Intro Geo Hamiltonm

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Periodica Mathematica Hungarica Vol. 48 (12), 2004, pp. 7782

THE GEOMETRY OF HAMILTON SPACES AN INTRODUCTION

Radu Miron (Iasi)

Dedicated to Prof. Dr. Lajos Tamassy on his 80th anniversary

Abstract

A short introduction of Hamilton, Cartan and Generalized Hamilton spacesis provided. The HamiltonJacobi equations are deduced by variational calculus.

A Hamilton space is a pair Hn = (M, H) formed by a differentiable manifoldM and a regular Hamiltonian H : TM . The notion was introduced by thepresent author in 1987 and published in the paper [1]. In the last five years manygeometers as: S. Watanabe, S. Ikeda, H. Shimada, M. Anastasiei, Gh. Atanasiu,D. Hrimiuc, M. Kirkovits and others have obtained new important results in this

field and its applications.In my lecture I shall present the notion of Hamilton space, its relationship

to Lagrange spaces, as well as to Cartan spaces (as duals of Finsler spaces) andgeneralized Hamilton spaces. Among the classes of these spaces and Riemann spaceswe have the inclusions

{Rn} {Cn} {Hn} {GHn}

which show the importance of these geometries. The variational problem is dis-cussed, too.

The recent book [3] contains part of these problems.

Mathematics subject classification number: 53C60.Key words and phrases: Hamilton spaces, metrical N-linear connection.

0031-5303/2004/$20.00 Akademiai Kiad o, Budapestc Akademiai Kiad o, Budapest Kluwer Academic Publishers, Dordrecht


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1. Nonlinear connections d-connections

Let M be a real n-dimensional manifold and

: T

M M its cotangentbundle. If (xi, pi) are the canonical coordinates of a point u TM, then a change

of coordinates on TM is given by

xi = xi(x1, . . . , xn), rank xixj = n,

pi = xjxipj .

(1.1)

On TM there are globally defined the Liouville 1-form

p = pidxi (1.2)and the natural symplectic structure

= dpi dxi

. (1.3)Let us denote by V = Ker T the vertical subbundle of the tangent bundle

T TM. If the base manifold M is paracompact, then there exist horizontal subbun-dles N of T TM such that

T TM = N V

holds, with the Whitney sum on the right hand side.The fibres of N determine a distribution N : u TM Nu TuTM,

which is supplementary to the vertical distribution V : u TM Vu TuTM :

TuTM = Nu Vu. (1.4)

A horizontal distribution N, which verifies (1.4) is called a nonlinear connection on

T

M. On a coordinate neighbourhood 1

(U) T

M there exists an adaptedbasis

xi

, pi

to N and V, where

xi=

xi+ Nji

pj

.

The functions Nji(x, p) are the coefficients of the nonlinear connection N.

Putting xi

= i,xi

= i,pi

= i we can write

i = i + Nji j. (1.5)

With respect to (1.1) we have i = xjxi j , i = xixj j .Let (dxi, pi) be the dual basis of (i, i):

pi = dpi Nij(x, p)dxj . (1.5)

Then

ij = Nij Nji (1.6)

is an antisymmetric d-tensor field on TM. If ij = 0 we say that N is symmetric.When ij = 0 we can write the symplectic structure in the form

= pi dxi.


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the geometry of hamilton spaces an introduction 79

This proves the geometrical character of .

Let us consider the d-tensor field:

Rkij = iNkj jNki.

A necessary and sufficient condition that the horizontal distribution N beintegrable is Rijk = 0.

Let h and v be the projectors on N and V. For a vector field X (TM) wecan write X = XH + XV, XH = hX, XV = vX.

A linear connection D on TM is called distinguished (briefly a d-connection) ifDXY

HV

= 0,

DXYVH

= 0.

We write

DhX = DXH , DvX = DXV

and say that Dh and Dv are the h- and the v-covariant derivations determined bythe d-connection D. Also, we denote by Th and Tv the h- and v-torsions of D,respectively.

2. Hamilton spaces

Definition 2.1. A Hamilton space is a pair (M, H) = Hn, where H :

TM is a function (a Hamiltonian) of class C on TM = TM \ {0},continuous on the null section, and having the metric tensor

ij

(x, p) =

1

2 i

j

H (2.1)

positively defined (or of constant signature and rankij = n).

H(x, p) is the fundamental function and ij(x, p) the fundamental tensor fieldof Hn. From (2.1) it follows that

rank ij(x, y) = n.

We can consider ij(x, y) and the tensor

Gh = ij(x, p)dxi dxj . (2.2)

Gh is called the metric tensor of the Hamilton space. We get:

Theorem 2.1. There exists a nonlinear connection N onTM determinedonly by the fundamental function H of the Hamilton space. N has the coefficients

Nij = 1

2jh

1

4ik

k{H, hH} + hiH

. (2.3)


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80 r. miron

This nonlinear connection is called canonical.

