δ/δxi,∂/∂pi}, {dxi,δpi} be an adapted frame (coframe) on T*M. Then the differential geometric objects on T*M can be expressed in terms of those of TM by using the Legendre transformation as follows:
(4)(φ-1)*(dxi)=dxi,(φ-1)*(δyi)=gijδpj,(φ-1)*(gij)=gij.


## 3. The Integral Curves and Metrics

In this section, we studied the relation between the integral curves of the Hamilton vector field onT*M and the geodesics on M. Then we obtained the pseudo-Riemann metric G on T*M by using two different methods. In addition, we defined an almost symplectic structure θ on T*M by using G and an almost product structure P. Finally, the fact that the total energy H is constant for each stage of the system as the system with n-particles moves with the effect of the gravitational field is reexpressed in terms of differential geometric objects on the cotangent bundle T*M of the Hamilton space M.Theorem 1.
LetH be the Hamiltonian given by (1). Let C be a curve in T*M, γ be a projection of C to M; that is, π∘C=γ, and let w be a 1-form associated with the tangent vector of curve γ(t).(i)
If the curveC is an integral curve of the Hamilton vector field V, the curve γ is geodesic.(ii)
The Hamiltonian of the gravitational fieldH is constant along the geodesic of the Hamilton space.Proof.
(i) LetV be the Hamilton vector field. V has local expression with respect to induced coordinate system on T*M(5)V=∂H∂pi∂∂xi-∂H∂xi∂∂pi.
The tangent vector field of C in T*M has local coordinate expression
(6)C*(ddt)=dxi(t)dt∂∂xi+dpi(t)dt∂∂pi
with respect to induced coordinate system of T*M. If the curve C is an integral curve of the Hamilton vector field V, the equation C*(d/dt)=VC(t) holds. From this equation, we obtain the following canonic Hamilton equations:
(7)dxi(t)dt=∂H∂pi=gijpj,dpi(t)dt=-∂H∂xi.
The right part of the above equations is expressed by
(8)ddt(gij∂H∂pj)+∂H∂xi=0.
Using the composite function differentiation, we get
(9)dpkdt+∂∂xk(gij∂H∂pj)dxkdt+∂H∂xi=0,
and by
(10)dxkdt=gkapa,
we get
(11)dpkdt+∂∂xk(gij∂H∂pj)gkapa+∂H∂xi=0.
Next, transvecting by gka, we get
(12)gkadpkdt+Sj(x,p)=0,
where
(13)Sj(x,p)=∂∂xk(gij∂H∂pj)pa+gka∂H∂xi.
We get a nonlinear connection on T*M defined by
(14)Njk=Sja=12gka∂2H∂xj∂pa,
where
(15)Njk=Sja=12∂Sj∂pa.
Then, we obtain
(16)∂Njk∂pb=12gka∂gab∂xj,
and since the following equation is satisfied:
(17)∂gab∂xj=∂∂xjg(dxa,dxb),
we get
(18)∂Njk∂pb=-Γjkb.
Subsequently we obtained that Njk=-Γjkbpb and Sj(x,p)=-Γjkbpbpa. If we substitute the above equation into (12), we get
(19)gkadpkdt-Γjkbpbpa=0,
and transvecting by gka, we get
(20)dpkdt-Γjkbpbgkapa=0.
Thus,
(21)dpkdt-Γjkbpbdxkdt=0.
Then we get
(22)∇(dxk/dt)(∂/∂xk)pjdxj=0.
Since w=pidxi is a 1-form associated with the tangent vector, γ˙=dγ(t)/dt of curves γ(t), and Riemann connection ∇ satisfies the following property:
(23)∇γ˙(w(γ˙))=2g(γ˙,∇γ˙γ˙)forw(γ˙)=g(γ˙,γ˙),
we get
(24)g(γ˙,∇γ˙γ˙)=0.
Therefore it can be seen that straightforward the curve γ(t) is a geodesic curve.
