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Lagrange multiplier and Wess-Zumino variable as large extra dimensions in the torus universe
Authors
Salman Abarghouei Nejad,Mehdi Dehghani
Majid Monemzadeh,Simon Bober,Pavel Zenon Sejas Paz,Yuanjing Mao,Paiboon Panase,Nadine Carol Schutte,Linda Lawton,Jerry Swan,Duanyun Cao,Laura Castaldi,Xuwei Chen,Zi Wang,Deok Won Kim,Guenter Gauglitz,Chris Ovenden,Soobin Kim,Xianghong Chen,Depeng Zhang,Efstratios Gallopoulos,Miroslaw Bober,Jesper Haglund,Elizabeth Searing,Ahmed Abdallah Elhashemy Zaki Mohamed,Yacine GUETTAF,Tetsurou Odatsu,Masoud Saeed,Kikuya Kato,yong zhang,Théo Mahut,Pedro Hallal,ilhan Ece,Yongjian Lu,Matic ,Sarah George,Piero Cappelletti,一平 方,Andrey Smelter,Pinaki Saha,Rosaleen Chun,Anna Vlot,Reena Patel,Jeffrey Parker,Lucas Rentschler,sara milner garcia,Kenji Kiuchi,Ruijia Zhang,Domenico Sambataro,Martha Brennich,Yasemin Bal Gunduz,Gayadhar Behera,Werner Grass,zhengcai lou,Zhou Tianfeng,Takao Sato,Mariwan Al-bajalan,Adam Tucker,Shan Chen,Ze Wang,Achille BASILE,Reina Coromoto Camacho Toro,Emily Silverman,hongxia li,Yang ,Andrea Gallo Rosso,Hossein Azari,Naima Boughalleb-M'Hamdi,Douglas Taatjes,Amin El-Heliebi,Peter Bates,Jeffrey Markert,Mohamad-Al-Fadl Rihani,Yong-Hoon Jang,Xing Wang,Li long,Krista Terry,ana raquel pengelly,Young-Hwa Kim,Eric Hurwitz,Thomas Scully,Vladimir Lobanov,Konstantin Korytov,James Lawson,Yi Shujian,Anders Kvellestad,Justin Richardson,Liliana MONIZ,Ali Larbi,Sthita Prajna Mishra,Joanna Sale,Robert Jarrow,Anne-Lotte Coolen,Hiroki Wakamatsu,Nagendra Singh,jayanta layek,Yang Li,Tami Swenson,Jenq-Shiou Leu,Cyril Mauffrey,宗佶 杨,Paweł Nicia,Maria Bokarewa,Chaniphun Butryee,Manuel Leonetti,Amir Sadeghian,Stefano Di Alesio,Janos Podani,Affirul Chairil Ariffin,Ding Du,Guohua Lv,Armando Saúl Ruiz Treviño,Wanghua Sui,Giulia Girolimetti,Vineeta Joshi,JAYARAMAN THANGAPPAN,Iker Zulaica-Hernández,Cristina Giannattasio,Richard Toro Araya,SLIMANE Abdelkader,Kazuhiro Imai,martijn schotanus,Julien Huen,Zhi Chen,Xupo Liu,Benjamin Farmer,Omar Matar,Jennifer Roebber,Ling Jiang,Robert Diaz Beveridge,LUCA SESSA,AJAY KUMAR,Jingshan Li,Darko Skorin-Kapov,Adam Orenstein,Abbas Ghebleh,Carlo Boldrighini,prathvi soni,Bo Liu,Leila Mansourian,Ludovit Hajnovic,Subrata Bera,Diana Costa Santos,Lavanya Murugan,Nie ,Yunjian Huang,Mansu CHO,Satomi ,Philippe Seil,Jasmin Raymond,Deena Al Asfoor,Seyed Arash Tehrani Banihashemi,Samuel Admassu,Ertan Düzgüneş,Soyoung Won,Milan Kostal,scott haldeman,Asem Khmag,Mohammad Ali Fazilati,Victor Virlogeux,Zi-lu ZHANG,Jean-Philippe Dionne,manish kumar,Rebeca Saborido Fiaño,爽 Shuang 陈 Chen,Enrico Rettore,Pimwara Tanvejsilp,Ursula Hiden,Rafael Moreno Cano,Qiaoshou Liu,Rosa Mª Peris Sancho,Xuan Zhou,Eve Robertson,Zhiliang Huang,Georgios Lioutas,Xiaotan Zhang,xinwei lu,Isabella Aboderin,Sajjad Sattari,Eva Bültmann,ELODIE COURTABESSIS,Dennis Allen,Caroline Ripat,chao liu,Qasim Janjua,Abdulrahman El-Nounu,Torgeir Waaga,Yuhao Yi,Hongrui Liu,Kei Yuen Chan,Celal Con,Donald Oswald,Arash Moazezi,Xiaogang Guo,Tanya Hathaway,Elaheh Alizadeh-Birjandi,Faisal Farooq,ELISABETTA TRONCI,Gerardo Vasquez,Nick Waltham,Isao Nozaki,K. 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arXiv:1508.01866v2 [hep-th] 14 Dec 2017 Lagrange multiplier and Wess-Zumino variable
