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THE EQUIVALENCE BETWEEN THE HAMILTONIAN AND LAGRANGIAN FORMULATIONS
FOR THE PARAMETRIZATION- INVARIANT THEORIES
S. I. MUSLIH
Received 27 March 2001 and in revised form 13 August 2001
The link between the treatment of singular Lagrangians as field systems and the canonical Hamiltonian approach is studied. It is shown that the singular Lagrangians as field sys- tems are always in exact agreement with the canonical approach for the parametrization invariant theories.
2000 Mathematics Subject Classification: 37K10, 37K05, 17B80.
1. Introduction. [2, 3, 4,5], the Hamilton-Jacobi formulation of constrained sys- tems has been studied. This formulation leads us to obtain the set of Hamilton-Jacobi partial differential equations (HJPDE) as follows:
Hα
tβ,qa, ∂S
∂qa, ∂S
∂tα
=0, α,β=0,n−r+1,...,n; a=1,...,n−r , (1.1)
where
Hα =Hα
tβ,qa,pa
+pα, (1.2)
andH0is defined as
H0=pawa+pµq˙µ|pν=−Hν−L
t,qi,q˙ν,q˙a=wa
, µ,ν=n−r+1,...,n. (1.3) The equations of motion are obtained as total differential equations in many vari- ables as follows:
dqa=∂Hα
∂padtα, dpa= −∂Hα
∂qadtα, dpβ= −∂Hα
∂tβ dtα, dz=
−Hα+pa∂Hα
∂pa
dtα; α,β=0,n−r+1,...,n, a=1,...,n−r ,
(1.4)
wherez=S(tα;qa). The set of equations (1.4) is integrable (see [4,5]) if
dH0=0, dHµ =0, µ=n−r+1,...,n. (1.5) If conditions (1.5) are not satisfied identically, we consider them as new constraints and again test the consistency conditions. Hence, the canonical formulation leads to
obtain the set of canonical phase space coordinatesqaandpaas functions oftα; be- sides, the canonical action integral is obtained in terms of the canonical coordinates.
The HamiltoniansHα are considered as the infinitesimal generators of canonical trans- formations given by parameterstα, respectively.
In [1], the singular Lagrangians are treated as field systems. The Euler-Lagrange equations of singular systems are proposed in the form
∂
∂tα
∂L
∂
∂αqa
−∂L
∂qa=0, ∂αqa=∂qa
∂tα, (1.6)
with constraints
dG0= −∂L
∂t dt, (1.7)
dGµ= −∂L
∂qµdt, (1.8)
where
L
tα,∂αqa,q˙µ,qa
=L
qa,qα,q˙a=
∂αqa
t˙α
, q˙µ=dqµ
dt , Gα=Hα
qa,tβ,pa= ∂L
∂q˙a
.
(1.9)
In order to have a consistent theory, we should consider the variations of the con- straints (1.7) and (1.8).
In this paper, we study the link between the treatment of singular Lagrangians as field systems and the canonical formalism for the parametrization invariant theories.
2. Parametrization-invariant theories as singular systems. In [4], the canonical method treatment of the parametrization-invariant theories is studied and will be briefly reviewed here.
Consider a system with the action integral as S
qi
= dtᏸqi,q˙i,t
, i=1,...,n, (2.1) whereᏸis a regular Lagrangian with Hessiann. Parametrize the timet→τ(t), with τ˙=dτ/dt >0. The velocities ˙qimay be expressed as
q˙i=qiτ,˙ (2.2)
whereqiare defined as
qi=dqi
dτ. (2.3)
Denotet=q0andqµ=(q0,qi),µ=0,1,...,n, then the action integral (2.1) may be written as
S qµ
= dτ˙tᏸ
qµ,qi
˙t
, (2.4)
which is parametrization invariant sinceLis homogeneous of first degree in the ve- locitiesqµwithLgiven as
L qµ,q˙µ
=˙tᏸ
qµ,qi t˙
. (2.5)
The LagrangianLis now singular since its Hessian isn.
The canonical method in [2,3,4,5] leads us to obtain the set of Hamilton-Jacobi partial differential equations as follows:
H0=pτ−L
q0,qi,q˙0,q˙i=wi
+piτqi+ptq˙0|pt=−Ht=0, pτ=∂S
∂τ, Ht=pt+Ht=0, pt=∂S
∂t,
(2.6)
whereHtis defined as
Ht= −ᏸqi,wi
+pτiwi. (2.7)
Here,piτandptare the generalized momenta conjugated to the generalized coordi- natesqiandt, respectively.
The equations of motion are obtained as total differential equations in many vari- ables as follows:
dqi=∂H0
∂pi dτ+∂Ht
∂pi dq0=∂Ht
∂pi dq0, (2.8)
dpi= −∂H0
∂qi dτ−∂Ht
∂qi dq0= −∂Ht
∂qi dq0, (2.9)
dpt= −∂H0
∂q0 dτ−∂Ht
∂q0dq0=0. (2.10)
Since
dHt=dpt+Ht (2.11)
vanishes identically, this system is integrable and the canonical phase space coordi- natesqiandpiare obtained in terms of the time(q0=t).