(ii) It is sufficient to show the HamiltonianH is not changed with respect to variable t in order to prove the theorem. We calculate
(25)dHdt=∂H∂xidxidt+∂H∂pidpidt.
If we take into account (3), we obtain dH/dt=0. Thus H is not dependent on value t.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 bundleT*M of the Hamilton space. Let us start by obtaining a metric on T*M. A pseudo-Riemann metric G on the cotangent bundle T*M of the Hamilton space is obtained by using two different ways. Firstly, the pseudo-Riemann metric on the cotangent bundle T*M is obtained as we were inspired by the paper of Willmore [3] as follows.Theorem 2.
The Levi-Civita connection∇ on M determines canonically a pseudo-Riemannian metric on T*M.Proof.
LetP be a point on T*M such that C(0)=P. Let X be a tangent vector to C at P. The image of the curve C(t) under the bundle projection map π is a curve γ(t) on M, passing through p=π(P)∈U. The curve C(t) can be regarded as a field of covariant vectors ω(t) defined along the curve γ(t). The covariant derivative (∇dπ(X)ω(t))t=0 is a covector at p which can be evaluated on the projected tangent vector dπ(X). This defines a quadratic differential form Q on T(T*M). From this we obtain a bilinear form Gon T*M at P by the usual formula
(26)G(X,Y)=Q(X+Y,X+Y)-Q(X,X)-Q(Y,Y),
where X and Y are tangent vectors to T*M at P. We shall consider that G corresponds to Riemann extension of ∇ on M. We choose a local coordinate system (xi), i=1,…,n, valid in some neighborhood U around p. Then a local coordinate system for π-1(U) is (xi,pj), where ω=pjdxj. The curve may be expressed locally by t→(xi(t),pj(t)) and the corresponding curve γ is t→(xi(t)). The vector X at P is given by (x˙i(0),p˙i(0)) and its projection dπ(X) by x˙i(0). Then, at whent=0, we have
(27)∇dπ(X)ω(t)=x˙i(0)(∇ipj(t))dxj=[dpjdt-Γijkpkx˙i(0)]dxj,
where Γijk are the connection coefficients of ∇. We evaluate this covector on dπ(X) to get the number
(28)Q(X,X)=(∇dπ(X)ω(t))(dπ(X))=-Γijkpkx˙ix˙j+p˙jx˙j.
If the equation which is obtained above taken into account in (26), we get for G(X,X)the following:(29)G(X,X)=Q(2X,2X)-2Q(X,X)=2Q(X,X)=-2Γijkpkx˙ix˙j+2p˙jx˙j=(-2Γijkpkdxidxj+2dpjdxj)(X,X).
Therefore G has local expression
(30)G=-2Γijkpkdxidxj+2dpjdxj
with respect to the induced coordinates (xi,pi) in π-1(U). Since the adapted dual frame on T*M is (dxi,δpi), where
(31)δpi=dpi-pkΓijkdxj,
we get G=2δpidxj.Secondly, the pseudo-Riemann metric on the cotangent bundleT*M is obtained as follows.Theorem 3.
LetM be a manifold with a (pseudo)Riemann metric g. Then the pseudo-Riemannian metric gC on TM corresponds to the pseudo-Riemannian metric G=2δpidxk on T*M.Proof.
Letg be a (pseudo)Riemannian metric on M then gC given by gC=2gjkδyjdxk is a pseudo-Riemann metric on TM. By using the equalities in (4), we get
(32)(φ-1)*(gC)=2(φ-1)*(gjk)(φ-1)*(δyj)(φ-1)*(dxk)=2gjkgji︸δkiδpidxk,
which gives a pseudo-Riemann metric G=2δpidxkon T*M.In order to understand the relation between the pseudo- Riemann manifold(T*M,G) and the symplectic manifold (T*M,θ), we need to define an almost symplectic structure θ on the cotangent bundle T*M of the Hamilton space and an almost product structure P on T*M. The definition of  P and θ  was obtained as we were inspired by the studies of Miron [10] for the Hamilton space.Definition 4.