as extra dimensions in the torus universe
Salman Abarghouei Nejad
Department of Physics,
University of Kashan, Kashan 87317-51167, I. R. Iran
Mehdi Dehghani
Department of Physics, Faculty of Science,
Shahrekord University, Shahrekord, P. O. Box 115, I. R. Iran
Majid Monemzadeh
Department of Physics,
University of Kashan, Kashan 87317-51167, I. R. Iran
Abstract
We study the effect of the the simplest geometry which is imposed
via the topology of the universe by gauging non-relativistic particle
model on torus and 3-torus with the help of symplectic formalism
of constrained systems. Also, we obtain generators of gauge trans-
formations for gauged models. Extracting corresponding Poisson
structure of existed constraints, we show the effect of the shape of
the universe on canonical structure of phase-spaces of models and
suggest some phenomenology to prove the topology of the universe
and probable non-commutative structure of the space. In addition,
we show that the number of extra dimensions in the phase-spaces
of gauged embedded models are exactly two. Moreover, in classical
form, we talk over modification of Newton’s second law in order
to study the origin of the terms appeared in the gauged theory.
1
as extra dimensions in the torus universe
Salman Abarghouei Nejad
Department of Physics,
University of Kashan, Kashan 87317-51167, I. R. Iran
Mehdi Dehghani
Department of Physics, Faculty of Science,
Shahrekord University, Shahrekord, P. O. Box 115, I. R. Iran
Majid Monemzadeh
Department of Physics,
University of Kashan, Kashan 87317-51167, I. R. Iran
Abstract
We study the effect of the the simplest geometry which is imposed
via the topology of the universe by gauging non-relativistic particle
model on torus and 3-torus with the help of symplectic formalism
of constrained systems. Also, we obtain generators of gauge trans-
formations for gauged models. Extracting corresponding Poisson
structure of existed constraints, we show the effect of the shape of
the universe on canonical structure of phase-spaces of models and
suggest some phenomenology to prove the topology of the universe
and probable non-commutative structure of the space. In addition,
we show that the number of extra dimensions in the phase-spaces
of gauged embedded models are exactly two. Moreover, in classical
form, we talk over modification of Newton’s second law in order
to study the origin of the terms appeared in the gauged theory.
1
2
1 Introduction
1.1 Why torus?
The torus universe model or the doughnut theory of the universe, is the
model which describes the universe as a doughnut, having surface with
topology 1 of a three dimensional torus. Historically, the first explana-
tions of the shape of the universe were proposed in the mid 60s, after
the discovery of CMB by Starobinsky and Zeldovich [1].
