DOI: 10.1515/auom-2015-0033
An. Şt. Univ. Ovidius Constanţa
Vol. 23(2),2015, 161–171
Ingarden mechanical systems with special
external forces
Otilia Lungu and Valer Nimineţ
Abstract
In the present paper we study a remarcable particular case of Finslerian mechanical system,
called Ingarden mechanical system. This is
P
defined by a 4-uple IF n = M, F 2 , N, Fe where M is the configuration space, F n = (M, F (x, y)) = (M, α (x, y) + β (x, y)) is an Ingarden
∂
space, N is the Lorentz nonlinear connection and Fe = aijk (x) y j y k ∂y
i
are the external forces.
P
One associates to this system
IF n a semispray S, or a dinamical
system on the velocity space TM. We write
P the generalized Maxwell
equations for the electromagnetic fields of IF n .
1
Introduction
The general theory of Finslerian mechanical systems was realized by R. Miron
[9 ], [10] and proceeds from the Finsler geometry. It started with Finsler’s
dissertation in 1918 and its study has been developed by geometers and physicists as: E.Cartan, H.Rund, L.Berwald, S.S.Chern, M.Matsumoto, R.Miron,
H.Shimada, G.S.Asanov, etc.
In this paper we introduce and investigate some geometric aspects of a
special kind of Finslerian
P mechanical systems. We define a 4-uple IF n = M, F 2 , N, Fe where M is the configuration
space, F = α + β is a Randers metric F 2 is the kinetic energy of the space, N
Key Words: Mechanical system, Ingarden space, Lorentz connection, Euler-Lagrange
equations.
2010 Mathematics Subject Classification: Primary 53C60, 53B40.
Received: November, 2013.
Revised: March, 2014
Accepted: March, 2014
161
INGARDEN MECHANICAL SYSTEMS WITH SPECIAL EXTERNAL FORCES
162
∂
is the Lorentz nonlinear connection and Fe = aijk (x) y j y k ∂y
i are the external
i
forces with ajk (x) a symmetric tensor on M of type (1, 2).
We call this 4-uple an Ingarden mechanical system with special external
forces and we determine the coefficients of the canonical nonlinear connection
MI
MI
P
N . We also construct the canonical metrical d-connection M IΓ( N ) of IF n
and we write the generalized Maxwell equations.
Let M be an n-dimensional, real C ∞ manifold. Denote by (T M, τ, M )
the tangent bundle of M and F n = (M, F (x, y)) be a Finsler space, where
F : T M → R+ is its fundamental function, i.e., F verifies the following axioms:
i) F is a differentiable function on T M̃ = T M \ {0} and it is continuous
on the null section of the projection τ : T M → M ;
ii) F is positively 1- homogeneous with respect to the variables y i ;
iii) ∀ (x, y) ∈ T M̃ the Hessian of F 2 with respect y i is positive defined.
∂2F 2
Consequently, the d-tensor field gij (x, y) = 21 ∂y
i ∂y j is positive defined. It is
called the fundamental tensor, or metric tensor of F n .
This definition can be extended to the case when the fundamental tensor
is of constant sygnature, when we imposed the condition det (gij (x, y) 6= 0) .
It is well known that a Randers metric
p is a deformation of a Riemannian or
pseudo-Riemannian metric α (x, y) = aij (x) y i y j , showing the gravitational
field, using a 1-form β (x, y) = bi (x) y i , representing the electromagnetic field.
Randers spaces are Finsler spaces F n = (M, F (x, y)) = (M, α (x, y) + β (x, y))
equipped with Cartan nonlinear connection. For F n , instead of the Cartan nonlinear connection, R. Miron introduced in [8] the Lorentz nonlinear
connection N determined by the Lorentz equations of the space F n with
the metric F (x, y) = α (x, y) + β (x, y). The local coefficients of N are
i
i
Nji = γjk
y k − Fji , where γjk
are the Christoffel symbols of the Riemannian
∂b
∂bs
j
structure a = aij (x) dxi ⊗ dxj and Fji (x) = ais Fsj , Fsj = ∂x
j − ∂xs .
n
The Finsler space F = (M, F (x, y)) = (M, α (x, y) + β (x, y)) equipped
with the Lorentz nonlinear connection N is called an Ingarden space. It is
denoted IF n = (F n , N ).
In Preliminaries we give some known results regarding the Lorentz nonlinear connection and Ingarden spaces.