Letw=pidxi be globally defined as 1-form on T*M. The exterior differential dw of the 1-form w is called an almost symplectic structure θ on the cotangent bundle T*M of the Hamilton space given by
(33)θ=dw=δpi∧dxi.Definition 5.
LetT*M be a 2n-dimensional manifold. A mixed tensor field defines an endomorphism on each tangent space of T*M. If there exists a mixed tensor field P which satisfies
(34)P∘P=I,
we say that the field gives an almost product structure to T*M.
We can consider the tensor field with type (1, 1) onT*M:
(35)P=δδxi⊗dxj-∂∂pi⊗δpj.Theorem 6.
P is an almost product structure on T*M.Proof.
We have(36)P(δδxi)=δδxi,P(∂∂pi)=-∂∂pi
from which P∘P=I.Theorem 7.
LetG be a pseudo-Riemann metric defined as Riemann extension of ∇ in M and letP be an almost product structure on T*M. θ is an almost symplectic structure associated with (G,P). Nondegenerate skew-symmetric 2-form θ on T*M is given by following equation:
(37)θ(X,Y)=G(PX,Y).Proof.
By using (33), the value of the vector fields X, Y on T*M under θ is
(38)θ(XV+XH,YV+YH)=δpi∧dxi(Xiδi+Xn+i∂i,Yjδj+Yn+j∂j)=δpi(Xn+i∂i)·dxi(Yjδj)-δpi(Yn+j∂j)·dxi(Xiδi)=Xn+iYi-Yn+iXi.
On the other hand, the value of G(PX,Y) is
(39)G(P(XV+XH),YV+YH)=G(XV-XH,YV+YH)=G(XV,YH)-G(XH,YV)=Xn+iYi-Yn+iXi.
Therefore, it is seen forward the accuracy of the claim of the theorem.Theorem 8.
LetM be Riemann manifold and let H be Hamiltonian. For any vector field X on T*M,
(40)dH(X)=θ(V,X).Proof.
d
H is a 1-form on T*M with local expression
(41)dH=δHδxidxi+∂H∂piδpi,
with respect to adapted local dual frame, and X has local expression
(42)X=Xjδδxj+Xn+j∂∂pj.
From (40) and (41), we get
(43)dH(X)=δHδxiXi+∂H∂piXn+i.
Since V is a Hamilton vector field, V has local expression with respect to adapted frame on T*M:
(44)V=dxjdtδδxj+δpjdt∂∂pj,
where δpi/dt=(dpi/dt)-pkΓijk(dxj/dt). Thus we have
(45)θ(V,X)=G(PV,X)=dxidtXn+i-(dpidt-pkΓijkdxjdt)Xi.
If we substitute the above equation into (3), we get
(46)θ(V,X)=∂H∂piXn+i+(∂H∂xi+pkΓijk∂H∂pj)︸δH/δxiXi.
From (43) and (46) it is easily seen that dH(X)=θ(V,X).
By using the differential geometric objectsH,G,θ, and P on the cotangent bundle of the Hamilton space considered in this section, we obtain
(47)dH(X)=θ(V,X)=G(PV,X)=∇dπ(PV)ω(dπ(X)),
where G is the pseudo-Riemannian metric defining the Riemann extension of the Levi-Civita connection on M. If we put V instead of X in the above equation, we obtain
(48)θ(V,V)=G(PV,V)=∇dπ(PV)ω(dπ(V))=dH(V)=V(H)=0.
Since V(H)=0, the Hamilton function which gives the total energy of each stage of the system is constant.


## 4. The Differential Geometry of(T*M,G)

In this section, we obtained that the coefficients of the Levi-Civita connection∇∘ and Riemannian curvature tensor K of (T*M,G) and we found the condition under which T*M is locally flat.Theorem 9.