In experimental point of view, data of cosmos radiation measure-
ments gathered by satellite COBE, shows small discrepancies in tem-
perature fluctuation. This shows that the universe consists of regions
of varying densities. Stenemse and Silk proposed that this paradox, i.e.
the isotropic universe with different regional densities, suggests that uni-
verse may have a complicated geometric structure [2]. In other words,
these fluctuations show that multiply connected universes are possible,
and the simplest one is a 3-torus [2, 3]. Also, simulations of CMB map
and the angular power spectrum of temperature fluctuations, consider-
ing the torus topology, and comparing them with the observations of
the COBE satellite in order to obtain the lower limit of universe size,
suggest that we live in a small universe [4, 5, 6, 7, 8].
On the other hand, although there is no statistically significant evi-
dence to support what the topology of the universe would be, there are
some suggestions which talk over a 3-torus as the probable shape of the
universe [9, 10, 11, 12, 13].
Moreover, data gathered by WMAP satellite shows more intense
CMB across one plane of the universe in comparison with others, which
forms a straight line in the universe. Where radiation surpasses its quota
for the size of the plane seen, one can say that the universe has over-
flowed in that direction and creates a plane in other directions. Thereby,
the invisible loop of a torus may have been created perpendicular to the
direction of the plane. Thus, the analysed CMB maps from data ob-
tained from WMAP has released some results in favour of a torus form
of the universe [1, 14, 15]. Measurements of WMAP shows that the
universe is flat with only 0.4% margins of error. On the other hand, flat
universes with boundaries or edges are not desired mathematically, and
thus, they are excluded from consideration. Although there are some
finite compact universe models without boundaries, the torus universe
is the one which explains an overall flat and a finite universe [16].
1The word topology in this article is used as the global shape and characteris-
tics of the universe and we do not intended its pure mathematical definition, where
properties of space that are preserved under continuous deformations are studies.
1 Introduction
1.1 Why torus?
The torus universe model or the doughnut theory of the universe, is the
model which describes the universe as a doughnut, having surface with
topology 1 of a three dimensional torus. Historically, the first explana-
tions of the shape of the universe were proposed in the mid 60s, after
the discovery of CMB by Starobinsky and Zeldovich [1].
In experimental point of view, data of cosmos radiation measure-
ments gathered by satellite COBE, shows small discrepancies in tem-
perature fluctuation. This shows that the universe consists of regions
of varying densities. Stenemse and Silk proposed that this paradox, i.e.
the isotropic universe with different regional densities, suggests that uni-
verse may have a complicated geometric structure [2]. In other words,
these fluctuations show that multiply connected universes are possible,
and the simplest one is a 3-torus [2, 3]. Also, simulations of CMB map
and the angular power spectrum of temperature fluctuations, consider-
ing the torus topology, and comparing them with the observations of
the COBE satellite in order to obtain the lower limit of universe size,
suggest that we live in a small universe [4, 5, 6, 7, 8].
On the other hand, although there is no statistically significant evi-
dence to support what the topology of the universe would be, there are
some suggestions which talk over a 3-torus as the probable shape of the
universe [9, 10, 11, 12, 13].
Moreover, data gathered by WMAP satellite shows more intense
CMB across one plane of the universe in comparison with others, which
forms a straight line in the universe. Where radiation surpasses its quota
for the size of the plane seen, one can say that the universe has over-
flowed in that direction and creates a plane in other directions. Thereby,
the invisible loop of a torus may have been created perpendicular to the
direction of the plane. Thus, the analysed CMB maps from data ob-
tained from WMAP has released some results in favour of a torus form
of the universe [1, 14, 15]. Measurements of WMAP shows that the
universe is flat with only 0.4% margins of error. On the other hand, flat
universes with boundaries or edges are not desired mathematically, and
thus, they are excluded from consideration. Although there are some
finite compact universe models without boundaries, the torus universe
is the one which explains an overall flat and a finite universe [16].