In section 3 we present main results and in section 4 some applications in
physical fields.
2
Preliminaries
Let F n = (M, F (x, y)) be a Finsler space with
p the fundamental function
F (x, y) = α (x, y) + β (x, y) where α (x, y) = aij (x) y i y j and β (x, y) =
bi (x) y i ; a = aij (x) dxi dxj is a pseudo-Riemannian metric on M and it gives
INGARDEN MECHANICAL SYSTEMS WITH SPECIAL EXTERNAL FORCES
163
the gravitational part of the metric F ; bi (x) is an electromagnetic covector
on M and β (x, dx) = bi (x) dxi is the electromagnetic 1-form field on M .
We consider the integral of action of the energy F 2 (x, y) along a curve c : t ∈
[0, 1] → c (t) ∈ M :
I (c) = ∫ 10 F 2 x,
dx
dt
dt = ∫ 10 α x,
dx
dt
+ β x,
dx
dt
2
dt
(1)
The variational problem for I (c) leads to the Euler-Lagrange equations:
Ei F 2
:=
∂(α+β)2
∂xi
−
2
d ∂(α+β)
dt
∂y i
= 0, y i =
dxi
dt .
(2)
The energy of F 2 is
2
2
2
2
2
εF 2 = y i ∂F
∂y i − F = 2F − F = F .
The covector field Ei F 2 is expressed by
∂α
Ei F 2 = Ei α2 + 2αEi (β) + 2 dα
dt ∂y i .
(3)
(4)
Let us fix a parametrization of the curve c, by natural parameter s with
respect to Riemannian metric α (x, y) . It is given by
ds2 = α2 x,
dx
dt
dt2 .
(5)
dx
It follows F 2 x, ds = 1 and dα
ds = 0.
Along to an extremal curve c, canonical parametrized by (5), Ei (β) is
expressed by
j
∂b
∂bi
dxj
dx
Ei (β) = ∂xji − ∂x
(6)
j
ds = Fij (x) ds .
One obtains [6]:
Theorem 2.1. (Miron-Hassan) In the canonical parametrization, the
2
Euler-Lagrange equations of the Lagrangian (α + β) are given by
Ei α2 + 2Fij (x) y j = 0, y i =
dxi
ds .
(7)
Theorem 2.2. The Euler-Lagrange equations (7) are equivalent to the
Lorentz equations:
d2 xi
ds2
i
+ γjk
(x)
dxj dxk
ds ds
◦
= Fji (x)
dxj
ds ,
(8)
◦
i
where Fji (x ) = a is Fsj (x ) and γjk
are the Christoffel symbols of the Riemannian metric tensor aij (x) .
INGARDEN MECHANICAL SYSTEMS WITH SPECIAL EXTERNAL FORCES
164
The Euler-Lagrange equations Ei F 2 = 0 determines a canonical semispray or a Dynamical System S on the total space of the tangent bundle :
i ∂
∂
S = y i ∂x
i − 2G ∂y i ,
(9)
where the coefficients Gi (x, y) are:
◦
i
2Gi (x, y) = γjk
(x) y j y k − Fji (x) y j .
(10)
Now we can consider the nonlinear connection N with the coefficients Nji =
i
∂G
∂y j
. Of course, we have
i
Nji = γjk
(x) y k − Fji (x) ,
(11)
◦
where Fji (x) = 21 Fji (x) .
Since the autoparallel curves of N are given by the Lorentz equations (8),
we call it the Lorentz nonlinear connection of the Randers metric α + β.
The nonlinear connection N determines the horizontal distribution, denoted by N too, with the property Tu T M = Nu ⊕ Vu , ∀u ∈ T M , Vu being the
natural vertical distribution on the tangent manifold T M .
The local adapted
to the horizontal and vertical vector spaces Nu
basis δ
∂
and Vu is given by δxi , ∂yi , i = 1, ..., n , where
◦
δ
∂
∂
∂
∂
∂
∂
δ
k
=
− Nik k =
− γis
(x) y s k + Fik k = i + Fik k (12)
i
i
i
δx
∂x
∂y
∂x
∂y
∂y
δx
∂y
◦
∂
k
s ∂
and δxδ i = ∂x
i − γis (x) y ∂y k .
The adapted cobasis is dxi , δy i , i = 1, ..., n with
◦
i
δy i = dy i + Nji dxj = dy i + γjk
(x) y k dxj − Fji dxj = δ y i − Fji dxj ,
(13)
◦
i
where δ y i = dy i + γjk
(x) y k dxj .
The weakly torsion of N is
i
Tjk
=
∂Nji
∂y k
−
∂Nki
∂y j
= 0.
(14)
The integrability tensor of N is
i
Rjk
=
δNji
δxk
−
δNki
δxj .
(15)
INGARDEN MECHANICAL SYSTEMS WITH SPECIAL EXTERNAL FORCES
165
Definition 2.1. The Finsler space F n = (M, F = α + β) equipped with the
Lorentz nonlinear connection N is called an Ingarden space. It is denoted
IF n .
The fundamental tensor gij of IF n is
gij =
F
α (aij
− ˜li ˜lj ) + li lj
(16)
∂F
∂α
˜
where ˜li = ∂y
i , li = ∂y i , li = li + bi .
The following results holds [8]:
2.3. There exists an unique N -metrical connection IΓ (N ) =
Theorem
i
i
Fjk
, Cjk
of the Ingarden space IF n which verifies the following axioms:
V
i) ∇H
k gij = 0; ∇k gij = 0;
i
i
ii) Tjk = 0; Sjk = 0.
The connection IΓ (N ) has the coefficients expressed by the generalized
Christoffel symbols:

 F i = 1 g is δgsjk + δgskj − δgjks
jk
2
δx
δx δx
(17)
 C i = 1 g is ∂gsjk + ∂gskj − ∂gjks ,
jk
2
∂y
∂y
∂y
where
3
δ
δxi
are given by (12).
Main results
For a manifold M , that is the configuration space of a Finslerian dynamical
system, let us consider the tangent bundle T M to which we refer to as the
velocity space. Suppose that there exists a Randers metric F = α + β on T M̃
and aijk (x) a symmetric tensor on the configuration space M , of type (1, 2).
Definition 3.1. An Ingarden mechanical system with special external forces
is a 4-uple
X
2
=
M,
(α
+
β)
,
N,
F
,
e
n
IF
∂
with N, the Lorentz nonlinear connection and Fe = aijk (x) y j y k ∂y
i the external
forces given as a vertical vector field globaly defined on T M .
We denote F i (x, y) = aijk (x) y j y k and we can state
Theorem
3.1. [9] For the Ingarden mechanical system
P
2
IF n = M, (α + β) , N, Fe the following properties hold good:
i) The operator S defined by
i
∂
1 i
S = y i ∂x
i − 2G − 2 F
∂
∂y i
(18)
INGARDEN MECHANICAL SYSTEMS WITH SPECIAL EXTERNAL FORCES
166
is a vector field, global defined on the velocityPspace T M .
ii) S is a semispray which depends only on IF n and it is a spray if Fe
are 2-homogeneous with respect to y i .
iii) The integral curves of P
the vector field S are the evolution curves given
by the Lagrange equations of IF n :
d2 xi
dt2
+ Γijk x,
dx
dt
dxj dxk
dt dt
= 21 F i x,
dx
dt
.
(19)
MI
The semispray S (18) has the coefficients Gi expressed by
MI
1
1
2 Gi = 2Gi − F i (x, y) = Γijk (x, y) y j y k − F i (x, y) .
2
2
(20)
MI
Thus,
Pthe canonical nonlinear connection N of the Ingarden mechanical
system IF n has the coefficients
MI
MI
Nji
1 ∂F i
1
∂ Gi
i
=
N
−
= Nji − aijk (x) y k .
=
j
j
j
∂y
4 ∂y
2
(21)
MI
This nonlinear connection N determines a direct decomposition of the
tangent space T M̃ into horizontal and vertical subspaces:
MI
Tu T M̃ = Nu ⊕Vu , ∀u ∈ T M̃ .
MI
∂
A local adapted basis to this decomposition is δxδ i , ∂y
i
(22)
where
i=1,n
◦
◦
1 j
∂
∂
δ
δ
δ
j
k
=
+
F
(x)
+
a
(x)
y
=
+ Aji j
i
δxi
δxi
2 ik
∂y j
δxi
∂y
(23)
1
Aji = Fij (x) + ajik (x) y k
2
(24)
MI
with
◦
k
− γis
(x) y s ∂y∂ k .
MI
i
i
The adapted cobasis is dx , δ y with
MI
◦
◦
1 i
i
i
i
k
y
=
y
−
F
+
a
(x)
y
dxj = δ y i − Aij dxj
δ
δ
j
jk
2
and
δ
δxi
◦
=
∂
∂xi
i
where δ y i = dy i + γjk
(x) y k dxj .
(25)
INGARDEN MECHANICAL SYSTEMS WITH SPECIAL EXTERNAL FORCES
MI
167
MI
i
i
We determine the torsion Tjk
and the curvature Rjk
of the canonical conMI
nection N by a direct calculation:
MI
MI
i
Tjk
MI
i
Rjk
=
MI MI
δ Nji
δy k
−
MI
∂ Nji
∂ Nki
=
−
=0
∂y k
∂y j
MI MI
δ Nki
δy j
◦
=
◦
δN
◦
i
Rjk
+
i
◦
∂Nji
Ajk j
∂y
(26)
−
∂Nki
Akj
k
∂y
!
,
(27)
Ni
i
where we have denoted Rjk
= δxkj − δδxjk .
Applying the theory from the book
[10] the following
theorem holds:
P
2
Theorem 3.2.
Let
M, (α + β) , Fe be an Ingarden meIF n =
MI
P
chanical system and N the canonical nonlinear connection
of IF n . There
!
MI MI
MI
i
i
exists an unique d-connection M IΓ N = Fjk
, Cjk
determined by the
following axioms:
MI
MI
V
i) ∇H
k gij = 0; ∇k gij = 0,
MI
MI
i
i
ii) Tjk
= 0; Sjk
= 0,
where
MI
MI
∇H
k gij =
MI
∇Vk gij =
δ gij
δxk
∂gij
∂y k
MI
MI
s
s
gsj − Fjk
gis
− Fik
MI
MI
.
We call this connection the canonical metrical d-connection of
of
(28)
s
s
− Cik
gsj − Cjk
gis
P
IF n .
Theorem 3.3.The local coefficients of the canonical metrical d-connection
P
IF n are