The Lie brackets of the horizontal base vector fieldsδ/δxi=∂/∂xi-Nij∂/∂pj,i,j=1,...,n and vertical base vector fields ∂/∂pi on T*M are given by(i)
[δ/δxi,δ/δxj]=-Rkij∂/∂pk,(ii)
[δ/δxi,∂/∂pj]=-Γikj∂/∂pk,(iii)
[∂/∂pi,∂/∂pj]=0, where Rkij=δNkj/δxi-δNki/δxj[2]. Rkij=-phRkijh for Nki=-Γkihph.Theorem 10.
Let(M,H) be Hamilton space, T*M the cotangent bundle of M, G a pseudo-Riemann metric defined as Riemann extension of Levi-Civita connection ∇ in M, and ∇∘ the Levi-Civita connection on T*M. Then the connection coefficients of the Levi-Civita connection of the pseudo-Riemannian metric G on T*M are given by
(49)∇∘δiδj=-Rkij∂k,∇∘δi∂j=-Γikj∂k,∇∘∂iδj=Γjki∂k,∇∘∂i∂j=0,
where
(50)δi=δδxi,∂i=∂∂pi.Proof.
LetX, Y, and Z be vector fields on T*M. According to the Koszul formula, we get(51)2G(∇∘XY,Z)=XG(Y,Z)+YG(Z,X)-ZG(X,Y)-G(X,[Y,Z])-G(Y,[Z,X])+G(Z,[X,Z]).
We put δi, δj, and δk instead of X, Y, and Z in (51); then we get
(52)2G(∇∘δiδj,δk)=Rijk+Rjki-Rkij.
By the equality ∑(i,jk)Rijk=0, we find
(53)G(∇∘δiδj,δk)=-Rkij,
and we put δi, δj, and ∂k instead of X, Y, and Z in (51). So we get
(54)2G(∇∘δiδj,∂k)=Γjik-Γijk.
Since the Levi-Civita connection ∇ which is defined on M is torsion-free, we have Γjik=Γijk. Subsequently we find
(55)G(∇∘δiδj,∂k)=0.
Thus we get
(56)∇∘δiδj=-Rkij∂k,
and the rest of the equalities can be obtained similarly.Theorem 11.
LetM be a Hamilton space, T*M the cotangent bundle of M, G a pseudo-Riemann metric defined as Riemann extension of Levi-Civita connection ∇ in M, ∇∘ the Levi-Civita connection on T*M, and K the Riemann curvature tensor on T*M. Then the components of the Riemann curvature tensor on T*M are given by
(57)K(δi,δj)δk=(-δiRhjk+δjRhik+RljkΓihl-RlikΓjhl+RlijΓkhl)∂h,K(δi,δj)∂k=(∂kRhij)∂h,K(δi,∂j)δk=δk(Γihj∘π)∂h,K(∂i,δj)δk=-δk(Γjhi∘π)∂h,K(δi,∂j)∂k=K(∂i,δj)∂k=K(∂i,∂j)δk=K(∂i,∂j)∂k=0.Proof.
LetX, Y, and Z be vector fields on T*M. Then
(58)K(X,Y,)Z=∇∘X∇∘YZ-∇∘Y∇∘XZ-∇∘[X,Y]Z.
If we put δi, δj, and ∂k instead of X, Y, and Zin (58), we get
(59)K(δi,δj)∂k=∇∘δi∇∘δj∂k-∇∘δj∇∘δi∂k-∇∘[δi,δj]︸∈VT*M∂k=-δi(Γjlk∘π)∂l+δj(Γilk∘π)∂l-ΓjlkΓihl∂h+ΓilkΓjhl∂h=(-∂Γjhk∂xi+∂Γihk∂xj-ΓjlkΓihl+ΓilkΓjhl)∂h=-Rhijk∂h.
By the equality Rhij=-pkRhijk, we obtain
(60)K(δi,δj)∂k=∂Rhij∂pk∂h.
The other coefficients of the curvature tensor can be obtained similarly.Theorem 12.
The pseudo-Riemann manifold(T*M,G) is flat if and only if the Riemann manifold (M,g) is Euclidean.Proof.
If the Riemann manifold(M,g) is Euclidean, the Christoffel symbols must be zero. Thus, Riemann curvature tensor R on M and K on T*M 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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