1The word topology in this article is used as the global shape and characteris-
tics of the universe and we do not intended its pure mathematical definition, where
properties of space that are preserved under continuous deformations are studies.
3
Theoretically, string theory and also theories considering extra large
dimensions suggest that we live in a universe with higher dimensions of
space-time and most of the modern cosmological models are founded on
such assumptions. Moreover, the problems of the standard cosmology
are avoided, considering higher dimensional spacetime, and also most
of the predictions of the inflation cosmology are fulfilled via these ap-
proaches [17, 18, 19].
In order to combine topological theories and extra large dimensions
universe, it has been shown that cyclic universe models can be acquired
in a toroidal spacetime which is embedded in a five-dimensional bulk
with large extra dimensions, and the three dimensional space has been
shown as a closed ring, moving on the surface of the torus [20, 21].
If we expect that the universe has topology of a torus, we can con-
struct a gauge theory, using the Lagrangian of a particle on the torus,
and quantize such a gauge theory, and extract its gauge transformation
relations. Our goal to study the motion of a non-relativistic particle
on a torus and gauging that model is to obtain a configuration space
with extra dimensions. As we know, studying the motion of a free par-
ticle is the most powerful laboratory, in which we can test whether the
torus universe exists or not. Making such gauge theories and studying
its Hamiltonian spectrum may help us to understand the real topology
of the universe. Moreover, with investigating the final obtained phase-
space, one can check the commutativity and non-commutativity of the
universe. In addition, we can determine the ratio of two diameters of
the torus.
Another point of view in which we can study the constructed classical
theory on a torus is the modification of Newton’s law, that talks over the
corrections added to the Newtonian classical mechanics. In the common
Poisson structure, Hamiltonian equations of motion and Newton laws
are equivalent [22]. In this article, we construct a classical theory which
has an unusual Poisson structure due to its constrained structure. This
Poisson structure adds some additional terms to the Hamiltonian and
consequently to the equations corresponding Newton’s second law which
can be studied via the MOND phenomenological theory.
Our tool to construct a gauge theory which reduces to a particle on
the torus after gauge fixing is the symplectic gauge analysis approach
will be discussed later.
1.2 Gauge theories and constraints
As we know, gauge invariance is one of the most significant and practi-
cal concepts in theoretical physics. This concept is the cornerstone of
Theoretically, string theory and also theories considering extra large
dimensions suggest that we live in a universe with higher dimensions of
space-time and most of the modern cosmological models are founded on
such assumptions. Moreover, the problems of the standard cosmology
are avoided, considering higher dimensional spacetime, and also most
of the predictions of the inflation cosmology are fulfilled via these ap-
proaches [17, 18, 19].
In order to combine topological theories and extra large dimensions
universe, it has been shown that cyclic universe models can be acquired
in a toroidal spacetime which is embedded in a five-dimensional bulk
with large extra dimensions, and the three dimensional space has been
shown as a closed ring, moving on the surface of the torus [20, 21].
If we expect that the universe has topology of a torus, we can con-
struct a gauge theory, using the Lagrangian of a particle on the torus,
and quantize such a gauge theory, and extract its gauge transformation
relations. Our goal to study the motion of a non-relativistic particle
on a torus and gauging that model is to obtain a configuration space
with extra dimensions. As we know, studying the motion of a free par-
ticle is the most powerful laboratory, in which we can test whether the
torus universe exists or not. Making such gauge theories and studying
its Hamiltonian spectrum may help us to understand the real topology
of the universe. Moreover, with investigating the final obtained phase-
space, one can check the commutativity and non-commutativity of the
universe. In addition, we can determine the ratio of two diameters of
the torus.