MI
MI
MI
MI

g
g

i
 Fjk
= 21 g is δδxksj + δδxgjsk − δδxsjk
(29)
MI


 C i = 1 g is ∂gsjk + ∂gskj − ∂gjks .
jk
MI
2
∂y
∂y
MI
i
i
In order to calculate Fjk
and Cjk
we have:
∂y
168
INGARDEN MECHANICAL SYSTEMS WITH SPECIAL EXTERNAL FORCES
MI
◦
δgsj
∂gsj
δ gsj
=
+ Ark r
k
k
δx
δx
∂y
(30)
◦
Denote ∇ the h-covariant derivative with respect to Levi-Civita connection:
k
◦
δ gsj
i
i
=
− γsk
gij − γjk
gsi .
δxk
(31)
◦
δ gsj
i
i
= ∇ gsj + γsk
gij + γjk
gsi
k
δx
k
(32)
◦
∇ gsj
k
We get
◦
Now we obtain
MI
◦
∂gsj
δ gsj
i
i
= ∇ gsj + γsk
(33)
gij + γjk
gsi + Ark r
k
δx
∂y
k
and we can state:
P
Theorem 3.4. The canonical metrical d-connection of IF n has the coefficients

MI

 F i = γi + Bi
jk
jk
jk
(34)
MI

 Ci = Ci ,
jk
jk
where
i
Bjk
1
= g is
2
◦
∇ gsj +
k
∂gsj
Ark r
∂y
◦
+ ∇ gsk +
j
∂gsk
Arj
r
∂y
◦
− ∇ gjk +
s
∂gjk
Ars r
∂y
(35)
Taking into account (33)
! we can express the curvature tensors of
MI MI
MI
i
i
M IΓ N = Fjk
, Cjk
:

MI


i

Rjkh
=



MI
i
 Pjkh =



 Mi I

Sjkh =
with
MI
i
Pjk
MI
=
∂ Nji
∂y k
MI MI
i
δ Fjk
δxh
MI
i
∂ Fjk
∂y h
−
MI MI
i
δ Fjh
δxk
MI MI
MI MI
MI MI
MI MI
(36)
i
i
s
− ∇H
k Chs + Cjs Pkh
MI
i
∂ Cjk
h
∂y
−
MI
i
− Fjk
.
MI
i
∂ Cjh
k
∂y
MI MI
s
i
s
i
i
s
+ Fjk
Fsh
− Fjh
Fsk
+ Chs
Rkh
MI MI
MI MI
s
i
s
i
+ Cjk
Csh
− Cjh
Csk
.
INGARDEN MECHANICAL SYSTEMS WITH SPECIAL EXTERNAL FORCES
4
169
Applications in physical fields
MI
In an Ingarden mechanical system the h-deflection tensor Dki of the canonical
metrical connection no vanishes. It give rise to an interior electromagnetic
tensor which is not coincident to the exterior electromagnetic tensor Fik (x)
provided by β.
MI
The h-deflection tensor Dki is given by
MI
Dki
=
MI
∇H
k
MI
MI
δ yi
i
j
i
j
i
+
F
kj y = Bjk y + Ak
δxk
i
y =
(37)
From the relation
◦
◦
1 is j ◦
1
∂gsk j
= g y ∇ gsj + ∇ gsk − ∇ gjk + g is Arj
y
2
2
∂y r
s
j
k
(38)
◦
◦
∂gsk j
1
1 is j ◦
g y ∇ gsj + ∇ gsk − ∇ gjk + g is Arj
y + Aik .
2
2
∂y r
s
j
k
(39)
i
Bjk
yj
we get
MI
Dki =
MI
The v-deflection tensor dik is
MI
dik
MI
= ∇Vk y i = δki .
(40)
The covariant h-tensor is
MI
Dsk =
MI
gis Dki
◦
◦
1 j ◦
1 ∂gsk j
= y ∇ gsj + ∇ gsk − ∇ gjk + Arj
y + gis Aik .
2
2
∂y r
s
j
k
(41)
and the covariant v-tensor is
MI
MI
dsk = gis dsk .
(42)
≈
The h-interior electromagnetic tensor F is
sk
MI
≈
1 MI
Fsk =
Dsk − Dks .
2
≈
and the v-interior electromagnetic tensor fsk is
(43)
INGARDEN MECHANICAL SYSTEMS WITH SPECIAL EXTERNAL FORCES
≈
fsk =
1
2
MI
MI
dsk − dks .
170
(44)
A direct calculus allows to formulate:
Theorem 4.1. TheP
h- and v- interior electromagnetic tensors of the Ingarden mechanical system IF n with respect to the canonical metrical connection
MI
N are given by
≈
◦
◦
Fsk = 21 y j ∇k gsj − ∇s gjk +
≈
fsk
1
2
gis Aik − gik Ais
(45)
= 0.
MI
MI
MI
MI
MI
MI
MI
s
s
s
We denote Rijk = gis Rjk
, Rijkh = gjs Rikh
, Pijk = gis Pjk
, Pijkh =
MI
s
gjs Pikh
.
By a direct calculus one proves:
≈
Theorem 4.2. P
The h- interior electromagnetic tensors Fij of the Ingarden
mechanical system IF n satisfies the following generalized Maxwell equations:
MI
≈
MI
≈
MI
≈
MI
≈
MI
≈
H
H
∇H
k Fij + ∇i Fjk + ∇j Fki =
MI
≈
V
V
∇V
k Fij + ∇i Fjk + ∇j Fki =
1
2
1
2
MI
MI
MI
MI
MI
MI
y r Rrijk + Rrjki + Rrkij − Rijk + Rjki + Rkij
MI
MI
MI
MI
MI
MI
yr
Prijk − Prikj + Prjki − Prjik + Prkij − Prkji
(46)