Now, we look at the Lagrangian (2.5) as a field system. Since the rank of the Hessian matrix isn, this Lagrangian can be treated as a field system in the form
qi=qi(τ,t), (2.12)
thus, the expression
qi=∂qi
∂τ +∂qi
∂t ˙t, (2.13)
can be substituted in (2.5) to obtain the modified LagrangianL: L=˙tᏸ
qµ,1
˙t ∂qi
∂τ +∂qi
∂t ˙t
. (2.14)
Making use of (1.6), we have
∂L
∂qi− ∂
∂t
∂L
∂
∂qi/∂t
− ∂
∂τ
∂L
∂
∂qi/∂τ
=0. (2.15)
Calculations show that (2.15) leads to a well-known Lagrangian equation as
∂ᏸ
∂qi− d dt
∂ᏸ
∂ dqi/dt
=0. (2.16)
Using (2.7), we have
Ht= −ᏸ+∂ᏸ
∂q˙i
q˙i. (2.17)
In order to have a consistent theory, we should consider the total variation ofHt. In fact,
dHt= −∂ᏸ
∂tdt. (2.18)
Making use of (1.8), we find that
dHt= −∂L
∂t dτ. (2.19)
Besides, the quantityH0is identically satisfied and does not lead to constraints.
We notice that (2.8) and (2.9) are equivalent to (2.15) and (2.16).
3. Classical fields as constrained systems. In the following sections, we study the Hamiltonian and Lagrangian formulations for classical field systems and demonstrate the equivalence between these two formulations for the reparametrization-invariant fields.
A classical relativistic field φi=φi(x,t) in four space-time dimensions may be described as the action functional
S φi
= dt d3x
ᏸφi,∂µφi
, µ=0,1,2,3;i=1,2,...,n, (3.1) which leads to the Euler-Lagrange equations of motion as
∂ᏸ
∂φi−∂µ
∂ᏸ
∂
∂µφi
=0. (3.2)
We can go over from the Lagrangian description to the Hamiltonian description by using the definition
πi= ∂ᏸ
∂φ˙i, (3.3)
then the canonical Hamiltonian is defined as H0= d3x
πiφ˙i−ᏸ. (3.4)
The equations of motion are obtained π˙i= −∂H0
∂φi, φ˙i=∂H0
∂πi. (3.5)
4. Reparametrization-invariant fields. In analogy with the finite dimensional sys- tems, we introduce the reparametrization-invariant action for the field system:
S= dτ ᏸRd3x, (4.1)
where
ᏸR=˙tᏸφi,∂µφi
. (4.2)
Following the canonical method [2,3,4,5], we obtain the set of [HJPDE], H0=πτ+πi(τ)dφi
dτ +πtdt
dτ−ᏸR=0, πτ= ∂S
∂τ, Ht=πt+Ht=0, πt=∂S
∂t,
(4.3)
whereHtis defined as
Ht= −ᏸφi,∂µφi
+πi(τ)dφi
dt , (4.4)
andπi(τ),πtare the generalized momenta conjugated to the generalized coordinates φiandt, respectively.
The equations of motion are obtained as follows:
dφi=∂H0
∂πi dτ+∂Ht
∂πidt=∂Ht
∂πi dt, (4.5)
dπi= −∂H0
∂φidτ−∂Ht
∂φi dt= −∂Ht
∂φidt, (4.6)
dπt= −∂H0
∂t dτ−∂Ht
∂t dt=0. (4.7)
Now the Euler-Lagrangian equation for the field system reads as
∂ᏸ
∂φi− ∂
∂xµ
∂ᏸ
∂
∂φi/∂xµ
=0. (4.8)
Again as for the finite-dimensional systems, (4.5) and (4.6) are equivalent to (4.8) for field systems.
5. Conclusion. As it is mentioned in the introduction, if the rank of the Hessian matrix for discrete systems is(n−r ), 0< r < n, then the systems can be treated as field systems [1]. The treatment of Lagrangians as field systems is always in exact agreement with the Hamilton-Jacobi treatment for reparametrization-invariant theo- ries. The equations of motion (2.8) and (2.9) are equivalent to the equations of motion (2.15) and (2.16). Besides, the variations of constraints (2.18) and (2.19) are identically satisfied and no further constraints arise.
In analogy with the finite-dimensional systems, it is observed that the Lagrangian and the Hamilton-Jacobi treatments for the reparametrization-invariant fields are in exact agreement.
References
[1] N. I. Farahat and Y. Güler,Singular Lagrangians as field systems, Phys. Rev. A (3)51(1995), no. 1, 68–72.
[2] Y. Güler,Canonical formulation of singular systems, Nuovo Cimento B (11)107(1992), no. 12, 1389–1395.
[3] ,Integration of singular systems, Nuovo Cimento B (11)107(1992), no. 10, 1143–
1149.
[4] S. I. Muslih and Y. Güler,On the integrability conditions of constrained systems, Nuovo Cimento B (11)110(1995), no. 3, 307–315.
[5] ,Is gauge fixing of constrained systems necessary? Nuovo Cimento B113(1998), 277–289.
S. I. Muslih: Department of Physics, Al-Azhar University, Gaza, Palestine E-mail address:[email protected]
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