In (2.3) the brackets { } are the Poisson brackets.

The canonical nonlinear connection is symmetric. We remark that N wasobtained from (2.3) by means of the Legendre transformation

xi = xi, pi =1

2

L

y i(2.4)

which transforms the Lagrangian L(x, y) into the Hamiltonian

H = L +piyi,

H =

1

2H, L =

1

2L

.

Now we take in Hn the canonical nonlinear connection N from (2.3) and consider

the canonical Hamiltonian vector field onTM :

i = dH.

In the adapted basis is expressed by

= (iH)i (iH)i.

The integral curves t c(t) of the vector field are the integral curves of theHamiltonJacobi equations

dxi

dt=

H

pi,

pi

dt=

H

xi. (2.5)

The law of conservation dHdt

= 0 is valid on the curves (2.5).

Theorem 2.2. 1) There exists a unique d-connection D onTM with the

property

DhXGh = 0, DvXG

h = 0, Th = Tv = 0.

2) In the adapted basis the coefficients of Dh and Dv are given by

Hijk =1

2ih(jhk + kjh hjk),

and

Cjk

i = 1

2ih

jhk + kjh hjk

,

(2.6)

respectively.

From here we can obtain the structure equations, geodesics, parallelism, h-paths, v-paths and the almost Kahlerian model on the bundle T TM.

We study in a recent book, [3], the variational problem for spaces H(k)n =(M, H), k > 1.

Up to now this problem has not been studied in Hamilton spaces.


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the geometry of hamilton spaces an introduction 81

Let us consider a curve c : t [0, 1] (xi(t), pi(t)) TM. For a differen-tiable Hamiltonian H(x, p) we define the integral of action I(c) by

I(c) =10

pi(t)

dxi

dt 1

2H

x(t), p(t)

dt.

A variation c(1, 2) = (xi + 1i, pi + 2i) of the curve c leads to the integralof action I(c). The necessary conditions in order that I(c) be an extremal value ofI(c) are

I(c)1

1=2=0

=I(c)

2

1=2=0

= 0. ()

So we have: The necessary condition that I(c) be an extremal value of thefunction I(c) is that the curve c be a solution of the HamiltonJacobi equations

dxi

dt=

1

2

H

pi,

dpi

dt=

1

2

H

xi.

The law of conservation holds: The Hamiltonian H is conserved along theintegral curves of the HamiltonJacobi equations.

The Jacobi method of integration of the HamiltonJacobi equations can beapplied. A Nother theorem for Hamilton spaces can be proved.

3. Cartan spaces

A particular case of Hamilton spaces are the Cartan spaces. A Cartan spaceCn = (M, K) is a Hamilton space Hn =

M, K2

, where K > 0 and positive 1-

homogeneous with respect to the momenta pi. All previous theory is particularized

for this case. For Cartan spaces Cn the canonical nonlinear connection and thecanonical N-metrical connection have some interesting analytical expressions andsome important properties.

Cartan spaces are dual to Finsler spaces via the Legendre transformation. Asone can see in the book [4], the Cartan spaces Cn have the same importance andbeauty as Finsler spaces.

Professor Dr. Lajos Tamassy has studied in [5] the deep relationship betweenCartan spaces and the areal spaces introduced by Elie Cartan.

Generalized Hamilton spaces are given by a pair GHn = (M, gij(x, p)) wheregij(x, p) is a nonsingular d-tensor field with constant signature.

In some particular cases GHn is reducible to a Hamilton space Hn or a Cartanspace Cn.

Consequently, the sequence{Rn} {Cn} {Hn} {GHn}

holds.By Hamiltonian geometries we mean the study of this sequence. The geometry

of GHn is presented in the book [4].


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82 r. miron

References

[1] R. Miron, Sur la geometrie des espaces Hamilton, C.R. Acad. Sci. Paris, Ser. I,306, No. 4, (1988), 195198.[2] R. Miron, Variational problems in Hamilton spaces, Proc. Non-Euclidean Geome-

try in Modern Physics, 2003.[3] R. Miron, The Geometry of Higher-Order Hamilton spaces Applications to Hamil-

tonian Mechanics, Kluwer, FTPH no. 132, 2003.

[4] R. Miron, D. Hrimiuc, H. Shimada and V. S. Sabau, The Geometry of Hamiltonand Lagrange Spaces, Kluwer Acad. Publ., FTPH no. 118, 2001.

[5] L. Tamassy, Area and metrical connections in Finsler spaces, Finslerian geometries(Edmonton, AB, 1998), Kluwer Acad. Publ., Dordrecht, 2000, 263280.

(Received: September 18, 2003)

Radu MironFaculty of MathematicsUniversity Al.I.Cuza IasiIasiRomania

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Intro Geo Hamiltonm