Conclusions. We defined in this paper a new kind of mechanical systems,
called Ingarden mechanical system with special external forces. We developed
the theory using the geometrical objects fields of the canonical metrical dconnection. After the calculation of the h- and v-interior electromagnetic
tensors, we got a new form for the generalized Maxwell equations. The same
theory can be also used to write the Einstein equation for the gravitational
fields.
References
[1] Anastasiei M.: Certain Generalizations on Finsler Metrics, Contemporary Mathematics, (1996), 61–170.
[2] Antonneli P.L., Ingarden R.S., Matsumoto M.: The Theory of
Sprays and Finsler Spaces with Applications in Physics and Biology,
Kluwer Acad. Publ. FTPH 58 (1993).
INGARDEN MECHANICAL SYSTEMS WITH SPECIAL EXTERNAL FORCES
171
[3] Bucătaru I.: Metric nonlinear connections, Differential Geometry and
its Applications, 35 (2007), no. 3, 335-343.
[4] Bao D., Chern S.S., Shen Z.: An Introduction to Riemann-Finsler
geometry, Graduate Text in Math., Springer, (2000).
[5] Lungu O., Nimineţ V. : General Randers mechanical systems, Scientific
Studies and Research. Series Mathematics and Informatics 22 (2012) no.1,
25-30.
[6] Miron R. , Hassan B.T.: Variational Problem in Finsler Spaces with
(α, β) -metric, Algebras Groups and Geometries, Hadronic Press, 20
(2003), 285-300.
[7] Miron R. , Hassan B.T.: Gravitation and electromagnetism in FinslerLagrange spaces with (α, β) -metrics,Tensor, N.S., 73 (2011) ,75-86.
[8] Miron R. : The Geometry of Ingarden Spaces, Rep. on Math. Phys., 54
(2004), 131-147.
[9] Miron R. : Dynamical Systems of Lagrangian and Hamiltonian Mechanical Systems, Advance Studies in Pure Mathematics, 48 (2007), 309-340.
[10] Miron R.: Lagrangian and Hamiltonian Geometries. Applications to Analytical Mechanics, Ed. Academiei Romne & Ed. Fair Partners, Buc.,
(2011).
[11] Shen Z. : Differential Geometry of Spray and Finsler Spaces, Kluwer
Academic Publishrs, Dodrecht, (2001).
[12] Shen Z. : Lectures in Euler Geometry, World Scientific, (2001).
Otilia LUNGU,
”Vasile Alecsandri” University
Department of Mathematics and Informatics
600115
Bacău
Romania
Email: otilia.lungu@ub.ro
Valer NIMINEŢ,
”Vasile Alecsandri” University
Department of Mathematics and Informatics
600115
Bacău
Romania
Email: valern@ub.ro
INGARDEN MECHANICAL SYSTEMS WITH SPECIAL EXTERNAL FORCES
172