Another point of view in which we can study the constructed classical
theory on a torus is the modification of Newton’s law, that talks over the
corrections added to the Newtonian classical mechanics. In the common
Poisson structure, Hamiltonian equations of motion and Newton laws
are equivalent [22]. In this article, we construct a classical theory which
has an unusual Poisson structure due to its constrained structure. This
Poisson structure adds some additional terms to the Hamiltonian and
consequently to the equations corresponding Newton’s second law which
can be studied via the MOND phenomenological theory.
Our tool to construct a gauge theory which reduces to a particle on
the torus after gauge fixing is the symplectic gauge analysis approach
will be discussed later.
1.2 Gauge theories and constraints
As we know, gauge invariance is one of the most significant and practi-
cal concepts in theoretical physics. This concept is the cornerstone of
4
the standard model of elementary particles. Gauge invariance is due to
the presence of the important physical variables which are independent
of local reference frames [23]. Whenever a change is applied in an ar-
bitrary reference frame, which makes changes in such variables, gauge
transformation occurs. Such physical variables are called gauge invariant
variables.
Generally, we deal with gauge invariance, or in other words, local
invariance, which produces gauge bosons in fundamental interactions.
As a physical law, the existence of (local) gauge symmetry in particle
physics is the sign of the presence of interactions [24].
It is very important to know that quantization of gauge theories
entails a particular prudence, because of the presence of gauge symme-
try exist some nonphysical degrees of freedom that must be eliminated
before and after the quantization is applied [25].
On the other hand, in a gauge theory, the equations of motion are
not able to determine the dynamics of the system thoroughly at every
moment. Thus, one of the most particular features of a gauge theory is
the emergence of arbitrary time dependent functions in general solutions
of the equations of motion. The emergence of such time dependent func-
tions is accompanied by the relations between phase-space coordinates,
which are called constraints [23, 26].
In order to quantize such systems, identities between phase-space
coordinates are classified into two main groups by Dirac [27]. The first
group are identities which are present in phase-space, similar to a co-
ordinate or a momentum variable. These identities, which transform
the physical system without any changes in phase-space, are called first-
class constraints, and according to Dirac’s guess are generators of gauge
transformations in phase-space. The second group are not related to
any degree of freedom and must be removed. Presence of such identi-
ties, which are called second-class constraints, indicates the absence of
gauge symmetry in the system. Therefore, to gauge a system, contain-
ing second-class constraints, we must transform them to first-class ones,
as a first step [28, 29].
There are some approaches to perform such a conversion, like BFT
method [30, 31, 32, 33, 34], the symplectic formalism [25, 35, 36, 37],
and the Noether dualization technique [38, 39, 40]. As we mentioned
before, in order to gauge a system with second-class constraints, we use
the symplectic approach in order to embed a non-invariant system in an
extended phase-space [41, 42, 43].
the standard model of elementary particles. Gauge invariance is due to
the presence of the important physical variables which are independent
of local reference frames [23]. Whenever a change is applied in an ar-
bitrary reference frame, which makes changes in such variables, gauge
transformation occurs. Such physical variables are called gauge invariant
variables.
Generally, we deal with gauge invariance, or in other words, local
invariance, which produces gauge bosons in fundamental interactions.
As a physical law, the existence of (local) gauge symmetry in particle
physics is the sign of the presence of interactions [24].
It is very important to know that quantization of gauge theories
entails a particular prudence, because of the presence of gauge symme-
try exist some nonphysical degrees of freedom that must be eliminated
before and after the quantization is applied [25].
On the other hand, in a gauge theory, the equations of motion are
not able to determine the dynamics of the system thoroughly at every
moment. Thus, one of the most particular features of a gauge theory is
the emergence of arbitrary time dependent functions in general solutions
of the equations of motion. The emergence of such time dependent func-
tions is accompanied by the relations between phase-space coordinates,
which are called constraints [23, 26].
In order to quantize such systems, identities between phase-space
coordinates are classified into two main groups by Dirac [27]. The first
group are identities which are present in phase-space, similar to a co-
ordinate or a momentum variable. These identities, which transform
the physical system without any changes in phase-space, are called first-
class constraints, and according to Dirac’s guess are generators of gauge
transformations in phase-space. The second group are not related to
any degree of freedom and must be removed. Presence of such identi-
ties, which are called second-class constraints, indicates the absence of
gauge symmetry in the system. Therefore, to gauge a system, contain-
ing second-class constraints, we must transform them to first-class ones,
as a first step [28, 29].
There are some approaches to perform such a conversion, like BFT
method [30, 31, 32, 33, 34], the symplectic formalism [25, 35, 36, 37],
and the Noether dualization technique [38, 39, 40]. As we mentioned
before, in order to gauge a system with second-class constraints, we use
the symplectic approach in order to embed a non-invariant system in an
extended phase-space [41, 42, 43].
5
1.3 Symplectic Formalism
Symplectic formalism was introduced by Faddeev and Jackiw [35], to
avoid consistency problems which spoil the Poisson brackets algebra
and consequently fail any quantization techniques in constrained systems
[44, 45]. The mathematics of this formalism is based on the symplec-
tic structure of the phase-space, and therefore, is different from other
approaches. Also, in the symplectic formalism there is no distinction
between the first and second-class constraints as in the case of the other
quantization procedures [37].
The starting point of the symplectic approach is a Lagrangian which
is first order in the time derivatives. All second order Lagrangian terms
can be converted to first order ones by enlarging the corresponding con-
figuration space so that it includes the conjugate momentum of the co-
ordinate variables [46]. Being dependent only on first order Lagrangian
makes the symplectic approach independent from the classification of
the constraints into primary, secondary, etc. [47]. In this approach, in-
stead of solving the constraints, one adds their time derivatives to the La-
grangian and considers corresponding Lagrange multipliers as additional
coordinates [49]. Also, to convert the nature of second-class constraints
to first ones, the phase-space would be extended with the help of Wess-
Zumino variables [50]. After such a conversion, choosing conventional
zero-modes which are generators of gauge transformations and obey par-
ticular boundary conditions, one can eliminate Wess-Zumino variables,
which makes the gauged model equivalent to the original system [51].
2 Gauging a non-relativistic particle model on
the torus
2.1 Particle on the torus
In the first part of this article we assume a non-relativistic particle on a
torus in a three-dimensional configuration space as a toy model. Consid-
ering this model, the particle lives on a two-dimensional configuration
space, effectively. After our gauging process, we will see that at least one
dimension is added to the previous configuration space, which makes its
space more realistic.
In all sections of this article we get the radii of the torus 1 and ς, in
order to use dimensionless coordinates. Thus, in spherical coordinates,
1.3 Symplectic Formalism
Symplectic formalism was introduced by Faddeev and Jackiw [35], to
avoid consistency problems which spoil the Poisson brackets algebra
and consequently fail any quantization techniques in constrained systems
[44, 45]. The mathematics of this formalism is based on the symplec-
tic structure of the phase-space, and therefore, is different from other
approaches. Also, in the symplectic formalism there is no distinction
between the first and second-class constraints as in the case of the other
quantization procedures [37].
The starting point of the symplectic approach is a Lagrangian which
is first order in the time derivatives. All second order Lagrangian terms
can be converted to first order ones by enlarging the corresponding con-
figuration space so that it includes the conjugate momentum of the co-
ordinate variables [46]. Being dependent only on first order Lagrangian
makes the symplectic approach independent from the classification of
the constraints into primary, secondary, etc. [47]. In this approach, in-
stead of solving the constraints, one adds their time derivatives to the La-
grangian and considers corresponding Lagrange multipliers as additional
coordinates [49]. Also, to convert the nature of second-class constraints
to first ones, the phase-space would be extended with the help of Wess-
Zumino variables [50]. After such a conversion, choosing conventional
zero-modes which are generators of gauge transformations and obey par-
ticular boundary conditions, one can eliminate Wess-Zumino variables,
which makes the gauged model equivalent to the original system [51].
2 Gauging a non-relativistic particle model on
the torus
2.1 Particle on the torus
In the first part of this article we assume a non-relativistic particle on a
torus in a three-dimensional configuration space as a toy model. Consid-
ering this model, the particle lives on a two-dimensional configuration
space, effectively. After our gauging process, we will see that at least one
dimension is added to the previous configuration space, which makes its
space more realistic.
In all sections of this article we get the radii of the torus 1 and ς, in
order to use dimensionless coordinates. Thus, in spherical coordinates,
6
the surface of a torus is defined by
x = (1 + ς cos θ) cos ϕ
y = (1 + ς cos θ) sin ϕ
z = ς sin θ (1)
The surface of the torus is described by primary constraint φ1 = 0
in configuration space for free particle on it.
φ1(r, θ) = r2 − 2ς cos θ − (1 + ς2), (2)
where, r2 = x2 + y2 + z2. In this coordinate canonical Hamiltonian for
unit mass is
Hc = 1
2 (p2
r + p2
θ
r2 + p2
ϕ
r2 sin2 θ ). (3)
In formal constrained analysis we arrive to secondary (final) con-
straint in phase-space as
φ2(r, θ, pr, pθ ) = 2(rpr + ςpθ sin θ
r2 ). (4)
The set of constraints form a second-class system with non-constant ∆
matrix as
∆12 = 4(r2 + ς2 sin2 θ
r2 ), (5)
which makes its embedding by BFT method problematic. This is the
reason that we use the symplectic approach, which is not affected by the
Poisson structure of second-class constraints.
2.2 Symplectic analysis of a particle on the torus
Constructing first-class models from a singular Lagrangian is more straight-
forward in the symplectic formalism than other similar approaches. This
is done by embedding the primary model in an extended phase-space.
In this model, the singularity nature of the free particle Lagrangian
due to its configuration constraint, φ1(r, θ), can be imposed by a new
dynamical variable (say undetermined Lagrange multiplier) λ, in such a
way that adds the constraints to the free Lagrangian,
L(0) = ˙rpr + ˙θpθ + ˙ϕpϕ − Hc − λ1φ1(r, θ). (6)
the surface of a torus is defined by
x = (1 + ς cos θ) cos ϕ
y = (1 + ς cos θ) sin ϕ
z = ς sin θ (1)
The surface of the torus is described by primary constraint φ1 = 0
in configuration space for free particle on it.
φ1(r, θ) = r2 − 2ς cos θ − (1 + ς2), (2)
where, r2 = x2 + y2 + z2. In this coordinate canonical Hamiltonian for
unit mass is
Hc = 1
2 (p2
r + p2
θ
r2 + p2
ϕ
r2 sin2 θ ). (3)
In formal constrained analysis we arrive to secondary (final) con-
straint in phase-space as
φ2(r, θ, pr, pθ ) = 2(rpr + ςpθ sin θ
r2 ). (4)
The set of constraints form a second-class system with non-constant ∆
matrix as
∆12 = 4(r2 + ς2 sin2 θ
r2 ), (5)
which makes its embedding by BFT method problematic. This is the
reason that we use the symplectic approach, which is not affected by the
Poisson structure of second-class constraints.
2.2 Symplectic analysis of a particle on the torus
Constructing first-class models from a singular Lagrangian is more straight-
forward in the symplectic formalism than other similar approaches. This
is done by embedding the primary model in an extended phase-space.
In this model, the singularity nature of the free particle Lagrangian
due to its configuration constraint, φ1(r, θ), can be imposed by a new
dynamical variable (say undetermined Lagrange multiplier) λ, in such a
way that adds the constraints to the free Lagrangian,
L(0) = ˙rpr + ˙θpθ + ˙ϕpϕ − Hc − λ1φ1(r, θ). (6)
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